On Interdependent Preferences and
Sequential Structures in Rent-Seeking Contests
Tobias Guse1
Inauguraldissertation
zur Erlangung des akademischen Grades
Doctor rerum politicarum
der Universita¨t Dortmund
July 22, 2010
1Lehrstuhl fu¨r Mikroo¨konomie, Universita¨t Dortmund, Fakulta¨t fu¨r Wirtschafts- und Sozial-
wissenschaften, Vogelpothsweg 87, D-44221 Dortmund, Germany. Email: tobias.guse@uni-
dortmund.de
Preface
This dissertation draws on research I undertook during the time in which I held a schol-
arship of the Deutsche Forschungsgemeinschaft (DFG) at the Graduiertenkolleg ”Al-
lokationstheorie, Wirtschaftspolitik und kollektive Entscheidungen”, and later while I
was a teaching and research assistant at the chair ”Mikroo¨konomik” at the Technische
Universita¨t Dortmund. The present thesis has strongly been influenced by and profited
from discussions with professors and fellow students of the Graduiertenkolleg, as well as
presentations as various national and international conferences. I am very grateful to all
supported my work in that way. Financial support by the Deutsche Forschungsgemein-
schaft is gratefully acknowledged.
In particular, I would like to thank Wolfgang Leininger, who supervised my disser-
tation, and my co-author Burkhard Hehenkamp. Moreover, I would like to thank Erwin
Amann, Christian Bayer, Damian Damianov, Frauke Eckermann, Johannes Mu¨nster,
Wolfram Richter, Sina Risse, Wendelin Schnedler and several anonymous referees for
helpful comments and discussion.
Furthermore, I want to thank my wife Marina and my family for their strong support,
such as Rayna and Keith Diekelman for their helpful comments on the final version of
this work.
1
Contents
1 Introduction 7
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 The History of Contest Theory . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 An Introduction to Interdependent Preferences . . . . . . . . . . . . . . . 11
1.4 Sequential Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 The Theory of Contests 16
2.1 Axiomatic Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 On the Returns to Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 The Role of Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . 21
I Interdependent Preferences 23
3 Strategic Advantage 24
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.1 Participation constraints . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.2 Characterizing equilibrium behavior . . . . . . . . . . . . . . . . . 31
3.3.3 Analyzing the strategic advantage . . . . . . . . . . . . . . . . . . 33
3.4 Two-player contests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5 Non-increasing marginal efficiency – The case r ≤ 1 . . . . . . . . . . . . 35
3.5.1 The special case of constant marginal efficiency (r = 1) . . . . . . 36
3.6 Increasing marginal efficiency - The case r > 1 . . . . . . . . . . . . . . . 37
3.6.1 Characterization of equilibrium types . . . . . . . . . . . . . . . . 39
3.6.2 Equilibrium multiplicity . . . . . . . . . . . . . . . . . . . . . . . 45
3.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 When Are Spiteful Players Better Off? 48
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Effort Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3 Effects on the Level of Material Payoff . . . . . . . . . . . . . . . . . . . 51
2
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Evolutionarily Stable Preferences 55
II Sequential Structures 59
6 Sequential Structures with Three Players 60
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2.1 The simultaneous game . . . . . . . . . . . . . . . . . . . . . . . . 62
6.2.2 Two players move early . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2.3 One player moves early . . . . . . . . . . . . . . . . . . . . . . . . 65
6.3 Comparing the subgame payoffs . . . . . . . . . . . . . . . . . . . . . . . 67
6.3.1 Decision Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.3.2 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.3.3 Comparison to the two-player-case . . . . . . . . . . . . . . . . . 70
6.4 Welfare Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7 Strategic Choice of Decision Timings 74
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.2 Setup of the game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.2.2 Timing decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.3 The Subgames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.3.1 Equilibrium Behavior in the Simultaneous Game . . . . . . . . . . 76
7.3.2 Equilibrium Behavior in Subgame L . . . . . . . . . . . . . . . . . 77
7.3.3 Equilibrium Behavior in Subgame E . . . . . . . . . . . . . . . . 78
7.4 Subgame-perfect Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.5 Welfare Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
8 Assumption of Simultaneous Moves in Contests 85
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
8.2 When will Late Move be Preferred? . . . . . . . . . . . . . . . . . . . . . 85
8.3 When will Early Move be Preferred? . . . . . . . . . . . . . . . . . . . . 88
8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
9 Conclusion 91
A Appendix 101
A.1 Definition of a strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.2 Supplement to Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.2.1 The subgames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3
A.2.2 The equilibrium path . . . . . . . . . . . . . . . . . . . . . . . . . 106
A.2.3 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
B Proofs 111
B.1 Comments on the Introduction . . . . . . . . . . . . . . . . . . . . . . . 111
B.2 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
B.2.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . 111
B.2.2 Non-increasing marginal efficiency . . . . . . . . . . . . . . . . . . 112
B.2.3 The case r > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
B.3 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
B.4 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
B.5 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
B.6 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
B.7 Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
B.8 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
B.9 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4
List of Symbols
a Effort level of players with independent preferences
b Effort level of players with interdependent preferences
c Relative effort level of players with independent preferences
d Marginal cost of effort
fi(x) Impact function
g Welfare gain of a subgame with the favorite moving early in contrast
to a simultaneous move game
i, j, k Index for a player
m Size of a subgroup of all players, especially the number of players with
independent preferences
n Number of players
pi Winning probability of player i
q := b−aa Relative additional effort of players with interdependent preferences
q1, q2, q3 Auxiliary functions
r Technology parameter
rI Upper limit for the value of r for which equilibrium type I can be observed
rIII Lower limit for the value of r for which equilibrium type III can be observed
s1, s2, s3 Parameter of a special preference function
t Point in time
u Utility function, depending on λ
xi Effort of player i
xˆ Effort of an underdog
x Highest effort chosen in a contest
x Vector of efforts
x−i Vector of the efforts of the opponents of player i
y Effort level of a mutant player
A Total aggregate effort of all players
A−1 Total aggregate effort of all players but player 1
A−i Total aggregate effort of all players but player 1 and player i, i 6= 1
B B(Z, λ) := br(Z, λ)
C C := ar
iE Subgame where player i moves earlier than her opponents
(who are assumed to choose simultaneously)
Fi Preference function of player i
I Set of players with independent preferences
J Set of players with interdependent preferences
iL Subgame where player i moves later than her opponents
(who are assumed to choose simultaneously)
5
M Set of a subgroup of all contestants
Nsym(Γ) Set of symmetric Nash-equilibria of the game Γ
Pr Subgame where the early mover preeempts all of her opponents
RI Set of all technology parameters with increasing returns to scale,
for which equilibrium type I can be observed
S Subgame with simultaneous moves
T T := (n− 2)2V 2 + 4(n− 1)x1V
V ; Vi Rent, prize to win in the contest; personal prize for player i
W i Welfare level of subgame i
Z Z := mar + (n−m− 1)br
αi Weight that player i puts on relative payoff compared to material payoff
β1, β2 Parameter of the linear preference function.
γ γ := β2β1+β2
θ Relative valuation of the prize for the favorite
κ Number of rounds in a tournament
λi Preference parameter of player i
ν Constant factor for adjusting winning probabilities
ξ Parameter of a special discontinuous preference function
pii Material payoff of player i
ρi Relative payoff function of player i
σi Parameter of the impact function of player i
τi Probability that player i moves early in a mixed strategy equilibrium
φ φ(n) := 4n−7+
√
16n2−40n+33
2(n−1)
Upper limit to which an underdog prefers not to move late
ϕ ϕ(z) := z
r
A V − z
Payoff function given a fixed aggregate effort
ψ Parameter representing valuations with identical payoffs in different subgames
ω(y) Loss function of a mutant, depending on her effort y
∆ Difference between payoff levels of players with independent
and interdependent preferences
Ξ Additional payoff of a favorite when switching from 1L to 1E
Ω Ω :=
∑n
i=2
1
Vi
T Set of feasible timing choices
V Set of feasible valuations
X Set of feasible effort choices
6
Chapter 1
Introduction
”In many market-oriented economies, government restrictions upon eco-
nomic activity are pervasive facts of life. These restrictions give rise to rents
in a variety of forms, and people often compete for these rents.” Anne O.
Krueger, 1974
1.1 Motivation
In many situations, we can observe that several individuals or institutions compete for an
exogenously given asset. In such situations they spend money or other valuable resources
to increase their winning probability, yet without exerting any influence on the value of
the asset. This asset can take many different forms. While the obvious example of an
asset is a fixed amount of money, we can also think of other examples: a job position, an
office (e.g. in a presidential election race), a patent, a license, a territorium (in a war),
a right (in some lawsuits), or a title (e.g. in sport contests). We call such situations
rent-seeking contests.
As already addressed, election campaigns are a standard example of contests. We
will now use one very prominent campaign as an example to demonstrate the relevance
of contests in economic and political reality. In November 2008, the election for the
president of the United States, presumably the most powerful office in the world, was
held. Of course, this office is a very valuable asset, and candidates are willing to pay
in order to increase their probability of winning this election. In such an election, it
would be far more efficient to simply inform voters about the political stance of each
candidate, and then elect the one with values matching those of the majority of the
electorate. However, each candidate has an interest in convincing voters to vote for
him. Consequently, a huge amount of resources is dissipated in the race between the
candidates for those votes that finally decide the election.
In the election campaign 2008, this race started long before the election itself. The
final decision about which candidates will be admitted to the actual election in November
has been made in the end of August for Barack Obama (by the Democratic Party) and
in the beginning of September for John McCain (by the Republican Party). Yet, in
advance, this attractive position initiated another contest, namely the contest for the
7
candidacy, the so-called primary elections. Every candidate in these primary elections
entered the race with the one goal: to become the next president of the United States.
While, by January 31 2008, there were still more than nine months to go to the
actual election, the top fifteen different candidates of the two major parties had spent a
total amount of not less than 546 millions US-$ (CNN 2008 a,b).1 The chairman of the
Federal Election Commission, Michael Toner, announced at the beginning of 2007 that he
expected the total expenditures on this year’s presidential election to exceed one billion
US-$.2 This example demonstrates how important a contest for a single position can be.
However, it is questionable whether or not these expenditures and the campaign itself
improve the candidates’ abilities and qualifications for the office of the US president.
Millions of dollars are spent with the sole aim of influencing the allocation of the asset
“presidential office”.
In this work, we focus on two different aspects of rent-seeking contests, namely inter-
dependent preferences and sequential structures. With respect to interdependent pref-
erences, we will restrict parts of our analysis to considering the impact of spite. From a
theoretical point of view, this is the most interesting form of interdependent preferences.
While it is hard to verify the presence of spite in an election campaign, some aspects of
the present presidential campaign indicate that candidates gain advantages from spiteful
behavior. In 2008, the most intensive duel between the candidates was fought between
Barack Obama, who became the designated candidate of the Democratic Party and fi-
nally was elected for the President of the United Stated, and Hillary Clinton. Although
both candidates represent the same party and thus represent similar ideas, there were
frequent news stories about how one of the candidates (or their staff members) had at-
tacked the other in order to damage his or her credibility. In this intra-party election
campaign, we find several examples that show how the candidates try to enhance their
winning chances through spiteful behavior. For example, Barack Obama reproached
Hillary Clinton that she would not care about social policy (FOX News, 2007, Wash-
ington Post, 2007) and lied concerning her attitude to the NAFTA trade agreement
(Die Welt, 2008). Clinton accused Obama of being unreliable in energy policy (NY
Daily News, 2008) or foreign affairs (Focus, 2007). Often, it was not the political issues
themselves that attracted attention, but rather, the sharpness of their personal attacks
(Focus, 2007, Die Zeit, 2008). Here, one could see that in the pre-election campaign it
was not only the aim of a candidate to enhance one’s own standing, but also to damage
the image of the opponent. We will allude to the problem of interdependent preferences
in contests in a more general framework in Chapters 3 to 5.
The second focus of this work lies on sequential structures. This part analyzes how
the timing of decisions changes far the incentives of a player to choose a certain strat-
egy. Think of a contest, like an election campaign, where two candidates present their
1This number does not include the cost incurred by candidates of smaller parties and independent
candidates.
2As a comparison: total expenditures from the three recent US-president election campaigns are
given by approximately 575 million US$ in 1996, 650 million US-$ in 2000 and a billion US-$ in 2004.
(CNN, 2008c)
8
manifesto one after another, in such a way that the second candidate can react to the
strategy of the first. Since the first player has already committed to her manifesto, she
cannot strategically react to the second candidates’ program, while the second candi-
date of course can react to her opponent’s manifesto. This decision situation is different
from one, where both candidates have to publish their manifesto simultaneously. For
this reason, it is advantageous for the opponents to consider the effects of this situation,
when they choose the date to publish their manifesto. For the addressed example of
the U.S. presidential election, when looking at the dates when the two most promising
candidates presented their manifesto, it is striking that in the last six decades, beginning
with the first post-war election campaign in 1948 up to this year’s presidential race, it
was always the candidate from the opposition who chose the earlier presentation date.
The candidate who represented the same party as the president in office always chose
the later timing (Morgan, 2003). As we will see, such a sequential structure is a rational
result for a contest between two candidates of different strength (cf. Leininger, 1993),
but not necessarily for a contest with more than two candidates. Situations where more
than two players join a sequential contest are the main issue of the Chapters 6 to 8.
1.2 The History of Contest Theory
Adam Smith (1776) was the first person to scientifically study economic problems. He
came to the result that in markets the individually rational behavior of every participant
colludes to a welfare maximum, driven by the famous invisible hand. This result, how-
ever, was only valid for perfect price competition. It was Cournot (1838) who first showed
that in a market competition where competitors do not decide about the price but about
the quantity, profit maximization by duopolists will induce a social loss. Applying the
equilibrium concept proposed by Nash (1951), that also drives Cournots result, we find
a large variety of games, where individual rationality maximizes neither welfare nor joint
output. In some games - especially the famous prisoners’ dilemma put forward by Luce
and Raiffa (1957) - individual rationality even fails to reach Pareto-efficiency, i.e. it would
be possible in the equilibrium outcome to increase one agent’s utility without decreasing
her opponents’ utility. In several papers, Tullock (1967, 1971, 1980) introduced the idea
that also competition for a fixed amount of money leads to an equilibrium outcome that
is not pareto-efficient if it induces wasteful efforts. The latter game is well-known as
“rent-seeking contest”.
The term ”rent-seeking” has first been put forward by Krueger (1974), but the idea of
rent-seeking as a welfare problem is some years older. In 1967, Gordon Tullock explored
that the possibility of tariff policy unintentionally creates incentives for firms in the
affected industry to use up resources for political lobbying. The scenario he examined
was the introduction of a tariff policy that is being discussed in the political process.
The tariff policy in question would be a mere redistribution without any productive
contribution. However, Tullock showed that such a situation creates incentives for firms
in the concerned industry to use up resources for political lobbying in order to either foster
or prevent the tariff depending on whether the firms gains or loses from its presence. To
9
sum up, in a rent-seeking contest, rational agents use up valuable resources for a contest
that cannot generate welfare.
Technically speaking, rent-seeking characterizes a situation, where two or more per-
sons can perform a certain action in order to receive a positive expected share of a fixed
amount of an asset that is distributed. The actions that are performed with the aim to
receive the rent are not productive.3 Yet, the problem is that, despite of the lack of pro-
ductivity, rent-seeking games, in general, create incentives to dissipate scarce resources
that could have been better spent in (more) productive ways.
As already mentioned, Tullock first addressed rent-seeking as a problem, still without
using the term ”rent-seeking”. In his early work where he discusses the effects of a
potential implementation of a tariff policy, he introduces two further similar examples,
namely a competition for a monopoly position and the prevention of theft.4 He showed
that, also in these situations, there are incentives for different parties to invest in a
contest for a payoff of fixed size.5 Tullock (1967, 1971) argues that whenever some kind
of welfare transfer is achievable that valuable resources are spent by several parties in
order to either obtain or prevent the transfer. In a further step, Anne Krueger (1974)
puts forward a formal model for the problem of competition for a tariff policy, applying
the problem to the specific situations of India and Turkey. With her model, she is the
first to analytically establish the welfare loss of rent-seeking. Besides, she also introduced
the term ”rent-seeking” into the literature.6
The scientific analysis of the problem of rent-seeking entered a new era when Tullock
(1980) developed a handy and simply-structured model of a rent-seeking contest, the so-
called Tullock-contest. This model is an important basis for a broad range of literature
(see Lockard and Tullock, 2001, for a compendium). This model will be the basis of all
the objective functions looked upon in this work. His major intention when introducing
this function is to demonstrate how rent-seeking generates dissipation. He shows the first-
order-conditions for some numerical examples to show that the amount of efforts spent
in a rent-seeking contest can be quite substantial. Besides the discussion of existence
of equilibria, his main issue is that resources are shown to be wasted in this setup.
Obviously, waste occurs whenever the chosen effort of some player is positive.
3In some examples like e.g. R&D-competitions effort can indeed be productive. Another example for
productive contests are sports contests, where the effort increases the utility of the spectators. However,
we will restrict our analysis to the assumption of non-productive efforts.
4Tullock analyzes a game between a thief and a potential victim, where the one invests in burgling
tools and the other one in respective security devices.
5Of course, Tullock pays tribute to the fact that the mere existence of a monopoly is detrimental
from a social point of view. However, he reckons that the size of the so-called Harberger triangle is
neglectable in contrast to the loss created by the rent-seeking activity.
6Krueger (1974), p. 291.
10
1.3 An Introduction to Interdependent Preferences
For decades, it has been standard in economic literature to assume that agents are homo
oeconomicus. This means that the players perfectly rationally maximize their own payoff,
without caring about the payoffs of other players. But looking into real life, we notice
that people indeed do care about what other people receive, in an altrusitic as well as
in a spiteful way. We call preferences that do not only depend on the own payoff of the
respective player but also on the payoffs of others interdependent preferences.
The role of interdependent preferences, notably altruism, in economic problems was
first addressed by Mill (1887): ”On the happiness of others, on the improvement of
mankind [...] followed not as a means, but as itself an ideal [...] they find happiness
by the way.”7 Younger approaches in analyzing interdependent preferences go back to
Hamilton (1971), an evolutionary biologist, whose work found first economic attention
by Schaffer (1988), and Gary Becker (1974, 1976), who contributed the concepts of so-
cial interactions (1974) and altruism (1976) to economic literature. Both Hamilton and
Becker emphasized the role, evolution played in revealing interdependent preferences
as stable. The branch of economic literature, where interdependent preferences have
found most application, is indeed evolutionary game theory. Further fields of economics
where the role of interdependent preferences has been analyzed are educational subsidies
(Lommerud, 1989), wealth taxes (Konrad, 1990) or risk decision (Konrad and Lom-
merud, 1993). Hopkins and Kornienko (2004) provide an analysis of a game between
status-oriented preferences to show that welfare is lower in the presence of negatively
interdependent preferences. Finally, Sobel (2005) gives an comprehensive overview over
the role of interdependent preferences in economic contexts and literature.
The first empirical evidence for interdependent preferences was provided in Duesen-
berry (1949). He explored the role of status-seeking for the theory of savings. In this
work, he showed “that the interdependence of consumer preferences affects the choice
between consumption and saving [...]”. Besides Duesenberry’s empirical basis, the pa-
pers by Bush (1994a,b) and Kapteyn et al. (1997) provide empirical support for the
economic relevance of interdependent preferences that is based on field data. Moreover,
experimental evidence has continuously been questioning the plausibility of independent
pay-off maximization over the past decade. For instance, Levine (1998) takes results
from ultimatum experiments and the final round of a centipede experiment in order to
pin down the distribution of altruistic and spiteful preferences. This distribution then
allows him to explain results, e.g. in public good contribution games or in early rounds
of the centipede game. Other examples for the presence of interdependent preferences
can be found in Davis and Holt (1993). Introspection commonly yields the same result.
A very important issue in the analysis of interdependent preferences is the question
of whether interdependent preferences yield a higher payoff than independent prefer-
ences. The works of Bester and Gu¨th (1998) and Possajennikov (2000) show that in a
certain class of games altruism can be evolutionarily stable. Beyond that, Kockesen et
al. (2000a) showed that spiteful behavior can also be advantageous in the sense that
7Mill (1887), p. 142, quoted via Seckler (1975).
11
maximization of these preferences provides the player a relatively higher payoff than her
opponents. While, in a biological context, this means a higher evolutionary fitness, in an
economic context, this represents a higher market share. Both interpretations demon-
strate that the spiteful player has a better chance to survive than her opponents, and
thus this behavior is evolutionarily more advantageous. For contests the evolutionary
stability of spite was first addressed by Hehenkamp et al. (2004) and Leininger (2003).
Kockesen et al. (2000b) interpret this evolutionary advantage as a strategic advantage
such that a player who consciously decides to choose a negative interdependent preference
(e.g. by employing a spiteful manager) yields a higher payoff than her opponents with
independent preferences. The work at hand shows that this also holds for rent-seeking
contests. Chapter 3 of our analysis establishes this strategic advantage for spiteful pref-
erences, while Chapter 4 sheds some light on the limits of this advantage and specifies
the parameter ranges for successful interdependent preferences. Overly spiteful behavior
can also be disadvantageous. Finally, we take a short glance at the question, which
preferences prevail in an evolutionary process.
1.4 Sequential Structures
In this section, we want to give a short introduction to the literature on sequential
structures in contests. In a sequential contest, players decide upon their effort at different
times, such that players moving late already know the choice of the opponents that move
early. This creates the opportunity for the early mover to commit to a certain effort
level and for the late mover to react to this commitment, which introduces an additional
strategic component into the game. In contrast to repeated contests, in a sequential
game, there is only one prize to be distributed while a repeated contest offers a prize in
every single round.8
The idea of sequential structures in a market game stems from the analysis of a
duopoly by Stackelberg (1934). He analyzed sequential games in a Cournot and a
Bertrand duopoly. In a sequential Cournot duopoly, the early moving player receives
a higher payoff than in a simultaneous move game while the late mover receives a lower
payoff.
In contest theory, sequential structures have been analyzed since the 1980’s. In a
seminal work, Dixit (1987) analyzed the impact of an exogenous order of moves on the
strategic effects a sequential structure introduces to a contest between two players. He
denotes the player who has the higher winning probability in Nash-equilibrium as the
favorite. Dixit now finds that the favorite’s equilibrium effort, when she is the first mover,
is higher than her equilibrium effort in a simultaneous move game. Yet, Baik and Shogren
(1992) showed that, in a subgame-perfect equilibrium of a model with an endogenized
order of moves, the stronger player will wait until the weaker player has chosen her action
in the early stage. If the sequential order is determined endogenously, both players will
8For literature on repeated contests see e.g. Amegashie (2005), Linster (1994), Mu¨nster (2004),
Netzer and Wiermann (2006).
12
expend less effort than in the simultaneous game. Leininger (1993) derived this result for
rent-seeking contests with a Tullock-payoff function, explicitly formulating effort choices
and equilibrium payoffs both in the simultaneous and the sequential game. In how far
the analyses of Baik and Shogren and Leininger can be converted from one to another,
has been shown by Nitzan (1994). Since Leininger’s work focuses on the special case of
rent-seeking contests while Baik and Shogren keep their results general, we will use the
article by Leininger as a basis for our analysis of the n-player-case in rent-seeking contests
with constant returns to scale. Unfortunately, we have to state that the analysis of a
fully endogenized order of moves is tractable only for the case of three players. However,
we find that even with this simple modification from the two-player- to the three-player-
case, results concerning equilibria and welfare change. Leininger (1993) found that, for
any combination of heterogenous players, the stronger one will always choose late, and
this will be welfare maximizing among the feasible subgames and thus an improvement
over the result of Tullock’s simultaneous move game. As we will see, these results cannot
be extended to the three-player-case to their full extent.
Further extensions of the analysis of sequential rent-seeking for the two-player-case
have been put forward by Morgan (2003) and Yildirim (2005). In a sequential contest,
Morgan allows for uncertainty about whether a player will have a high or a low valuation
at the moment when she decides upon her decision timing. He shows that, in equilib-
rium, players always move sequentially, although this need not necessary mean that the
stronger player will move late. Morgan can replicate the result that both players have
a higher expected payoff from the sequential game compared to the simultaneous deci-
sion. However, the total expected amount of effort spent in the contest will be higher
than in the simultaneous move game.9 Yildirim (2005) concentrates on the question of
how equilibria change if players are allowed to increase their effort bids after their first
bid. He establishes the existence of multiple equilibria, among them one equilibrium in
the simple simultaneous game and another one that is consistent with a Stackelberg-
equilibrium. But since our research aims at an extension of the player set, let us now
throw a glance at works that deal with more than two players.
The first approach to analyze contests with more than two players in sequential
structure has been proposed by Gradstein and Konrad (1999) who seek for an effort-
maximizing contest structure under the assumption of homogenous players. They find
that a contest designer can maximize effort with a simultaneous move contest for de-
creasing returns to scale, while, for increasing returns to scale, the contests (with n
players) should be played in κ rounds of two-player contests, where only the winner
enters the next round and κ = log2 n. Yet, Gradstein and Konrad assume simultaneous
choice in each subcontest. A first attempt to analyze three-player-contests with truly
sequential structures is proposed by Glazer and Hassin (2000) who analyzed the game
where three homogenous players compete for a rent while deciding upon effort in three
different points of time, one after another. The aggregate effort is, in this case, higher
9At the first glance this might seem contradictory. However, the higher costs of the contest are
outweighed by a higher probability of assigning the rent to the player with higher valuation, which
means a higher expected return of the contest.
13
than in the simultaneous game, and thus welfare is lower.
Finally, Konrad and Leininger (2007) proposed a sequential analysis of an n-player
all-pay-auction.10 They find that in a sequential all-pay-auction with endogenous moves
there exists an equilibrium with a pareto-efficient outcome. No wasteful effort will be
exerted. In this equilibrium each but the strongest player will decide in the first stage
not to make any positive effort in the subgame-perfect equilibrium. The favorite will
then get the rent for free in the second stage, reducing the dissipation to zero, which is
surprisingly good news from a welfare point of view.
1.5 Outlook
Let us now sketch our proceeding. Before starting the main part of our work, we will
first present the basic model and its most important features. The main part of this
work can be split in two parts. The first is concerned with interdependent preferences,
while the second concentrates on sequential structures.
In the first part, we deal with interdependent preferences. We begin by putting up a
model where players of two different types are faced with a rent-seeking contest. While
the first type (individualists) is only interested in maximizing absolute payoff, the second
type (status-seekers) also cares about the relative payoff. In this scenario, we analyze
participation conditions for both types and put up the question of which combinations
of parameters provide the status-seekers a higher material payoff. In a further step,
we relax the assumptions on the set of preference parameter and allow for altruistic
and a much broader range of spiteful preferences, distinguishing between weakly spiteful
preferences that are less spiteful than relative payoff maximization, and strongly spiteful
preferences. We analyze the incentives of these players to spend positive effort. The
main focus of this chapter of our work is to outline the conditions under which more
spiteful players experience a strategic advantage. Finally, the question arises, whether
there is an optimal preference level. It was Schaffer (1989) who first draws the relation
between evolutionarily stable preferences and relative payoff maximization. Later, Eaton
and Eswaran (2003) showed for rent-seeking contests with constant returns to scale that
maximizing the relative payoff is an evolutionarily stable preference. We will generalize
the latter for a much broader range of rent-seeking contests, including those with slightly
increasing returns to scale.
The second part concentrates on multi-player-contests with an endogenous order of
moves. The articles by Baik and Shogren (1992) and Leininger (1993) show that for two-
player-contests, the stronger player will always decide to choose late, while the weaker
one will decide to choose early. Leininger, furthermore, shows that this is the optimal
order of moves if the aim is to reduce the effort spent by the contestants. The second
part of our work seeks to generalize these results to a larger number of players. In
the first chapter on sequential structures, we assume a two-stage-game in which three
10To see the relationship between all-pay-auctions and Tullock-contests with constant returns to scale,
see Section 2.2.
14
players can decide at the first stage whether to decide upon their effort early or late,
and, in the second stage, they play a sequential (or simultaneous) rent-seeking contest
depending on the chosen timings of the first stage. We provide efforts and payoffs of
every feasible subgame and compare them. In the next step, we analyze a game where
one favorite faces (n− 1) equal opponents. These opponents are assumed to be weaker,
in the sense that they have a lower valuation for the prize. In this setting, we look
for the endogenous order of moves for two different cases: one with a first-mover-right
for the strongest player and one without. This enables us to evaluate, from a welfare
point of view, whether such a first-mover-right should be granted. Finally, we analyze
a general n-player-contest and ask under which conditions players have an incentive to
deviate from a contest with simultaneous moves. As a result we will be able to state a
general condition under which players will decide upon a simultaneous move contest in
a subgame-perfect equilibrium.
15
Chapter 2
The Theory of Contests
This chapter gives an introduction to the technical framework of this work. In the prior
chapter, we mentioned that Tullock (1980) has advanced the theory of rent-seeking by
introducing a handy model. In this model, he described a contest for a rent of fixed size
V , where a player i exerts a non-negative effort xi in order to receive a non-negative
payoff. The payoff function has the following form:1
pii(x) =



xri∑n
j=1
xrj
V − xi if ∃j = 1, ..., n : xj > 0
1
n if xj = 0, ∀j = 1, ..., n,
(2.1)
with x = (x1, ..., xn). The parameter r represents the returns to scale of the underlying
contest technology and n denotes the number of contestants. Typically, we assume that
the term x
r
i∑n
j=1
xrj
depicts the probability that player i wins the total amount of the prize
V . Therewith, we implicitly assume that a player maximizing (2.1) is risk-neutral. This
term can also be interpreted as displaying the share of the prize that player i receives
depending on her effort. Yet, although there is no difference from an analytical point of
view, we will restrict the discussion of our results to the interpretation of this term as a
winning probability.
Tullock calculates the equilibrium and finds that the first-order-condition is given
by: xi = rn−1n2 V . This effort will be chosen in the Nash-equilibrium of a contest between
homogenous players if the second-order-condition is fulfilled, which holds for r < nn−1 (cf.
Perez-Castrillo and Verdier, 1992). The aggregate effort is given by rn−1n V and, from
a welfare point of view, these efforts are unproductive waste. In the remainder of this
section, we will allude to some features that can occur in rent-seeking contest in different
settings.
We should, furthermore, point out that our analysis uses the playing-the-field assump-
tion, which is common in contest theory and has implications mostly for evolutionary
questions. ”Playing-the-field” means that, if a population of players is observed playing
1We extended the original model such that the payoff is also defined, if no player exerts a positive.
While it is conceivable, that if no player applies for the rent, no player will receive it, we assume that
in this case the winning probability is equally distributed among the players.
16
a game, then all players participate in the game together. In contrast to analyses where
repeatedly e.g. two players are chosen out of a population to play a game, under the
playing-the-field assumption evolutionary stability yields equivalent outcomes regardless
of whether evolution works at strategy or at preference level for a large variety of games
(see Eaton and Eswaran, 2003). We will show in Chapter 5 that this also holds for all
rent-seeking contests with decreasing and for some with increasing returns to scale.
2.1 Axiomatic Foundation
While this work concentrates on the Tullock-contest, for which the winning probability
has a logit form and the impact function is a power function, we should, nevertheless,
stress that the theory of contests goes beyond this easily tractable model. An extensive
overview of the field of contests in general is given by Konrad (2007). Let us start our
discussion by referring to models where not only the assumption that the contest success
function is a power function is dropped, but also the assumption of the logit form, namely
the works of Dixit (1987) and Malueg and Yates (2004).2 Both these analyses look only
at contests between two players. Dixit’s work focuses on a game with an exogenous
structure where one player moves earlier than the other. His result for the general case
is that, for a positive cross derivative of her winning probability, a player will exert higher
effort than in a simultaneous-move Nash-equilibrium.3
Malueg and Yates assumed the existence of an interior solution of the simultaneous
move game and found some additional results concerning the comparative statics. In
particular, they pointed out how the influence of valuations on the effort levels, the
winning probabilities and the final payoff depend on the cross derivative of the winning
probabilities. All of these results have been previously adapted to Tullock-contests (see
e.g. Baik, 1994, Nti, 1999, Stein, 2002).
Yet, most of the literature concentrates on contests with a logit form. On the first
glance this seems to be arbitrary. However, Skaperdas (1996) has proposed a system of
axioms that are reasonable for the winning probability in a contest with homogenous
players and immediately imply a logit contest success function. His axioms are:
1. The winning probabilities are true probabilities, i.e., they lie in the interval [0, 1]
for every player, and sum up to one for all players.4
2. Raising one player’s effort strictly increases this player’s winning probability, while
it decreases the other players’ winning probabilities.
2Baik and Shogren (1992) analyze their sequential game not only for a logit but also for a probit
form. Yet their analysis does not cover contests as general as Dixit and Malueg and Yates.
3In the case of a Tullock-contests, this is for player i: ∂
2pi
∂xi∂xj
= ∂
2
∂xi∂xj
(
xri
xri+x
r
j
)
= ∂∂xj
(
rxr−1i x
r
j
(xri+xrj)
2
)
= r(xixj)
r−1
(xi+xj)3
(
xri − x
r
j
)
. This is positive if player i exerts the higher effort in equilibrium.
4For an example violating this axiom, see Nti (1997).
17
3. Skaperdas called for anonymity. This means that each player can be exchanged
with any other player without harming the results.
4. Let pi(x) denote the winning probability of player i in the contest among all play-
ers, and pMi (x
M) denote the winning probability of player i in a contest among
a subgroup M of the initial group of players, say of size m. Then, the winning
probabilities of the players in this subgroup will not depend on M , if effort levels
do not change, i.e. p
M
i (x
M )
pMj (x
M ) =
pi(x)
pj(x)
.5
5. Whenever the contest is played only among a subgroup of the initial group of
players, the winning probabilities of the participating players do not depend on
non-participating players.
Skaperdas showed that any probability function that satisfies these axioms is a logit
function, i.e., pi(x) =
f(xi)∑n
j=1
f(xj)
. In a further step, he introduced a sixth axiom, namely
the one of zero-homogeneity.
6. Increasing the effort of all players by the same factor does not alter the winning
probability of any player. Specifically, this means that the contest success function
does not depend on the unit of efforts, which can be interpreted as the currency if
prize and efforts are monetary variables.
Skaperdas established that the probability function used in the Tullock-contest is the
only one that fulfills the Axioms 1 through 6. This means that, if one accepts the
axioms of Skaperdas, it is sufficient to analyze a Tullock-contest instead of a general
contest.
Note that Skaperdas’ axiom 3 requires homogenous players. This axiom has been
regarded critically in economic literature. Thus, Kooreman and Schoonbeek (1997) and
Clark and Riis (1998) extended the analysis of Skaperdas by dropping Axiom 3 and came
to the result that the axioms 1, 2, 4, 5 and 6 are equivalent to:6
pi(x) =
σixri
∑n
j=1 σjx
r
j
.
However, for our analysis, the results of Skaperdas are sufficient since we do not assume
players to be heterogenous with respect to their contest success function but only with
respect to their preferences or to their valuations. The functional form of the winning
probability is not affected by this heterogeneity.
5See appendix for a short discussion.
6Kooreman and Schoonbeek (1997) analyzed this case for two players, while Clark and Riis (1998)
looked at the n-player-case.
18
2.2 On the Returns to Scale
In this section, we want to focus on the role that the technology parameter r plays in
Tullock’s payoff function. While we will restrict the analysis in the main part of this
work to small values of r and, mostly, to r = 1, it is, nevertheless, worthwhile discussing
what will happen if r becomes large.
The parameter in the power function represents how the marginal effectiveness of the
efforts develops. While, for small values of r < 1, the first marginal effort is infinitively
effective for an effort level of zero and marginal effectiveness decreases in effort, higher
values of r > 1 represent a contest function with increasing returns to scale.
Let us first take a look at the limit cases. On the one hand, if r takes the value 0, the
impact function xr is reduced to 1 for every positive effort level x. We thus have a fair
lottery where every player has an equal chance of winning, regardless of her effort. Since
a player receives the prize with probability pi = 1n even if she chooses zero effort, an
optimal reaction of a player cannot be to choose a positive effort level xˆ. In equilibrium
x = (0, ..., 0).
The converse limit case, r = ∞, has a far more important interpretation. In this
case, every player pays the costs for her effort, but the player with the highest effort
choice, if unique, wins the prize with certainty. This form of contest is better known
as an all-pay-auction. Under the assumption of complete information and simultaneous
decisions, Hillman and Riley (1989) showed that, in an all-pay-auction with heterogenous
players, only the two players with highest valuations are active, in the sense that they
choose a positive effort with a positive probability. In equilibrium, the player with the
highest valuation (V1) plays a mixed strategy, where she chooses her effort level from
an equal distribution between zero and the second highest valuation (V2). The player
with the second highest chooses a strictly positive effort with probability V2V1 , and with the
probability V1−V2V1 she chooses an effort of zero. In the case that she exerts a positive effort,
she also chooses her effort from an equal distribution on [0, V2]. Players with a lower
valuation remain inactive. As a result, the player with the highest valuation receives
an expected payoff of V1 − V2, while all the other players receive an expected payoff
of zero. Baye et al. (1996) provide a complete analysis of all equilibria in the all-pay-
auction, where they relax the implicit assumption that the highest and the second highest
valuation are unambiguous. Especially if the two highest valuations take equal value, no
player receives a positive payoff. In this case, we have full dissipation. Overdissipation
will never occur. Konrad and Leininger (2007) propose a model of an all-pay-auction,
where they find an equilibrium in which no dissipation exists, as we already discussed in
detail in Section 1.4.
Let us now concentrate on positive but finite values of r, which we will focus on for
the rest of this work. In general, this class of games can be divided into four different
cases each with different properties. The first and most prominent is the one with r = 1
that represents a contest technology with constant returns to scale. Many works on
the field of contest theory (e.g., Leininger, 1993, Linster, 1994, Stein, 2002, Schoonbeek
and Winkel, 2006) concentrate on this special case, among others, because of its easy
19
tractability.7 Apart from its prominence with concern to manageability, this case also
takes a special role from the point of view of results. As we will see, it depicts the border
betweeen two further areas of interest with different properties.
For the case of decreasing returns to scale where r < 1, we find an interesting contrast
to constant returns in the sense that the reaction function is always positive. While,
in a setting with r = 1 and heterogenous players, as we will see in Chapter 4, some
players might abstain from participation, every player has an incentive to make a positive
investment if returns to scale are decreasing. The reason for this is that the marginal
return on invested effort converges to infinity as the effort itself converges to zero (cf.
Cornes and Hartley, 2005). This means that, for the case of non-participation (i.e. an
effort level of zero), the marginal gain of the first bit of effort is infinitely high, while the
marginal cost is a finite (and, in general, constant) number. Thus, positive investment
always pays.
Next, let us focus on the case of slightly increasing returns to scale meaning that
r ∈
(
1, nn−1
]
. In contrast to the case where r = 1 for increasing returns to scale (r > 1),
the general properties of the contest, such as existence and uniqueness of (pure-strategy)
equilibria, depend on the number of players n in the contest. Hence, we split the analysis
for increasing returns to scale in two subcases where the threshold nn−1 is a function of
n. For a contest with homogenous players, this is the threshold up to which maximizing
payoff leads to non-negative payoff in an interior Nash-equilibrium. As a consequence,
in the presence of increasing returns to scale, the number of players must be known in
order to determine the type of equilibrium that can occur. As we will see in section
3.6, the uniqueness cannot even be ensured for r ∈
(
1, nn−1
]
if the assumptions of payoff
maximizing and homogeneity are relaxed (see the discussion below).
In his seminal paper, Tullock (1980) showed that, if returns to scale are high enough,
the first-order-conditions for a number of numerical examples hint at effort levels with a
dissipation level higher than the prize (overdissipation).8 Indeed, these numerical exam-
ples are those characterized by r > nn−1 . But since payoff maximization is assumed to be
the major concern of players in the standard rent-seeking contest, overdissipation cannot
occur in equilibrium, because every player could, by reducing her effort to zero, ensure
that she receives zero payoff. As a consequence, the equilibrium analysis becomes more
complex for r > nn−1 ,
9 since the second-order-condition is no longer fulfilled. However,
this challenging case has hardly been dealt with in literature. The first important contri-
bution in this field was made by Perez-Castrillo and Verdier (1992). They characterize
pure-strategy equilibria for r > nn−1 , and show that, for 2 ≤
r
r−1 (which is equivalent
to r ≤ 2), a Nash-equilibrium is given when m players compete actively in the contest
(where 2 ≤ m ≤ rr−1 < n), while the remaining players stay outside the contest. Note
that this means that r ≤ mm−1 .
The only work that analyzes equilibria for the case of r > 2 is the one of Baye
7E.g., while the first derivative of the impact function in general is rxr−1, for r = 1 it is 1!
8Yet, for these effort levels, the second-order-conditions of a payoff maximization are not fulfilled.
9As we will see later, overdissipation can indeed occur, if the standard assumptions are relaxed.
20
et al. (1994) who restricted their work to the two-player-case.10 They calculate some
Nash-equilibria for parameter constellations, where continuous choice of effort is not
feasible, but effort choice must take integer values. They do not give a general solution
to the problem. Yet, they find that, in their setting, overdissipation will never occur in
Nash-equilibrium.
2.3 The Role of Heterogeneity
In his standard model, Tullock (1980) had assumed that all players were identical. In real
life, we observe that, in many contests, players differ in several aspects such as valuation
for rent (Chapter 6 to 8), costs in exerting efforts, effectiveness of efforts or objective
functions (Chapter 3 to 5). Throughout this work, we will assume that players can be
heterogenous in at least one of these points.
For the introductory motivation, let us focus on heterogeneity that can arise from
the payoff function. The further chapters will then address the questions of how results
change if other variables than the own material payoff function, especially the payoff of
the opponents, play a role for the decision of the contestant or if we introduce different
timing aspects into the decision structure.
To illustrate the three possible levels of heterogeneity in the material payoff function
let us write a general payoff function as:11
pii(x) =
σixri
∑n
j=1 σjx
r
j
· Vi − dixi. (2.2)
Different values of Vi represent heterogeneity in the valuation of the rent, a difference in
di between the players signals a heterogenous cost structure for the effort and different
values of σi represent differences in the effectiveness of effort.
The first-order-condition for the maximization of (2.2) is given by:
∂pii(x)
∂xi
!= 0⇔ r · σi · x
r−1
i


∑
k 6=i
σkx
r
k

 ·
Vi
di
=


n∑
j=1
σjx
r
j


2
. (2.3)
The only way that valuation and cost of effort influence this reaction function is via
the factor Vidi . Thus, an increase (decrease) of the valuation and a decrease (increase) of
the effort costs by the same factor induce the same change in the reaction function of
player i.
With respect to effectiveness, however, results are no longer equivalent. Comparative
statics of both heterogeneity in valuation and in effectiveness have been put forward by
10In a comment, Tullock (1995) claims, that Perez-Castrillo and Verdier (1992) have already looked
upon this case. However, Perez-Castrillo and Verdier restrict their analysis to r ≤ 2.
11Although heterogeneity with respect to the technology parameter r would be also conceivable, this
issue has not been addressed in the literature so far. Except for the huge reduction in calculatory
complexity, a motivation for assuming homogenous technology scales is given by the axiomatic theory
of heterogenous players in contests, cf. Section 2.1.
21
Stein (2002). He shows that, while an increase in valuation will always increase individual
and total effort, a change in effectiveness has ambiguous effects. An increase in σi only
increases total effort if σi is relatively small. In this case, the players become more similar,
which intensifies competition in the contest. If a strong player becomes more effective,
total effort decreases. In a similar way, changes in valuation and effectiveness have very
different effects on the player’s own effort. While an increase in valuation always incites
her to choose a higher effort level, increasing one’s effectiveness eases up the player’s
situation, such that she reduces her effort in most situations. The effect of changes in
valuation and effectiveness on winning probabilities, payoffs and the opponents’ effort
levels are, however, comparable.
22
Part I
Interdependent Preferences
23
Chapter 3
Strategic Advantage of Negatively
Interdependent Preferences1
3.1 Introduction
In this chapter, we explore the implications of negatively interdependent preferences in
the strategic context of rent-seeking contests. The focus is on heterogeneous populations
where part of the players maximize material payoff while others are additionally con-
cerned with their relative position. To this aim, we will introduce the notion of relative
payoff, that represents the difference between one’s own material payoff and the average
material payoff of all players.
In this setup, we will allude to a number important issues that help to characterize
the equilibria of this game. We start with discussing the conditions that have to be
fulfilled such that players actively participate in this game. Beyond this, we present
some results that illustrate how the relative size of effort can determine the relative size
in material payoff.
In the core of this chapter, we seek for conditions that are necessary and sufficient to
obtain the result that players with interdependent preferences receive the higher material
payoff in equilibrium. Here, we first concentrate on the two-player-contest, an ubiquitous
example for which we find that the spiteful player will always receive a higher material
equilibrium payoff. Thereafter, we show that this result generalizes to multi-player-
contests if returns to scale are non-increasing. Finally, we turn to increasing returns to
scale. Here, the results are not as straightforward if there are more than two players.
We distinguish four types of equilibria that are characterized by (a) whether or not
interdependent preferences lead to a higher effort level and (b) whether both types of
players choose a positive effort level or only one type. We find that in general all four
types of equilibria can occur for certain parameter ranges. Even more, we find that
at least three types of these equilibria can occur for the same parameter constellation,
which immediately implies that non-uniqueness of equilibrium can occur for games with
more than two players. Examples are provided.
1This chapter represents joint work with Burkhard Hehenkamp, cf. Guse and Hehenkamp (2006).
24
A related analysis to ours has been presented by Kockesen et al. (2000a, 2000b).
They establish the strategic advantage of negatively interdependent preferences for cer-
tain classes of supermodular and submodular games, which do not include general rent-
seeking contests. Individuals, who have an additional concern for their relative position,
act more aggressively than players who just care about their own material pay-off. This
gives them higher pay-off in equilibrium. However, Kockesen et al’s results do not cover
rent-seeking contest. First, this is because they focus on cases where either strategic
complements or strategic substitutes are present. In rent-seeking contests, by contrast,
payoff functions have both a range of strategic complementarity and a range of strategic
substitutability. Second, Kockesen et al. restrict their analysis to non-negative payoff
functions, a premise that, in the context of rent-seeking contests, would simply assume
the issue of dissipation away. It thus remains an open question whether the strategic ad-
vantage of negatively interdependent preferences and the phenomenon of overdissipation
prevail in the economically important context of rent-seeking contests.
Whether status-seeking preferences have a strategic advantage over individualistic
ones or not, has a number of interesting applications, two of which we will introduce
here.
For the first one, imagine a situation, where a decision-maker is about to delegate
a task that involves play of a rent-seeking contest to one out of a group of possible
delegates - which one should he select? Then - provided the type of rent-seeking contest
gives a strategic advantage to status-seeking preferences - he should pick the person that
will presumably be most concerned about his relative performance in the contest.
The second application is related to the first. Consider the decision whether to
delegate or not and, in the case of delegation, which incentives to provide the delegate
with. This framework was originally proposed and developed by Vickers (1985) and
Fershtman and Judd (1987) in the context of oligopoly. There is an increasing amount
of literature on strategic delegation in rent-seeking contests (see e.g. Baik and Kim,
1997, Kra¨kel and Sliwka, 2002, or Baik, 2003), all of which, among other restrictions,
restrict to the rather special case of a constant returns to scale technology. Notice that
we deal with a much more general class of payoff functions, covering the whole range of
decreasing, constant and increasing returns to scale. In the conclusion, we will comment
on the implications of our results.
3.2 The model
In our model, we examine rent-seeking contests, where the material payoff is character-
ized by (2.1). The effort of each player is non-negative. With slight abuse of notation,
we write x = (xh, x−h) when it is convenient to single out player h’s strategy.
We consider a population where the set of players is divided into two groups. Each
member of the first group, i ∈ I ≡ {1, ...,m}, picks her effort in order to maximize
material payoff, pii(x), where x is the vector of effort levels of all n players. Members of
the first group have independent preferences. We call them individualists.
Members of the other group, j ∈ J ≡ {m+1, ..., n}, additionally care about their own
25
material performance relative to that of the other players. We call them status-seekers.
The function F below represents their (negatively) interdependent preferences with
respect to absolute and relative material payoff. More precisely, they choose their effort
level in order to maximize
Fj(x) = F (pij (x) , ρj (x)) , (3.1)
where we assume F to be strictly increasing in both arguments and where the relative
payoff, ρj (x), is given by
ρj (x) := pij (x)−
1
n
n∑
h=1
pih (x) . (3.2)
Since these types of preferences do not only depend on absolute material payoff, but also
on relative material payoff, we call these preferences (negatively) interdependent.
Alternative specifications of relative payoff are also possible. For example, a player
might compare her own payoff with the average payoff of the other players:
ρ˜j (x) := pij (x)−
1
n− 1
∑
h6=n
pih (x) . (3.3)
However, although the equilibrium slightly changes, the class of solutions does not. Thus,
the general results we establish with (3.2) could also be implemented with (3.3).
Notice that, is important to distinguish between material payoff, which determines
the material success of players and their strategies, and utility, which represents the
players’ preferences that can be independent or interdependent. While, for individualists,
the notions of material payoff and utility coincide, they differ for status-seekers. The
following example introduces the class of linear interdependent preferences.
Example 1 Consider the following interdependent utility function, which assumes pref-
erences to be linear in absolute and relative material payoff,
F (pij(x), ρj (x)) = β1pij(x) + β2ρj (x) =
(
β1 + β2
n− 1
n
)
pij(x)−
β2
n
∑
h6=n
pih(x)
= (β1 + β2)pij(x)−
β2
n
[
V −
n∑
h=1
xh
]
, (3.4)
where β1, β2 > 0 guarantee the mentioned monotonicity properties. Notice that for fixed
β1 > 0 and β2 = 0 maximizing (3.4) is equivalent to maximizing (2.1). Similarly, given
any fixed β2 > 0, maximizing (3.4) is equivalent to maximizing relative material payoff
if β1 = 0.
As is standard in game theory, we assume that, in equilibrium, each player takes the
other players’ efforts as given, i.e., we search for a Nash-equilibrium of the game that is
induced by the independent and interdependent preferences. As in Kockesen et al., our
26
focus is on intragroup symmetric equilibria, where all players with identical preferences
choose the same effort level, i.e., where xi = x1,∀i ∈ I, and xj = xn,∀j ∈ J . We denote
these equilibria by xˆ = ([a]m,[b]n−m) ∈ Nsym (ΓF (m)).
For our further analysis, we need to define the notions of strategic advantage and
dissipation:
Definition 1 Consider a game ΓF (m) for some k ∈ {1, ..., n − 1}. Then, (negatively)
interdependent preferences yield a (strict) strategic advantage over independent prefer-
ences, if and only if
pij (xˆ)
(>)
≥ pii (xˆ) , ∀i ∈ I ∀j ∈ J ∀xˆ ∈ Nsym (ΓF (m)) .
Definition 2 An equilibrium xˆ displays overdissipation (full or underdissipation), if and
only if
∑n
h=1 xh > (= or <) V, respectively.
It follows that in situations where status-seekers have a strategic advantage over individ-
ualists there will be no overdissipation. If the strategic advantage is strict, this implies
underdissipation. To see this, notice that an individualist will only exert positive effort
if her material payoff would be non-negative, i.e., pii(xˆ) ≥ 0. Otherwise, she could do
better with xi = 0. In case of a strategic advantage, it follows that pij(xˆ) ≥ pii(xˆ) ≥ 0.
Consequently, the material payoff of these players is non-negative. This implies that
the sum over all material payoffs is also non-negative,
∑n
h=1 pih(xˆ) ≥ 0. Because of∑n
h=1 pih(xˆ) = V −
∑n
h=1 xh, this is equivalent to V ≥
∑n
h=1 xh. Thus, the sum of all
effort levels is lower than the prize at stake. There will be no overdissipation.
In case of a strict strategic advantage, we have pij(xˆ) > pii(xˆ) ≥ 0. Therefore,
∑n
h=1 pih(xˆ) > 0 and V >
∑n
h=1 xh, which excludes the case of full dissipation. Accord-
ingly, we must have underdissipation whenever a strict strategic advantage is present.
We have shown:
Proposition 1 Suppose interdependent preferences yield a strategic advantage over in-
dependent preferences. Then, dissipation is, at most, full, i.e.,
∑n
h=1 xh ≤ V . If the
strategic advantage is strict, we have underdissipation, i.e.,
∑n
h=1 xh < V .
3.3 Preliminary results
We start with deriving general conditions under which individualists and status-seekers
optimally participate in the contest. Subsequently, we derive first order conditions that
characterize equilibrium behavior. We conclude this section with a preliminary lemma
that helps to analyze the strategic advantage of negatively interdependent preferences.
3.3.1 Participation constraints
We, first, give the conditions under which individualists optimally participate and under
which they optimally stay out of the contest. Subsequently, we turn to the respective
conditions for status-seekers.
27
Individualists
On the one hand, individualists i ∈ I optimally participate in the contest (i.e. a > 0)
whenever doing so yields non-negative payoff in equilibrium (zero payoff is what they
would earn leaving the contest):
pii(xˆ) ≥ pii(0, xˆ−i) ⇔
ar
mar + (n−m)br
V − a ≥ 0. (PCI)
On the other hand, non-participation is optimal to individualists when no positive
effort level gives positive material payoff, i.e., when pii(xˆ) = 0 ≥ pii(xi, xˆ−i), for all
effort levels xi ≥ 0. The following lemma provides an equivalent characterization of
non-participation:
Lemma 1 Fix xˆ = ([a]m, [b]n−m) ∈ Nsym(ΓF (m)). If a = 0 then xi = 0 is optimal to an
individualist i ∈ I if and only if r > 1 and
(r − 1)r−1
rr
V r ≤ (n−m)br. (NPCI)
Proof. To prove this lemma, we simply refer to Perez-Castrillo and Verdier (1992).
They show in their first proposition that non-participation in a contest is optimal when-
ever (r−1)
r−1V r
rr is lower than the aggregate effort of the other contestants, which is here
given here by (n−m)br.
Status-seekers
Status-seekers j ∈ J participate in the contest whenever this gives higher utility with
respect to F (·) than non-participation would. Contrary to individualists, status-seekers
additionally evaluate their relative payoff that would result from non-participation. Cor-
respondingly, a status-seeker’s participation constraint reads:
F (pij(xˆ), ρj(xˆ)) ≥ F (pij(0, xˆ−j), ρj(0, xˆ−j)). (PCJ)
Without knowing the concrete utility function F (·), we can hardly compare the two
expressions in (PCJ). Therefore, it is not possible to give a general characterization
of a status-seeker’s participation constraint. To get a grip on this problem, we first
investigate the relationship between absolute and relative payoff. Lemma 2 summarizes
our findings.
Lemma 2 Let x′,x′′ ∈ Xn be two arbitrary effort profiles that only differ in their j :th
component, i.e., x′ = (x1, .., x′j, .., xn), x
′′ = (x1, .., x′′j , .., xn) and x
′
j < x
′′
j . Then, the
following implications hold:
(i) pij(x′) ≤ pij(x′′) ⇒ ρj(x′) < ρj(x′′),
(ii) ρj(x′) ≥ ρj(x′′) ⇒ pij(x′) > pij(x′′).
28
Proof. Inserting the definitions of ρj(·) and pij(·), one obtains
ρj(x′)− ρj(x′′) =
[
pij(x′)− 1n
∑
h pih(x
′)
]
−
[
pij(x′′)− 1n
∑
h pih(x
′′)
]
= [pij(x′)− pij(x′′)]−
[
1
n
∑
h pih(x
′)− 1n
∑
h pih(x
′′)
]
= [pij(x′)− pij(x′′)]− 1n [x
′′
j − x
′
j],
where the last equality follows from
∑
h pih(x) = V −
∑
h xh. Then, the two claims hold
because of x′′j > x
′
j.
Notice that Lemma 2 does not only cover switches between participation and non-
participation, but, more generally, analyzes the effects of an increase in effort (or a
decrease). Part (i) says that any increase in effort that increases absolute payoff will
also increase relative payoff, while part (ii) states the converse: Any decrease in effort
that increases relative payoff leads to an increase in absolute payoff.
It, thus, follows from Lemma 2 that an effort profile xˆ = ([a]m, [b]n−m) can only
represent an intragroup symmetric equilibrium if, for any status-seeker j ∈ J , (i) no
higher effort level xj > b increases her absolute payoff pij and (ii) no lower effort level
xj < b increases her relative payoff ρj. So, if one of the two conditions applied, i.e., if
(i) xj > b increased absolute payoff pij or if (ii) xj < b increased relative payoff ρj, then
Lemma 2 would imply that both absolute and relative payoff increase, which would lead
to a higher utility level with respect to F (·) – the former choice of xˆj = b would not have
been optimal. We have thus established the following corollary:
Corollary 1 Fix xˆ = ([a]m, [b]n−m) ∈ Nsym(ΓF (k)). Then, for all status-seekers j ∈ J ,
we have (i) pij(xˆ) ≥ pij(xj, xˆ−j), for all xj > b and (ii) ρj(xˆ) ≥ ρj(xj, xˆ−j) for all xj < b.
Returning to the problem of characterizing the status-seekers’ participation con-
straint, we can now derive a necessary and a sufficient condition for the optimal partic-
ipation of status-seekers. The necessary one says that non-participation may not lead
to higher relative payoff ρj(0, xˆ−j), since by Lemma 2, this would also imply higher ab-
solute material payoff pij(0, xˆ−j) and, hence, higher utility with respect to F (·). Thus,
non-participation would be optimal in this case. On the other hand, the sufficient con-
dition says that status-seekers will optimally exert positive effort when participation
results in higher absolute payoff than non-participation. By Lemma 2, this also implies
higher relative payoff and thus higher utility with respect to F (·). The corollary below
sums up our findings:
Corollary 2 (Optimality of status-seekers’ participation)
Fix xˆ = ([a]m, [b]n−m) ∈ Nsym(ΓF (m)). Then, for all status-seekers j ∈ J , (i) b > 0
implies ρj(xˆ) ≥ ρj(0, xˆ−j) and (ii) pij(xˆ) ≥ pij(0, xˆ−j) implies b > 0.
In contrast, non-participation is optimal to a status-seeker when any positive effort
level does not increase her utility with respect to F (·), i.e., when
F (pij(xˆ), ρj(xˆ)) ≥ F (pij(xj, xˆ−j), ρj(xj, xˆ−j)) (NPCJ)
29
for xˆ = (0, xˆ−j) and for all xj ≥ 0. Again, without knowing the functional form of
F (·), we can only give a necessary and a sufficient condition.
Non-participation can only be optimal for a status-seeker j when participation with
an arbitrary effort level xj > 0 does not increase her absolute material payoff above the
zero payoff margin (which results from non-participation). Otherwise, a positive effort
level does not only increase j’s material payoff, but by Lemma 2, also decreases all other
players’ material payoff. Again, both absolute and relative payoff increase so that utility
increases with respect to F (·) – non-participation would not have been optimal.
Similar to the above, non-participation is, in fact, optimal when any positive effort
level xj > 0 reduces relative payoff. By Lemma 2, this also implies a reduction in absolute
payoff and hence in utility. We have established:
Corollary 3 (Optimality of status-seekers’ non-participation)
Fix xˆ = ([a]m, [b]n−m) ∈ Nsym(ΓF (m)). Then, (i) b = 0 implies pij(xˆ) ≥ pij(xj, xˆ−j), for
all xj > 0 and (ii) ρj(xˆ) ≥ ρj(xj, xˆ−j) implies b = 0, for all xj > 0.
We conclude the analysis of participation constraints with the example of linear
interdependent utility functions. For this class of status-seeking preferences, we can
determine an equivalent characterization.
Example 1 (continued) Let xˆ = ([a]m, [b]n−m) denote an intragroup symmetric Nash
equilibrium and xˇ = (0, xˆ−j) the alternative strategy profile where status-seeker j opts
out of the contest. Recall that
F (pij(x), ρj (x)) = β1pij(x) + β2ρj (x) = (β1 + β2)pij(x)−
β2
n
[
V −
n∑
h=1
xh
]
. (3.5)
Consequently, the participation constraint of status-seekers, (PCJ), is equivalent to
Fj(xˆ) ≥ Fj(xˇ),
(β1 + β2)pij(xˆ)−
β2
n
[
V −
n∑
h=1
xˆh
]
≥ (β1 + β2)pij(xˇ)−
β2
n
[
V −
n∑
h=1
xˇh
]
,
(β1 + β2)
[
br
mar + (n−m)br
V − b
]
−
β2
n
b ≥ 0,
br
mar + (n−m)br
·
n(β1 + β2)
nβ1 + (n− 1)β2
V − b ≥ 0. (3.6)
Here, the third row follows from pij(xˇ) = 0 and xˆh = xˇh for h 6= j, while the fourth
row is a multiple of the third one. Setting
λ =
n(β1 + β2)
nβ1 + (n− 1)β2
, (3.7)
30
we can interpret λV as the implicit personal value a status-seeker ascribes to the material
prize V. (A status-seeker with personal value λV will participate in the contest with ma-
terial value V if and only if an individualist would participate in a contest with material
value λV .) Notice that β1, β2 > 0 implies λ ∈ (1, nn−1). Put verbally, the monotonicity
assumptions put an upper boundary on the implicit personal value λV. We will allude to
this issue in more detail in the following chapter.
3.3.2 Characterizing equilibrium behavior
For individualists i ∈ I, the first order condition of a local absolute payoff maximum at
xi = a > 0 can be derived from (2.1) as
∂pii
∂xi
(xˆ) = 0 ⇔ rar−1[(m− 1)ar + (n−m)br]V = [mar + (n−m)br]2. (FOCI)
As is well known in the rent-seeking literature (cf. Pe´rez-Castrillo and Verdier, 1992),
any solution to the first order condition that satisfies the participation constraint (PCI)
also satisfies the second order condition for a local maximum. Moreover, the only other
candidate for a global maximum is xi = 0. Hence, for any individualist i ∈ I, an effort
level xi > 0 is optimal and leads to non-negative payoff pii(xˆ) ≥ 0 if and only if xi = a
satisfies (FOCI) and (PCI).
A status-seeker j’s effort level xˆj = b > 0 is optimal when it maximizes utility with
respect to F (·), i.e., when
F (pij(xˆ), ρj(xˆ)) ≥ F (pij(xj, xˆ−j), ρj(xj, xˆ−j)),
for all xj ≥ 0. Unfortunately, we cannot derive general conditions without knowing the
particular functional form of F (·). However, we can build on the insight of Lemma 2 in
order to derive a lower and an upper boundary on the derivative of j’s material payoff
function. To this end, let us first determine the first derivative of j’s absolute and relative
payoff function, respectively:
∂pij
∂xj
(xˆ) =
rxˆr−1j
∑
h 6=j
xˆrh
[
∑
h
xˆrh]
2 V − 1, (3.8)
∂ρj
∂xj
(xˆ) =
rxˆr−1j
∑
h 6=j
xˆrh
[
∑
h
xˆrh]
2 V − 1 + 1n =
∂pij
∂xj
(xˆ) + 1n . (3.9)
Obviously, ∂pij∂xj (xˆ) ≥ 0 implies
∂ρj
∂xj
(xˆ) > 0 and, conversely, ∂ρj∂xj (xˆ) ≤ 0 implies
∂pij
∂xj
(xˆ) <
0. These implications represent the marginal analogue to Lemma 2. Similar to the
corresponding corollary, (i) ∂pij∂xj (xˆ) > 0 and (ii)
∂ρj
∂xj
(xˆ) < 0 cannot hold in any intragroup
symmetric Nash equilibrium. Therefore, in the former case, any status-seeker j ∈ J could
strictly increase her interdependent utility F (·) exerting slightly higher effort b+ε, while,
in the latter case, he could gain from a slightly lower effort level b−ε (for ε > 0 sufficiently
small). Therefore, in any intragroup symmetric Nash equilibrium xˆ = ([a]m, [b]n−m) ∈
Nsym(ΓF (m)), we have
31
∂pij
∂xj
(xˆ) ≤ 0 and
∂ρj
∂xj
(xˆ) =
∂pij
∂xj
(xˆ) +
1
n
≥ 0.
Inserting xˆ = ([a]m, [b]n−m) in (3.8) and (3.9), the inequalities imply
n− 1
n
<
rbr−1[mar + (n−m− 1)br]
[mar + (n−m)br]2
V < 1,
which can be written as
rbr−1[mar + (n−m− 1)br]λV = [mar + (n−m)br]2, (FOCJ)
for some λ ∈ (1, nn−1). Comparing (FOCI) and (FOCJ), we obtain the same intuitive
interpretation of λ that we introduced in Example 1: Given λ, the expression λV repre-
sents the implicit personal value a status-seeker ascribes to the material prize V . From
λ > 1, we have λV > V, that is, a status-seeker associates a higher personal value with
the prize V than does an individualist (whose personal value coincides with the material
value). Just as in the example, the feasible range of values, λ ∈ (1, nn−1), is bounded
from above for the analysis in this chapter. However, as our analysis will later show, this
does not always prevent status-seekers from wasteful competition.
Example 1 (continued) Consider the example of linear interdependent utility func-
tions defined by (3.4). Calculating the first order condition of status-seekers, we obtain
n(β1 + β2)
nβ1 + (n− 1)β2
rxˆr−1j


∑
h6=j
xˆrh

V =
[
∑
h
xˆrh
]2
, (3.10)
for any intra-group symmetric Nash equilibrium xˆ = ([a]k, [b]n−k). Not surprisingly, equa-
tion (3.10) reduces to (FOCJ) for λ = n(β1+β2)nβ1+(n−1)β2 .
Notice that the class of linear interdependent utility functions spans the range of
implicit personal values of λV ∈ (V, nn−1V ). Moreover, for utility functions from this
class, the interpretation as implicit personal value λV does not only apply in equilibrium
(such as in the first order condition, (3.10)), but also describes optimal behavior out of
equilibrium (such as seen in the participation constraint, (3.6)). In contrast, the out-
of-equilibrium value of λ will depend on the absolute and relative payoff for alternative
specifications of interdependent utility.
In case both groups take active roles in equilibrium, i.e., a, b > 0, we can combine
the equations (FOCI) and (FOCJ) to obtain
ar−1[(m− 1)ar + (n−m)br] = λbr−1[mar + (n−m− 1)br] .
We divide both sides of this equation by b2r−1 and get
cr−1[(m− 1)cr + (n−m)] = λ[mcr + (n−m− 1)] (FOCIJ)
32
where we have substituted c := ab . Together with the participation constraints (PCI)
and (PCJ), this combined first order condition (FOCIJ) completely characterizes the
intragroup symmetric equilibrium. Having determined c, we can then use the first order
conditions (FOCI) and (FOCJ) to find the equilibrium solutions of a and b. These are
given by
a =
cr[(m− 1)cr + n−m]
[kcr + n−m]2
rV and (SOLA)
b =
[mcr + n−m− 1]
[mcr + n−m]2
rλV =
cr−1[(m− 1)cr + n−m]
[mcr + n−m]2
rV. (SOLB)
Unfortunately, the combined first order condition (FOCIJ) does not allow the calcula-
tion of a general solution (which, of course, would depend on m,n, λ and r). Therefore,
we have to fall back on numerical methods to find an equilibrium value of c. Fortunately,
however, we are able to establish the strategic advantage of interdependent preferences
without relying on numerical solutions.
3.3.3 Analyzing the strategic advantage
The following Lemma helps us to characterize the set of intragroup symmetric Nash
equilibria that display the strategic advantage of interdependent preferences.
Lemma 3 (i) For any m ∈ {1, . . . , n − 1}, any technology parameter r > 0, and any
intragroup symmetric Nash equilibrium x ∈ Nsym(ΓF (m)), we have
pij(x)
(>)
≥ pii(x) ⇒ xj
(>)
≥ xi
for all i = 1, . . . ,m and all j = m+ 1, . . . , n.
(ii) For m = n−1, for any technology parameter r > 0, and any intragroup symmetric
Nash equilibrium x ∈ Nsym(ΓF (n− 1)), we have
pin(x)
(>)
≥ pii(x) ⇔ xn
(>)
≥ xi.
for all i = 1, . . . , n− 1.
Proof. See appendix.
Part (i) provides a necessary condition for the presence of a strategic advantage
of interdependent preferences. Status-seekers (or interdependent preferences) can only
be better off than individualists (or independent preferences) if they choose a higher
effort level xj ≥ xi. Part (ii) shows that a higher effort level is also sufficient for the
strategic advantage of interdependent preferences in cases where only one player has
interdependent preferences. Notice that part (ii) also covers the ubiquitous case of two-
player contests.
33
The proof of Lemma 3 reveals that the individualists play the crucial role in deriving
the necessary condition for a strategic advantage of interdependent preferences. Only
when the individualists choose lower effort levels, x1 ≤ xn, their choice x1 can be optimal
against xn if there is a strategic advantage. Otherwise (i.e. if x1 > xn), the individualist
could do strictly better with choosing the lower effort level xn instead of x1, which
contradicts x representing an equilibrium.
In contrast, in part (ii), it is the single status-seeker who takes the key role in es-
tablishing the result. Even if there were no strategic advantage, the status-seeker nev-
ertheless would choose the higher effort level xn > x1. Without a strategic advantage,
the status-seeker has negative relative payoff. If she mimicked the individualists, she
would realize zero relative payoff. This means, in order to be optimal in terms of her
interdependent preferences, the effort level xn must yield higher material payoff than
effort level x1, in contradiction to the non-existence of the strategic advantage. Thus, a
higher effort level in equilibrium implies status-seekers have a strategic advantage.
3.4 Two-player contests
In this chapter, we focus on the ubiquitous special case of two-player contests. Typical
examples include legal conflicts such as lawsuits or most contests in political lobbying.
One general result for two-player contests is the following theorem, which identifies
two-player contests as the case where the status-seeker always experiences a strategic
advantage over the individualist.
Theorem 1 For any two-player contest, n = 2, any technology parameter r > 0, and
any Nash equilibrium x ∈ Nsym(ΓF (1)), we have pi2(x) ≥ pi1(x).
The proof builds on part (ii) of Lemma 3. We establish that, in any equilibrium x
where the status-seeker exerts lower effort than the individualist, she could increase both
her relative and her material payoff by slightly increasing her effort level. Consequently,
x cannot represent an equilibrium.
Proof. Let r > 0, fix x = (x1, x2) ∈ Nsym(ΓF (1)), and suppose to the con-
trary that pi2(x) < pi1(x). From Lemma 3, it follows that x2 < x1. Moreover, we have
x2 > 0, because otherwise x1 cannot be a best response for player 1 against x2. Since
pii(x1, x2), i = 1, 2, is differentiable for x1, x2 > 0, it follows that
∂pi1(x1,x2)
∂x1
= 0, which is
equivalent to
rxr−11 x
r
2V = (x
r
1 + x
r
2)
2. (3.11)
But then x1 > x2 implies that
∂pi2(x1, x2)
∂x2
=
rxr1x
r−1
2 V
(xr1 + x
r
2)2
− 1 =
x1
x2
− 1 > 0, (3.12)
where the last equality follows from (3.11). Because of ρ2(x1, x2) = 12 [pi2(x1, x2) −
pi1(x1, x2)] and
∂pi1(x1, x2)
∂x2
= −
rxr1x
r−1
2 V
(xr1 + x
r
2)2
< 0,
34
inequality (3.12) implies ∂ρ2(x1,x2)∂x2 > 0. Hence, there exist some xˆ2 ∈ (x2, x2 + ε) such
that
pi2(x1, xˆ2) > pi2(x1, x2) and ρ2(x1, xˆ2) > ρ2(x1, x2),
in contradiction to x2 being optimal for player 2 against x1 (in terms of player 2’s
interdependent preferences). Thus, pi2(x) ≥ pi1(x).
As a consequence of Theorem 1, notice that no overdissipation can ever occur in
two-player contests where an individualist and a status-seeker interact.
We conclude by investigating the two-player case for the class of linear interdependent
preferences. In this case, we can solve for the equilibrium explicitly.
Example 1 (continued) Combining the first order conditions of the two players, we
obtain b = aλ or, equivalently, c = ab =
1
λ . Replacing c in (SOLA) and (SOLB) by
1
λ ,
one can determine the equilibrium values of a and b, respectively. Building on these
values, it is possible to calculate material equilibrium payoffs and to derive participation
constraints that hinge upon the parameter values of λ and r only.
3.5 Non-increasing marginal efficiency – The case
r ≤ 1
After looking at the case of two-player contests, we now extend our analysis to general
n-player contests. Throughout the rest of this chapter, we assume n > 2. For these
general n-player contests, the case of increasing marginal efficiency (r > 1) can result
in non-participation by one of the two groups. Therefore, we split our analysis into two
parts: the case of non-increasing marginal efficiency, presented in this section, and the
case of increasing marginal efficiency, which we postpone to the next section.
The following theorem establishes the strategic advantage of status-seeking prefer-
ences for the case of non-increasing marginal efficiency. More precisely, we show (i)
that at most one intragroup symmetric equilibrium exists, (ii) that status-seekers exert
higher effort in equilibrium, and (iii) that this implies a strategic advantage of negatively
interdependent preferences.
Theorem 2 Fix r ≤ 1 and any m ∈ {1, . . . , n − 1} and let xˆ = ([a]m, [b]n−m) ∈
Nsym(ΓF (m)). Then, (a) xˆ is the only intragroup symmetric equilibrium; (b) interdepen-
dent preferences yield a strategic advantage over independent preferences, i.e. pin(x) ≥
pi1(x); (c) all players exert positive effort levels, status-seekers a higher one, i.e., b ≥
a > 0; and (d) if b > a then the strategic advantage is strict, pin(x) > pi1(x). Finally, (e)
if F is differentiable, then b > a, in fact, holds so that the strategic advantage is strict,
i.e., pin(x) > pi1(x).
Proof. See appendix.
Although, in general, it is not possible to determine the equilibrium strategies for
arbitrary values of r ≤ 1, one can solve the first order conditions for the special case of
linear contest technologies, i.e., where r = 1.
35
3.5.1 The special case of constant marginal efficiency (r = 1)
In order to illustrate Theorem 2, we determine the equilibrium for the class of preferences
introduced in Example 1:
Example 1 (continued) Consider again the example of linear interdependent prefer-
ences. The first order condition of the maximization problem max
xj
Fj(x) is:
(
n∑
h=1
xh
)2
=
V (β1 + β2)
β2 (n− 1) + β1n
∑
h6=n
xh, (3.13)
as derived in the appendix. Assuming intragroup symmetry, the first order condition of
status-seekers becomes
(ma+ (n−m) b)2 =
V (β1 + β2)
β2 (n− 1) + β1n
(ma+ (n−m− 1) b) , (3.14)
while the first-order condition for individualists reads
(ma+ (n−m) b)2 =
V
n
((m− 1) a+ (n−m) b) . (3.15)
In equilibrium, both (3.14) and (3.15) hold, such that the respective right hand sides have
to be equal to each other, which implies
c =
a
b
=
β1n+ β2m
β2 (n+m− 1) + β1n
(3.16)
as derived in the appendix. By Theorem 2, the strict strategic advantage is strict if a < b
or β1n+β2mβ1n+β2m(n+m−1)+β1n < 1, which is equivalent to
β1n+ β2m < β2 (n+m− 1) + β1n⇔ 0 < β2 (n− 1)⇔ n > 1.
Thus, in an intragroup symmetric linear rent-seeking contest with linear interdependent
preferences, a status-seeker always has a strict strategic advantage over an individualist,
since the last condition, n > 1, is trivially satisfied. The equilibrium values of a and b
can be explicitly calculated by inserting c into (SOLA) and (SOLB), respectively.
Finally, we illustrate that we cannot skip the restriction of differentiability of F in
Theorem 2. To this end, we present a discontinuous and hence non-differentiable but
increasing utility function F , where a status-seeker cannot gain by increasing her effort b
above a. Suppose the equilibrium is ([a]m, [a]n−m), where both groups of players choose
strategy a.
Proposition 2 ∃ξ = V
2
εn , with ε ∈]0; a[, such that for the utility function
Fj (pij(x); ρj(x)) =
{
ξ + ρj(x) if pij(x) ≥ Vn − a
ρj(x) if pij(x) < Vn − a
(3.17)
in an intragroup symmetric equilibrium, the status-seekers will choose b = a.
36
Proof. Notice that Vn − a is the maximum value that absolute payoff takes in Nash-
equilibrium. Hence, with higher effort xj = b, the absolute payoff shrinks so that the
lower line of equation (3.17) is the relevant one to any player j. Furthermore, for a = b,
we have ρj(x) = 0, since all players choose the same strategy. The value of (3.17) in this
situation is Fj = ξ.
The question, we address next, is how much is the maximum that player j can get by
increasing her effort. To this aim, we maximize ρ subject to b > a. The maximization
problem ∂ρj(x)∂b
!= 0 implies
b =
√
V an− a (n− 1) , (3.18)
which is larger than a in the case of underdissipation (see appendix). The second-order
condition is satisfied. Inserting (3.18) in (2.1) and (3.2) yields
ρj(x) = n−1n pij(x)−
1
n
∑
h6=j pih(x)
= n−1n
(
V√
V an
− 1
) (√
V an− an
)
(3.19)
<
(√
V an
an − 1
)
(V − an)
≤
(
V
εn − 1
)
V < V
2
εn = ξ,
where the second equality is established in the appendix. Thus, there exists a ξ, such that
no player j with utility function (3.17) can reach a higher utility level when increasing
her effort b above the effort level a of her opponents.
3.6 Increasing marginal efficiency - The case r > 1
We finally turn to investigating the case of increasing marginal efficiency, i.e. r > 1. In
this case, we make out four different types of equilibria depending on (a) which group
exerts higher effort and on (b) whether both groups or just one group take active roles
in equilibrium. Having characterized each equilibrium type separately, we conclude with
illustrating that different equilibrium types can, in fact, result from the same parameter
constellation.
More specifically, for r > 1 sufficiently small, it turns out that both groups always
participate actively in equilibrium. Depending on the interdependent utility function
F there might now exist an additional equilibrium, where individualists choose higher
expenditure levels than status-seekers. In this equilibrium, individualists will earn higher
payoff. Moreover, we illustrate that the existence of this equilibrium requires that status-
seekers put a higher weight on relative payoff when they opt out of the contest than when
they participate actively in equilibrium. Put intuitively, when status-seekers’ preferences
display a strong aversion to leaving a contest, this will be costly to them in equilibrium.
However, if status-seekers’ relative weight between absolute and relative payoff is
constant, then there is only one equilibrium where both groups take active roles. More-
over, for r > 1 sufficiently small, there is no other equilibrium type. In that case,
interdependent preferences unambiguously yield a strategic advantage over independent
preferences.
37
For larger values of r, we observe two additional types of equilibria, where only
one group of players – either status-seekers or individualists – takes an active role in
equilibrium. Here, the relative size of the two groups, mn−m , determines the size of the
parameter regions associated with these two additional types of equilibria.
In the equilibrium where status-seekers drive individualists out of the contest, the
strategic advantage prevails, provided that r > 1 is relatively small. However, for higher
values of r, the competition within the group of status-seekers intensifies, resulting in
a negative material payoff. Notice that opting out is not an option for these players
(provided r is not too large). Opting out by one status-seeker would raise the other
status-seekers’ absolute payoff above the zero payoff level, while it fixes her own payoff
and the payoff of individualists to zero. In this type of equilibrium, since opting out will
reduce a status-seeker’s own relative payoff, the status-seeker would reject this option.
Finally, if r is too large, opting out becomes profitable (and there is no longer any such
equilibrium in pure strategies).
In the equilibrium where only individualists choose positive effort levels and status-
seekers refrain from competition, individualists realize non-negative payoff, while status-
seekers end up with zero payoff. Basically, this parameter region is characterized by the
fact that all active players realize very small payoffs: Any status-seeker that entered with
some arbitrary positive effort level would earn negative material payoff.
Finally, if r > 1 is too large, then no intragroup symmetric equilibrium in pure
strategies exists and (presumably) we are “back to the bog” of mixed strategy equilibria.
Altogether, we have to deal with the four above-mentioned types of equilibria. A
fifth type of effort profile, where both groups remain inactive, does not represent an
equilibrium, since, in that case, any player has an incentive to deviate. By exerting
slightly positive effort she could gain the full prize.
We call the four possible equilibrium types
Type I: b ≥ a > 0 – both groups are active in equilibrium, status-seekers choose higher
effort levels;
Type II: a > b > 0 – both groups are active in equilibrium, individualists choose
higher effort levels;
Type III: b > a = 0 – only status-seekers engage in the contest;
Type IV: a > b = 0 – only individualists are active.
Below, we establish these results and address whether status-seekers experience a
strategic advantage over individualists. Subsequently, we relate the equilibrium types to
the contest parameter r > 1. We find the following four equilibrium ranges:
Type I: RI := (1, rI) rI ≤ nn−1 ≤
3
2 ,
Type II: RII := (1, rII) rII ≤ min{ mm−1 ,
1
λ
n2
(n−1)2} ≤
9
4 ,
Type III: RIII := (rIII , rIII) rIII > 1 and rIII ≤
1
λ
n
n−1
n−m
n−m−1 ≤ 3,
Type IV: RIV := (rIV , rIV ) rIV > 1 and rIV ≤
m
m−1 ≤ 2.
38
We will show that n − m > 1 is necessary for equilibria of type III and m > 1
for equilibria of type IV so that the corresponding expressions above are well-defined.
For equilibria of type II and in case of m = 1, one has to replace the inequality with
rII ≤ n
2
λ(n−1)2 .
Finally, for r > max{rI , rII , rIII , rIV }, there is no equilibrium in pure strategies. In
this work, we refrain from working out the equilibria in mixed strategies that presumably
occur in this parameter region.
3.6.1 Characterization of equilibrium types
We start with characterizing equilibrium types I and II, where both groups of players
choose positive expenditure levels. Moreover, as the subsequent characterization of the
other equilibrium types III and IV will show, equilibria of type I and II are the only
equilibrium candidates for r > 1 sufficiently small.
Equilibrium Type I
Let xˆ = ([a]m, [b]n−m) denote an intragroup symmetric equilibrium with b ≥ a > 0.
Equilibria of type I are present whenever r > 1 and r is sufficiently close to one.
Before we state the theorem, we rewrite the participation constraints as functions
of c = ab . By assumption, we have c ≤ 1. Individualists i ∈ I participate if and only if
pii(xˆ) ≥ 0, i.e., if
ar
mar + (n−m)br
V − a =
cr
mcr + n−m
V −
cr[(m− 1)cr + n−m]
[mcr + n−m]2
rV ≥ 0,
which is equivalent to
cr[m− r(m− 1)] ≥ (r − 1)(n−m), (3.20)
or (given that m > 1) to
r <
m
m− 1
and c ≥ c :=
[
(r − 1)(n−m)
m− r(m− 1)
] 1
r
. (3.21)
Notice that the restriction c ≤ 1 implies c ≤ 1 or (r− 1)(n−m) ≤ m− r(m− 1), which
is equivalent to r ≤ nn−1 . Since m < n implies
n
n−1 <
m
m−1 , it follows that individualists
participate if and only if r ≤ nn−1and c ≥ c. Moreover, we have that rI ≤
n
n−1 .
Similarly, status-seekers participate only if ρj(0, xˆ−j) ≤ ρj(xˆ), which is equivalent to
pij(xˆ) + bn =
V
mcr+n−m −
n−1
n b ≥ 0. Inserting (SOLB), we obtain
n [mcr + n−m] ≥ rλ(n− 1) [mcr + n−m− 1] , (3.22)
which provides another upper boundary on r (if m < n− 1), because c > 0 implies
r ≤
n [mcr + n−m]
λ(n− 1) [mcr + n−m− 1]
<
n(n−m)
λ(n− 1)(n−m− 1)
. (3.23)
39
Combining the two upper boundaries on r, we conclude that
rI ≤ min
{
n(n−m)
λ(n− 1)(n−m− 1)
,
n
n− 1
}
=
n
n− 1
if m < n− 1. The equality holds because λ < nn−1 implies
n(n−m)
λ(n−1)(n−m−1) >
n−m
n−m−1 >
n
n−1 .
Finally, since in any equilibrium of type I both groups participate, the first order
condition is given by (FOCIJ).
Theorem 3 Fix r > 1, any m ∈ {1, . . . , n−1} and let xˆ = ([a]m, [b]n−m) ∈ Nsym(ΓF (m))
such that b ≥ a > 0. Then, status-seekers have a strategic advantage over individualists.
Proof. See appendix.
While equilibria of type I represent the type of equilibrium that one would expect
after the analysis in the preceding sections, it turns out that other types of equilibria can
be present, depending on the contest parameter r and, as in the following subsection, on
the type of status-seeking preferences.
Equilibrium type II
Under equilibrium type II, both groups are active in equilibrium and individualists choose
higher expenditure levels than status-seekers. Accordingly, let xˆ = ([a]m, [b]n−m) denote
any intragroup symmetric equilibrium such that a > b > 0. From part (i) of Lemma
3, it directly follows that individualists will always earn higher payoff in equilibrium.
The question remains whether there exists such an equilibrium. Our first result in this
paragraph shows that equilibria of type II do not exist if the interdependent utility
function of status-seekers, F, is linear in both absolute and relative material payoff.
Theorem 4 Fix r > 1 and m ∈ {1, . . . , n − 1}. Suppose F (pi, ρ) = β1pi + β2ρ, where
α, β > 0. Then, no intragroup symmetric equilibrium of type II exists.
To establish the Theorem, the following Lemma is helpful. We replace the aggregate
behavior of other players in the first order condition of status-seekers (FOCJ), that is
mar + (n− k − 1)br, with Z and obtain
rbr−1ZλV = [Z + br]2 . (3.24)
Then, Lemma 4 derives properties of the function b(Z, λ), which is implicitly defined by
(3.24). Similarly, we define B(Z, λ) by setting B(Z, λ) := [b(Z, λ)]r = br(Z, λ).
Lemma 4 Fix r > 1 and any m ∈ {1, . . . , n−1} and let xˆ = ([a]m, [b]n−m) ∈ Nsym(ΓF (m))
with a, b > 0. Then, we have
(i) ∂B(Z,λ)∂Z ≥ −1 ⇔ Z ≤
r
r−1B,
(ii) ∂b(Z,λ)∂λ > 0 ⇔ Z <
r+1
r−1B,
(iii) Z ≤ rr−1B ⇒ a ≤ b.
40
Proof of the Lemma. See appendix.
Parts (i) and (ii) derive conditions that characterize partial derivatives of B(Z, λ)
and b(Z, λ), respectively, which then, in part (iii), are shown to be sufficient for the
non-existence of equilibrium type II. Therefore, to prove Theorem 4, it is sufficient to
establish that, for any intragroup symmetric equilibrium, the sufficient condition in part
(iii) is satisfied.
Proof of Theorem 4. Let xˆ = ([a]m, [b]n−m) ∈ Nsym(ΓF (m)), where F (pi, ρ) =
β1pi + β2pi and β1, β2 > 0. Without loss of generality, let player n be a representative
status-seeker and set xˇ = ([a]m, [b]n−m−1, 0). Set γ =
β2
β1+β2
. Then, player n will partici-
pate in equilibrium, i.e., b > 0, if and only if
Fn(xˆ) ≥ Fn(xˇ)
⇔ (1− γ)pin(xˆ) + γρn(xˆ) ≥ (1− γ)pin(xˇ) + γρn(xˇ)
⇔ pin(xˆ)−
γ
n
[
V −
n∑
h=1
xˆh
]
≥ 0−
γ
n
[
V −
n∑
h=1
xˇh
]
⇔
brV
mar + (n−m)br
−
n− γ
n
b ≥ 0,
where the last equivalence follows from xˇh = xˆh for all h 6= n. Substituting (SOLB),
c = ab , and λ =
n
n−γ , we can equivalently express this as
V
mcr + n−m
≥
1
λ
mcr + n−m− 1
(mcr + n−m)2
rλV
⇔ mcr + n−m ≥ r [mcr + n−m− 1] .
Finally, inserting c = ab gives us
mar + (n−m)br ≥ r [mar + (n−m− 1)br]
⇔ Z +B ≥ rZ,
which implies Z ≤ 1r−1B <
r
r−1B and, by Lemma 4 (iii), a ≤ b. Thus, no equilibrium of
type II exists.
Notice that the Proof of Theorem 4 does not fully exploit the power of Lemma 4,
since the lemma also implies non-existence of type II equilibria for larger values of Z
(namely for Z ∈ [ 1r−1B,
r
r−1B]).
Theorem 4 confirms that, for interdependent utility functions with constant relative
weight between absolute and relative material payoff, there is only one equilibrium type
with both groups taking active roles, namely, equilibrium type I. Thus, in that case,
status-seekers unambiguously experience a strategic advantage over individualists. How-
ever, the following two examples illustrate that an equilibrium of type II can, in fact,
result if we allow for utility functions that put higher relative weight on relative material
payoff under opting out than under participation in equilibrium.
Example 2 Consider (r,m, n, V ) = (1.41, 2, 4, 1) and solve (FOCIJ) for λ = 1.01 nu-
merically. This gives c = 2.9884 and the corresponding equilibrium values of a and b
41
as (a, b) = (0.3416, 0.1143). The absolute material payoffs of individualists and status-
seekers amount to (pii, pij) = (0.0704,−0.0263), respectively. The relative (material)
payoff in equilibrium xˆ is ρj(xˆ) = −0.0484 and under ”opting out”, xˇ, it amounts to
ρj(xˇ) = −0.0506. Consequently, the necessary condition for participation of status-
seekers is satisfied (i.e. ρj(xˆ) ≥ ρj(xˇ)). Then, it turns out that
F (pi(x), ρ(x)) = s1pi(x)2s3+1 + s2pi(x) + ρ(x) (3.25)
for s1 = 2.378 · 10441, s2 = 0.000001 and s3 = 140, represents an interdependent utility
function that
(a) gives rise to a first order condition (FOCJ) that, for λ = 1.01, has the same solution
(a, b) = (0.3416, 0.1143) as the first order condition (FOCIJ) above and that
(b) satisfies the necessary and sufficient participation constraint of status-seekers, (PCJ),
Fj(xˆ)− Fj(xˇ) = 0.00001240 ≥ 0.
Example 3 Set (r,m, n, V ) = (1.01, 2, 4, 1) and solve (FOCIJ) for λ = 1.000001. This
results in c = 1016.5738. The corresponding equilibrium values of a and b can be cal-
culated using (SOLA) and (SOLB). An interdependent utility function implying λ =
1.000001 and satisfying the participation constraint of status-seekers is then given by the
above example, (3.25), for s1 = 9.576 · 1067244, s2 = 0.000001 and s3 = 7293.
Example 2 shows that equilibria of type II might exist even when equilibria of type
I do not (notice that r = 1.41 > nn−1 =
4
3). In contrast, Example 3 illustrates that type
II equilibria also exist if r > 1 is close to one (possibly for some different utility function
of status-seekers). From each example separately, it follows that Theorem 4 cannot be
generalized to interdependent utility functions that are additively separable in absolute
and relative payoff.
Let us finally comment on the conditions that characterize equilibria of type II. First,
participation by individualists requires that inequality (3.20) is satisfied. Since c now
exceeds one, we can no longer conclude that c ≤ 1. This is actually the case in Example
2, where c ' (1.3898)
1
1.41 ' 1.2630 and c ' 2.9884. It is hence compatible with equilibria
of type II that r > nn−1 (like in Example 2). Accordingly, it is now the conditions r <
m
m−1
(for m > 1) and c ≥ c that characterize participation by individualists (for m = 1, this
participation constraint reduces to c ≥ c).
While the participation constraint of individualists has changed, the necessary condi-
tion for participation of status-seekers (3.22) and the first order condition characterizing
equilibrium behavior (FOCIJ) remain the same as under equilibrium type I. Similarly,
the necessary participation constraint of status-seekers provides an upper boundary on
r:
r ≤
n [mcr + n−m]
λ(n− 1) [mcr + n−m− 1]
≤
n2
λ(n− 1)2
,
where the second inequality follows now from c ≥ 1. We can thus conclude that rII ≤
min{ mm−1 ,
1
λ
n2
(n−1)2}.
42
Equilibrium Type III
We continue with equilibrium type III, where status-seekers choose such high effort
levels that they drive individualists out of the contest. Let xˆ = ([a]m, [b]n−m) denote a
corresponding intragroup symmetric equilibrium with b > a = 0.
On the one hand, if there is more than one status-seeker, i.e., if n−m > 1, then the
first order condition of status-seekers FOCJ gives us the equilibrium value of b,
b =
(n−m− 1)
(n−m)2
rλV. (3.26)
Non-participation is optimal to individualists if
(r − 1)r−1
r2r
≤ λr
(n−m− 1)r
(n−m)2r−1
. (3.27)
Participation of status-seekers can only be optimal if ρj(xˆ) ≥ ρj(0, xˆ−j), which is equiv-
alent to pij(xˆ) + bn ≥ 0. Inserting (3.26) we obtain
r ≤
1
λ
n
n− 1
n−m
n−m− 1
. (3.28)
It follows that rIII ≤ 1λ
n
n−1
n−m
n−m−1 . Interdependent preferences yield a strategic advantage
if and only if pij(xˆ) ≥ 0 = pii(xˆ) or, equivalently, if
r ≤
1
λ
n−m
n−m− 1
. (3.29)
Notice that
1
λ
n−m
n−m− 1
<
1
λ
n
n− 1
n−m
n−m− 1
,
so that a negative payoff to status-seekers is compatible with their participation in equi-
librium.
On the other hand, for n −m = 1, there is no equilibrium of type III. In this case,
for instance, xˇn = b2 > 0 would imply both higher absolute payoff, pin(xˆ) < pin([0]n−1,
b
2),
and, by Lemma 2, higher relative payoff. Consequently, xˆ = ([0]n−1, b) cannot represent
an intra-group symmetric Nash equilibrium.
Theorem 5 summarizes our findings:
Theorem 5 Fix r > 1 and any k ∈ {1, . . . , n − 2} and let xˆ = ([0]m, [b]n−m) ∈
Nsym(ΓF (m)) with b > 0. Then, conditions (3.26), (3.27) and (3.28) hold. Interde-
pendent preferences yield a strategic advantage if and only if condition (3.29) applies.
For m = n− 1 no equilibrium of type III exists.
Observe that constraint (3.27) provides a lower bound on r, denoted by rIII , that
strictly exceeds one. To see rIII > 1, take the limit r → 1 on both sides of (3.27). While
43
the left hand side goes to one, the right hand side converges to λn−m−1n−m , which is strictly
lower than one because of
λ
n−m− 1
n−m
≤
n
n− 1
n− 2
n− 1
< 1.
Hence, in the limit, constraint (3.27) is not satisfied. On the other hand, for r = n−mn−m−1 ≤
rIII this constraint holds. Therefore, it follows from continuity of both sides in (3.27)
that there exists an rIII ∈ (1,
n−m
n−m−1) such that there is no equilibrium of type III in the
interval (1, rIII).
Equilibrium type IV
Finally, consider the remaining type of intragroup symmetric equilibria xˆ = ([a]m, [b]n−m),
where a > b = 0. Similar to the above, there is no equilibrium of type IV if m = 1.
However, for m > 1, it follows from the first order condition of individualists, (FOCI),
that
a =
m− 1
m2
rV, (3.30)
where individualists participate if and only if pii(x) ≥ 0. Participation is equivalent to
r ≤
m
m− 1
, (3.31)
which implies rIV ≤ mm−1 .
Obviously, given this type of equilibrium, status-seekers never experience a strict
strategic advantage. They optimally remain out of the contest if pij(xj, xˆ−j) < pij(xˆ) = 0,
for all positive effort levels where xj > 0. Inserting the equilibrium value of a, we obtain
that non-participation of status-seekers can only be optimal if
(r − 1)r−1
r2r
≤
(m− 1)r
m2r−1
. (3.32)
Theorem 6 Let r > 1, m ∈ {2, . . . , n− 1} and xˆ = ([a]m, [0]n−m) ∈ Nsym(ΓF (m)) with
a > 0. Then, we have r ≤ mm−1 and the conditions (3.30) and (3.32) apply. Individualists
realize higher payoff. For m = 1, no equilibrium of type IV exists.
Observe that each type IV equilibrium, such as characterized in the above theorem,
exactly corresponds to a Nash equilibrium of the standard Tullock contest between m
players, where all of them have standard (i.e. individualistic) preferences. Since these
contests have a unique equilibrium for any r ≤ mm−1 , the same holds for equilibria of type
IV.
Corollary 4 For any r > 1, there exists at most one equilibrium of type IV.
Similar to the remark in the end of the discussion of Equilibrirm Type III, it can be
shown that constraint (3.32) provides a lower bound rIV < rIV such that no equilibrium
of type IV exists in the interval (1, rIV ). Thus, for r > 1 sufficiently small, namely for
r ∈ (1,min{rIII , rIV }), we can only have equilibria of types I and II.
44
3.6.2 Equilibrium multiplicity
In order to illustrate that the equilibrium types I, III and IV can occur for the same
constellation of parameter values, we refer to the example characterized by the parameter
values λ = 1.001, n = 10; m = 5 and r = 1.1. Without loss of generalit, we set V = 1.
First let us investigate equilibrium type I, where members of both groups are active
and b > a. Solving (FOCIJ) numerically for c yields a ratio of expenditure levels of
c ≈ 0.9560. Inserting c in (SOLA) and (SOLB) gives us the effort level of each group,
a = 0.0968 and b = 0.1013. Then, we have to ensure participation of each group.
Individualists optimally participate because they receive positive absolute payoff under
participation, pii = 0.0007 > 0. Status-seekers’ absolute payoff is pij = 0.0012 > pii.
Since opting out would lead to zero absolute payoff, it follows from Corollary 2 that
status-seekers also optimally participate in this contest.
Equilibrium type III can occur for this parameter constellation, provided inequalities
(3.26) to (3.28) hold. Inequality (3.26) applies for an effort of b = 0.176. Inserting the
above parameter values in (3.27) yields 0.644 ≤ 0.667, which tells us that there is no
incentive for the individualists to join the game. Participation of the status-seekers is
guaranteed by (3.28) because this inequality is satisfied as well: 1.1 ≤ 1.388. Finally,
inequality (3.29) holds with 1.1 ≤ 1.249, which implies the strategic advantage of the
status-seekers. Each status-seeker experiences an absolute material payoff of pij = 0.024.
For an equilibrium of type IV, three similar conditions have to apply. First, inequality
r ≤ mm−1 becomes 1.1 ≤ 1.25 for our set of parameter values and hence is satisfied.
Second, the optimality condition of individualists (3.30) is fulfilled if they choose a =
0.176. Third, inequality (3.32) holds because of 0.644 ≤ 0.666, ensuring that status-
seekers do not participate. As mentioned earlier, individualists earn higher payoff, since
status-seekers opt out of the contest, while individualists experience non-negative payoff
of pii = 0.024.
Having illustrated that different equilibria can occur for the same parameter con-
stellation, we content ourselves with noting that the instance of equilibrium multiplicity
creates an equilibrium selection problem. This has to be taken into account when ap-
plying our results.
3.7 Concluding remarks
In this chapter, we have examined symmetric rent-seeking contests where two groups
of players are heterogenous with respect to preferences. One group has independent
preferences, while the other group acts according to interdependent preferences. The
focus of our analysis was whether status-seekers experience a strategic advantage over
individualists. We found that such a strategic advantage has two major implications in
rent-seeking contests. First, whenever status-seekers experience a strategic advantage,
aggregate effort never exceeds the value of the rent at stake, i.e., there is no overdis-
sipation. Second, a strategic advantage of status-seekers is not compatible with them
exerting lower effort than individualists. In cases where one status-seeker only interacts
with individualists, the converse holds.
45
Subsequently, we investigated for which classes of rent-seeking contests the strategic
advantage is present in equilibrium. It turned out that the ubiquitous case of two-
player contests and the class of n-player contests with non-increasing marginal efficiency
(r ≤ 1) display this property. This result extends to n-player contests with increasing
marginal efficiency (r > 1), if status-seekers maximize some convex combination of
absolute and relative material payoff and if r > 1 and r is sufficiently small. However,
for general interdependent utility functions, there can exist another type of equilibrium,
where individualists exert higher effort and earn higher material payoff. (This type can
exist for r > 1 sufficiently small as well as for larger values of r). For larger values of
r, we also observe two additional types of equilibria, where only one group of players
are active in equilibrium and the other group of players exert no effort at all. Thus,
for general contests with increasing marginal efficiency, we can no longer conclude that
status-seekers (or negatively interdependent preferences) have a strategic advantage over
individualists (or independent preferences).
Next, we comment on the two suggested applications to the theory of strategic delega-
tion. Without further investigations, we can only address delegation problems, where the
delegating decision-maker is mainly concerned with relative performance (e.g. market
share). Due to the payoff structure of rent-seeking contests, it seems obvious, however,
that our conclusions (at least locally) extend to delegation problems where absolute
payoff objectives are in the foreground.
Let us first consider the problem whether to delegate or not (and, in case of delegation,
which incentives to provide the delegate with). From our results, it follows that, for
rent-seeking contests with two players (n = 2) or with non-increasing marginal efficiency
(r ≤ 1), any subgame-perfect equilibrium will involve a positive delegation decision. A
delegating firm can perform better than its opponents if she compensates her delegate
according to relative rather than to absolute performance. For contests with increasing
marginal efficiency (r > 1), there is no definitive answer. One first has to solve the
equilibrium selection problem.
Given a positive delegation decision in the above delegation problem, the second
delegation problem arises. Which individual should one select from a pool of delegates
that are heterogeneous with respect to the degree of interdependency in utility. In this
second application, we refer to intrinsic rather than to extrinsic incentives in delegation.
Again, we find a positive answer for the case of two players or for contests with non-
increasing marginal efficiency. In these two cases, the decision-maker should always pick
the status-seeker, when she can choose between a status-seeker on one hand and an
individualist on the other hand.
However, in the end, we find a type of prisoner’s dilemma for both rent-seeking del-
egation games. Comparing the equilibrium, where all other firms choose to compensate
according to relative performance rather than absolute performance (or to pick status-
seekers rather than individualists), material payoff will be lower than in the situation
where all firms refrain from doing so.
The theory shows us that, in a two-player-contest, the player that maximizes relative
payoff and, therefore, shows spiteful behavior will be more successful. The presence of
the strategic advantage we pointed out can, therefore, explain why two candidates for
46
the same position have a strategic incentive to behave spitefully against each other in
an election campaign. If one candidate acts spitefully while the other one does not, the
candidate with the spiteful behavior is more likely to win the race. This also held for the
race for between Barack Obama and Hillary Clinton. Although both these candidates
show respect for each other, as they say after the race, and even consider a cooperation
(Su¨ddeutsche 2008), they did not avoid an aggressive battle against each other. Since
the only aim of the campaign was to prove to be more capable than the opponent, the
candidates did not only gain from presenting themselves as especially qualified, but also
from discrediting the opponent and worsening his or her position. In the next chapters
we will allude to the limits of this result and show that overly spiteful behavior will not
benefit a candidate.
47
Chapter 4
On Heterogeneity in Interdependent
Preferences
4.1 Introduction
In Chapter 3, we established that, in a broad range of situations, players with spiteful
preferences achieve higher material payoff in Tullock-contests (Tullock, 1980) than play-
ers with standard independent preferences, especially for non-increasing returns to scale.
However, we focused above on a model with only two different types of players. In this
chapter, we restrict our analysis to additively separable preferences and non-increasing
returns to scale. For this case, we generalize the model in so far as allowing for complete
heterogeneity among the players rather than confining our analyses to two types of char-
acters. Furthermore, we allow for all levels of altruism and non self-destructive spite.
However, this strategic advantage no longer prevails when too strong spiteful preferences
induce overdissipation and thus negative material payoff for all active players. We find
that, for constant returns to scale, overdissipation can occur and induce a higher material
payoff for less spiteful players. Furthermore, we look at the question which incentives
players have to join the contest at all. In the presence of overdissipation, only spiteful
players will choose a strictly positive effort.
Another important result of this chapter is that interdependent preferences with
different levels are equivalent to different valuations of the players. Therefore, we will
also refer to the literature that deals with asymmetric valuations. Asymmetric valuations
have been looked upon in rent-seeking literature in different contexts (e.g. Hillman
and Riley 1989, Leininger, 1993, Baik, 1994). Nti (1999) was the first to analyze the
game with asymmetric valuations (and independent preferences) in detail. However, his
results are still restricted to the two-player-case. They show that the player with higher
valuation will spend more effort and the ratio of their expenditure to their valuation
is the same for both players. He also finds that the effort of the player with higher
valuation increases in both valuations, while the effort of the player with lower valuation
increases only in her own valuation but decreases in her opponents’ valuation. Note
that, in contrast to our analysis, Nti’s results do not only hold for non-increasing but
also for increasing returns to scale. Stein (2002) analyzed asymmetric valuations in an
48
n-player context with constant returns to scale. His result that the own effort is higher
if the own valuation is higher is equivalent to our results that more spiteful players exert
higher efforts, and thus our results generalize his result to the case of decreasing returns
to scale. However, it holds for both analyses that the equivalence to our work ends when
looking at the material payoff, since the prize players compete for in our setting is not
the prize they can receive in the end. This leads us to the result that, in contrast to
the models of Nti (1999) and Stein (2002), more spiteful players do not necessarily gain
the higher expected payoff. Indeed, if the players in the game are sufficiently spiteful,
we are able to find that the less spiteful players receive the higher material payoff. For
the specific case of constant returns to scale, this condition reduces to overdissipation.
We show that overdissipation can occur in the presence of spiteful preferences. From
a technical point of view, we exploit the fact that rent-seeking contests are aggregative
games. This is in line with the findings of Possajennikov (2003), Alos-Ferrer and Ania
(2005) and Leininger (2006).
4.2 Effort Choice
The model we present here builds on the analysis in Chapter 3. The material payoff
function considered here is also given by (2.1). Let us now point out the differences in
the assumptions between the models in this chapter and the previous one. At first, we
restrict the values for r considered here to r ∈]0, 1]. Next, we concentrate on a linear
objective function of the form
Fi (x) = pii (x) + αi
n∑
h=1
1
n
pih(x), (4.1)
which can be seen as a specification of (3.1).1 Both these restrictions are made to avoid
technical inconvenience. Chapter 3 captures an analysis with objective functions of the
form (4.1) in so far as αi ∈ (−1, 0) for the status-seekers and αi = 0 for the individualists.
We take this as a starting point to show how an extension of the set of players to an
arbitrary number of different players can generalize or respectively change the results of
Chapter 3, when allowing for any values of αi > −n. The latter restriction ensures that
the utility function of each player positively depends on her own payoff. If player i is
characterized by αi > 0, we call her an altruist. In contrast, if player i is characterized
by αi < 0, we stick to the notation status-seeker. For the case where αi = 0, equation
(4.1) reduces to equation (2.1).
To simplify the analysis we introduce the following parameter:
λi :=
n
n+ αi
.
From αi ∈] − n,∞[, it follows that λi ∈ IR
+. In the next sections, we will distinguish
the players according to the preference parameter λi. Note that αi < 0 implies λi > 1
1The proposed function is equivalent to (α+ 1)pii − αρi.
49
and αi > 0 leads to λi < 1. So λi > (<)1 represents the preference of a status-seeker
(altruist).
From Fang (2002), we know that a rent-seeking game with non-increasing returns
to scale has a unique Nash-equilibrium for any positive prize V . This result also holds
true for asymmetric valuations. Therefore, another problem with the same optimality
condition will have the same solution. We exploit this to state
Lemma 5 In an n-player rent-seeking contest with material payoff function (2.1) and
r ≤ 1, a player maximizing an interdependent utility function in the form of (4.1) with
αi > −n acts according to the same reaction function as if she would maximize material
payoff competing for a prize of λiV . Uniqueness of the Nash-equilibrium implies that in
equilibrium the same effort is chosen as if competing for λiV .
Proof. See appendix.
As already adressed, the game where two players compete for prizes of different
size was already analyzed by Nti (1999). Instead of deriving Nti’s results again in our
framework, we will renounce a specific analysis of two-player-contests, and remark that,
except for the final proposition concerning the expected payoff, Nti’s results can be
transferred to our analysis as the special case of two-player-contest, even for increasing
returns to scale.
Next, we will compare the properties of the behavior of active players with different
levels of preference parameters. We define:
Definition 3 A player i is called active in equilibrium xˆ, if and only if the ith component
of xˆ is given by xi > 0.
Before analyzing active players in detail, we shall discuss up front the incentives
for a player to be active in a rent-seeking contest with non-increasing returns to scale.
For rent-seeking contests with decreasing returns of scale (r < 1), Cornes und Hartley
(2005) already established that every player in the game has an incentive to exert strictly
positive effort. On the contrary, for constant returns to scale, relatively altruistic players
may have an incentive not to invest in the contest. We find:
Lemma 6 Let n players decide whether to join a rent-seeking contest with material
payoff function (2.1) and r = 1. Let each player i be characterized by αi > −n such that
she acts like competing for λiV . In equilibrium, player i will only choose a positive effort
xi > 0, if in equilibrium it holds that
λiV >
n∑
j=1
xj.
Proof. See appendix.
Now consider a game where λiV can be interpreted as the personal valuation of player
i in a game with heterogenous valuations and independent preferences. In this case one
can interpret the result of Lemma 6 in the following way: a player only chooses positive
50
effort in the contest if she considers the aggregate effort spent in the game to represent
underdissipation in the game. Put differently, a player in a contest with heterogenous
valuations will only exert positive effort in equilibrium if the aggregate effort does not
exceed her own valuation.
Let us now assume for the rest of this chapter that all n players are spiteful enough
to participate in equilibrium. Now, we analyze which effect the preference parameter λi
has on the level of effort a player exerts. Every player i maximizes a utility function
that can be written as a function of λi ∈ IR
+, ∀i = 1, ..., n, since equation (4.1) can be
transformed to
ui
(
(pij(x))j=1,...,n |λi
)
= pii(x) +
( 1
λi
− 1
) n∑
j=1
pij(x).
Given this individual preference parameter λi, we can rewrite the first-order-condition
of player i as
rxr−1i
∑
h6=i
xrhλiV =
(
n∑
h=1
xrh
)2
=: A2, (4.2)
where A represents the aggregate effort of all n players. Note that in every situation all
players face the same aggregate effort. So for any pair of players i and j, i, j = 1, ..., n,
we can conclude from (4.2) that
rxr−1i
∑
h6=i
xrhλiV = rx
r−1
j
∑
h6=j
xrhλjV. (4.3)
From (4.3), we deduce the following proposition, which generalizes the result of The-
orem 2 (c) in Chapter 3.
Proposition 3 Consider a rent-seeking contest with material payoff function (2.1) and
r ≤ 1. Then for efforts xi and xj of two players i and j, characterized by λi and λj
respectively, we find that in equilibrium λi > λj ⇔ xi > xj.
Proof. See appendix.
Obviously, these results are in line with Stein (2002) and those of Chapter 3. However,
they do not imply that a higher value of λi also leads to higher material payoff. We will
show in the next section that this need not necessarily hold in our general setting.
4.3 Effects on the Level of Material Payoff
In order to more clearly distinguish the cases where spiteful preferences induce a higher
payoff, let us introduce the following notions:
Definition 4 Define a preference function of form (4.1) as ”weakly spiteful” if 0 > αi ≥
−1, or equivalently if 1 < λi ≤ nn−1 . In contrast, define a preference function of form
(4.1) as ”strongly spiteful” if αi < −1, i.e. if λi > nn−1 .
51
Chapter 3 showed that a preference function is weakly spiteful if it increases in both
absolute and relative payoff ρi(x) = pii(x)−pi(x). We found for certain classes of contests
with weakly spiteful preferences that more spiteful players experience the higher absolute
payoff in equilibrium. Like in Chapter 3, the aim of this analysis is to show under which
conditions more spiteful preferences can receive a higher material payoff. We will see that
the above result cannot be generalized to the entire set of interdependent preferences,
since the absence of strongly spiteful players played an important role in Chapter 3. For
games with a player set that either contain no or only strongly spiteful players, we are
able to find:
Theorem 7 Assume that n players play a rent-seeking contest with non-increasing re-
turns to scale, where it holds that λi ≤ (≥) nn−1 , ∀i = 1, ..., n. Then players with a more
spiteful preference realize a higher (lower) material payoff in equilibrium, i.e.
λi > λj ⇒ pii > (<)pij, ∀i, j = 1, ..., n, i 6= j.
Proof. The proof follows the line of Theorem 2. It is given in the appendix.
Without loss of generality, say that λ1 ≤ ... ≤ λn. Then, we can conclude from
Theorem 7:
Corollary 5 Assume that n players play a rent-seeking contest with non-increasing re-
turns to scale. Then, it holds for the material payoff in equilibrium that
pi1(x) ≤ (≥)pi2(x)... ≤ (≥)pin(x), if λn ≤
n
n− 1
(
λ1 ≥
n
n− 1
)
with pii(x) < (>)pii+1(x), whenever λi < λi+1, ∀i ∈ 1, ..., n− 1.
Corollary 6 If players with the same preference parameter participate in a rent-seeking
contest, they will choose the same effort level and thus receive the same material payoff.
Corollary 6 supports the assumption of Chapter 3 where we restrict the analysis to
intragroup symmetric equilibria. The corollary now tells us that these equilibria are the
only one to arise in a rent-seeking game with utility functions of the type (4.1). Let us
next state a lemma that gives a more extensive characterization of a strategic advantage.
Lemma 7 Assume a rent-seeking game with payoff function (2.1) and utility function
(4.1). For two active players i and j, we find that, in equilibrium, λi > λj results in
pii(x)
(>)
≥ pij(x), if and only if
xri − x
r
j
xi − xj
(>)
≥
∑n
h=1 x
r
h
V
.
Proof. From Proposition 3, we know that xi > xj has to hold. Then, the condition
follows from equivalence to pii(x) ≥ (>)pij(x).
Let us now focus on the case of constant returns to scale. By inserting r = 1 in the
analysis of Lemma 7, we find
52
Proposition 4 Consider a rent-seeking game with r = 1. Then in equilibrium it holds:
{λi > λj ⇔ pii(x) > (<) pij(x)} ⇔
n∑
h=1
xh < (>)V ⇔ pii(x) > (<) 0 ∀i = 1, ..., n
Proof. The first part follows directly from Lemma 7. The latter equivalence sign
follows directly from rearranging pii > (<) 0 for r = 1.
Proposition 4 says that the less spiteful players can indeed earn a higher material
payoff in certain settings. Note that Stein (2002) finds that the player with higher
valuation will always perform better in terms of material payoff in a simultaneous rent-
seeking game with constant returns to scale. The difference from our analysis is the
following: while Stein assumes the higher value of the prize to be actually based on a
difference in the material payoff function, in our setting, the higher valuation is only felt
by the player and is not reflected in the actual material payoff.
Proposition 4 shows that there are limits to the strategic advantage of negatively
interdependent preferences even in contests with non-increasing returns to scale. The
reason for this is that players with strongly spiteful preferences concentrate more on
reducing their opponents’ payoff than on increasing their own. We will use the following
example to illustrate that the strategic advantage can indeed be on the side of the less
spiteful player.
Example 4 Assume a contest with n = 2 and r = V = 1, where the players are
characterized by λ1 = 2 and λ2 = 3. In equilibrium, they will choose x1 = 1225 and
x2 = 1825 , resulting in aggregated effort of 1.2V , which indicates overdissipation. The
material payoffs are given by pi1 = − 225 and pi2 = −
3
25 , which means pi1 > pi2.
Of course, the result of Example 4 depends on player 2 being a status-seeker. At first
glance, it is not obvious that player 1 also has to be a status-seeker. However, we find:
Corollary 7 Consider an equilibrium of an n-player rent-seeking contest with constant
returns to scale, i.e. r = 1, where overdissipation - and thus a strategic advantage for
the less spiteful player - occurs. Then all active players will be status-seekers.
Proof. From Proposition 4, we know that every active player will receive a negative
material payoff if overdissipation occurs in equilibrium. To show that in this case a
positive effort cannot be the best response of a player i characterized by λi ≤ 1, we refer
to the appendix.
Note that this does not necessarily imply full or overdissipation for cases where two
status-seekers meet in a rent-seeking contest with constant returns to scale. A coun-
terexample can be directly derived from Theorem 7 and Proposition 4.
4.4 Conclusion
We present an analysis that captures the range of all levels of interdependent preferences
that attach an overall positive utility to own material payoff. Self-destructive preferences
53
are excluded. We find that, for Tullock-contests with non-increasing returns to scale for
each pair of players, the player with the more spiteful, preference exerts higher effort in
equilibrium. As an implication, the results that Chapter 3 points out for contests with
non-increasing returns to scale do not depend on the assumption of intragroup symmetric
equilibrium. Furthermore, we find that, in an equilibrium with underdissipation, the
player with the most spiteful preference will earn the highest material payoff. For the
case of two-player-contests, we can refer to Nti (1999) to show that this result is not
restricted to non-increasing returns to scale.
As another important insight, we saw that results change with a too high level of
spite. The results that spiteful preferences yield a strategic advantage in a broad range of
games have to be qualified with respect to the assumptions behind analyses of Kockesen
et al. (2000) and Chapter 3. As we saw, relaxing these assumptions can reverse the
result.
The results of this chapter also have implications for contests with asymmetric val-
uations. Mathematically, this problem is equivalent to ours for the most part. Results
concerning effort choice or winning probability in equilibrium can be smoothly trans-
ferred from the one framework to the other. Of course, this means that the additional
findings we obtain for the effort choice in multi-player games with decreasing returns to
scale can also be applied to games with asymmetric valuations. On the other hand, this
analogy is restricted only to some of our results. Since players in games with interdepen-
dent preferences and with asymmetric valuations interpret the role of the heterogeneous
prize competed for in a different way, results concerning the material payoff cannot be
transferred. Hence, we can find that spiteful preferences can induce overdissipation while
we know that asymmetric valuations cannot.
54
Chapter 5
Evolutionarily Stable Preferences
The analysis of heterogenous preferences in rent-seeking contests put forward in the
previous chapter brings about the question which preference parameter is evolutionarily
stable. A preference is evolutionarily stable if a population of players maximizing this
preference cannot be successfully invaded by a mutant maximizing a different preference
function. This means that a mutant will never receive a strictly higher payoff than the
players with evolutionarily stable preferences. This chapter shows that, for a game where
players receive a material payoff determined by (2.1), players with a preference parameter
of λi = nn−1 , or respectively αi = −1, will receive an at least weakly higher payoff against
any mutant in a playing-the-field contest with complete information, as long as r is
not too large. As we know from Chapter 4, this preference parameter constitutes the
border between weakly and strongly spiteful preferences. For this case, (4.1) reduces to
ρi (x) := pii(x) − 1n
∑n
h=1 pih(x). The maximization of this equation is equivalent to the
maximization of the relative payoff function
ρi (x) := pii(x)−
1
n− 1
∑
h6=i
pih(x). (5.1)
With respect to the first-order-condition, it has already been shown by Schaffer (1989)
that relative payoff maximizers, i.e. players maximizing (5.1), have evolutionarily stable
preferences. Eaton and Eswaran (2003) showed that, for constant returns to scale,
relative payoff is indeed an evolutionarily stable preference function. Furthermore, they
show that, under the playing-the-field assumption, the evolutionarily stable preferences
leads to playing the evolutionarily stable strategy. Hehenkamp et al. (2004) showed that
the strategies played by relative payoff maximizers are evolutionarily stable, if evolution
works on the level of strategies. Let us now show that relative payoff maximization
is evolutionarily stable in rent-seeking contests for a much broader set of parameters
than only for constant returns to scale, including every level of decreasing returns to
scale. This means we want to focus on the question under which circumstances such a
population can be successfully invaded by a mutant j with the preference function
Fj (x) 6= ρj (x) , (5.2)
in the sense that the mutant’s preference function brings about a higher material payoff.
Here, we stick to the assumption of playing-the-field. An analysis of evolutionary stability
55
in Tullock-contests that relaxes the playing-the-field assumption is given by Leininger
(2008).
To establish evolutionary stability in our setup, we calculate the reaction function
of a player with objection function (5.1) if she reacts rationally and optimally, to the
behavior of her opponents, and we show that the mutant cannot receive a higher material
payoff.
Without loss of generality, we denote the mutant by n, and her equilibrium effort by
xn. The first-order-condition of each incumbent player i is given by
∂ρi(x)
∂xi
!= 0
⇔


∑
j 6=i
xrj

xr−1i rV
n
n− 1
=
(
n∑
h=1
xrh
)2
, (5.3)
as we can see from λi = nn−1 and equation (4.2). Since we know that (5.3) displays the
equilibrium behavior of all players i = 1, ..., n− 1, we can conclude that, for each pair of
players i and j with i, j = 1, ..., n− 1, it has to hold that


∑
h6=i
xrh

xr−1i rV
n
n− 1
=


∑
h6=j
xrh

xr−1j rV
n
n− 1
or equivalently
∑n
h=1 x
r
h − x
r
i
∑n
h=1 x
r
h − x
r
j
=
xr−1j
xr−1i
. (5.4)
As we can see from the proof of Proposition 3, which can be generalized to r > 1, we
conclude from (5.4) that
xi = xj = x1, ∀i, j = 1, ..., n− 1. (5.5)
Now inserting (5.5) in (5.3) and denoting x1 =: x and xn =: y yields
((n− 2)xr + yr)xr−1rV
n
n− 1
= ((n− 1)xr + yr)2
⇔
V
(n− 1)xr + yr
=
(n− 1)xr + yr
(n− 2)xr + yr
n− 1
xr−1rn
. (5.6)
The population is stable against invading mutants if, for every mutant strategy y gener-
ated by preference of type (5.2), it holds that
pi1(x) ≥ pin(x)
⇔
xr
(n− 1)xr + yr
V − x ≥
yr
(n− 1)xr + yr
V − y
⇔
xr − yr
(n− 1)xr + yr
V ≥ x− y. (5.7)
56
This condition ensures that the incumbents receive the (at least weakly) higher material
payoff. Inserting (5.6) we can transform (5.7) to
ω(y) := (n− 1)
xr − yr
xr−1rn
·
(n− 1)xr + yr
(n− 2)xr + yr
− (x− y) ≥ 0 (5.8)
Whenever (5.8) holds, the mutant does not obtain a strictly higher payoff than the
incumbents. We can now show for a certain range of r that (5.8) always holds, meaning
that incumbents will receive the higher material payoff regardless of which particular
strategy y maximizes (5.2). Note that from a mutant’s point of view ω displays the
relative loss she has to bear in contrast to the incumbents. We will now show that
even when minimizing this difference, or equivalently maximizing her advantage against
the incumbents, she will not end up with a strictly higher payoff in equilibrium. This
becomes intuitively clear, since the incumbents also maximize the difference of their
material payoff to their opponents. However, formally we still have to show that ω ≥ 0
holds. For this, we derive the first and the second derivative of ω, given by
ω′(y) = 1−
(y
x
)r−1 n− 1
n ((n− 2)xr + yr)
[
(n− 1)xr + yr +
x2r − xryr
(n− 2)xr + yr
]
and
ω′′(y) = −
n− 1
xr−1n
·
yr−2
((n− 2)xr + yr)3
{
(r − 1)
[(
n3 − 5n2 + 9n− 6
)
x3r (5.9)
+
(
3n2 − 11n+ 11
)
x2ryr + (3n− 6)xry2r + y3r
]
− 2r(n− 1)x2ryr
}
.
Next we continue by showing that
1. ω(x) = 0,
2. ω′(x) = 0 to show us that y = x is a candidate for a local minimum of ω(y),
3. ω′′(y) > 0,∀y > 0 to ensure that y = x is both local and global minimum of ω(y).
Since we then have shown that ω(x) = 0, we can conclude that ω is weakly positive
for all positive values of y.
At first, inserting y = x into ω(y) yields
(n− 1)
xr − xr
xr−1rn
·
(n− 1)xr + xr
(n− 2)xr + xr
− (x− x) = 0.
Next inserting y = x into f ′(y) yields
1−
(x
x
)r−1 n− 1
n ((n− 2)xr + xr)
[
(n− 1)xr + xr +
x2r − xrxr
(n− 2)xr + xr
]
= 1− 1
n− 1
n(n− 1)xr
[
nxr +
0
(n− 1)xr
]
= 1− 1 = 0.
57
The third step is not that straightforward. However, it is easy to see that q1 :=
n−1
xr−1n ·
yr−2
((n−2)xr+yr)3
is positive. Furthermore it can be shown that for n ≥ 2 the term
q2 :=
[(
n3 − 5n2 + 9n− 6
)
x3r +
(
3n2 − 11n+ 11
)
x2ryr + (3n− 6)xry2r + y3r
]
is positive. From this, we conclude
Lemma 8 Assume a rent-seeking contests with n players. Then there exists a technology
parameter r∗ > 1, such that for every technology with r < r∗ it holds that f ′′(y) > 0.
Proof. See appendix.
Now, since we have shown that the difference between the material payoff of an incumbent
and a mutant is minimized by y = x, and is not negative for r < r∗, we can conclude
the following theorem:
Theorem 8 Assume a rent-seeking contest with n players. Then there exists an r∗ > 1
such that, for every rent-seeking contest with a technology r < r∗, a population of status-
seekers maximizing (5.1) cannot be successfully invaded by any other preference type.
With this, we could show that relative payoff maximization is an evolutioanrily stable
preference for all rent-seeking contests with non-increasing returns to scale and also for
some contests with increasing returns to scale, provided the scale effects are not too
large.
Our analyses of Chapters 3 to 5 provides strong support for the anaylsis put forward
by Hehenkamp et al. (2004). We find that, as long as all players are not or only weakly
spiteful, then a more spiteful preference yields a higher payoff for the respective player if
the technology is characterized by non-increasing returns to scale. For increasing returns
to scale, we point out that several equilibria with different properties may occur, if only
two types of players are present, where one type is characterized by independent and the
other one by weakly interdependent preferences.
Furthermore, we showed that preferences that maximize relative payoff are stable
against other intruding preferences and, thus, evolutionarily stable. In a world where
different types repeatedly interact with each other, it will be the relative payoff maxi-
mizers who survive in the long run. However, while spiteful players that act according to
relative payoff maximization are more successful compared to other preferences, we have
shown that the absolute payoff reduces to zero if all players act according to relative
payoff maximization. In homogenous populations with less spiteful preferences, players
can receive positive absolute payoff though. This creates a preference dilemma. From a
welfare perspective, the aggregate absolute payoff that the players will get in equilibrium
is larger if the players are less spiteful. For a homogenous population, this means that
every player gains in absolute terms without losing relative payoff if all players behave
less spitefully. Yet, there are individual incentives to deviate from this behavior, so in a
preference equilibrium all players will behave as relative payoff maximizers.
58
Part II
Sequential Structures
59
Chapter 6
Sequential Structures in
Rent-Seeking Contests: The
Three-Player-Case
This part of our work now concentrates on sequential structures in contests with more
than two players. Each of the analyses presented in this part have a common structure.
In the first stage of the game, all contestants decide (simultaneously) about a point in
time when they will choose their effort. At the end of this stage, the results of the
timing decision are publicly announced and are common knowledge for the rest of the
game. Starting in the second stage, the players will commit to their effort sequentially,
depending on which stages were chosen in the timing decisions of the first stage.1 Once
a player has committed to her effort choice, it is common knowledge among all players.
Finally, in the last stage, the payoffs of the players are determined by Tullock-contest
with a payoff function of the form:
pii(x) =



xi∑n
j=1
xj
Vi − xi if ∃j = 1, ..., n : xj > 0
1
n if xj = 0, ∀j = 1, ..., n.
(6.1)
This payoff function is derived from the Tullock function (2.1), but with constant returns
to scale (r = 1) and the possibility of heterogenous valuations (i.e. Vi instead of V ). For
two players, Leininger (1993) presented such an analysis with two timings, finding that
in the subgame-perfect equilibrium the player with the higher valuation Vi will always
decide to move late, while the player with the lower valuation will choose to move early.
As a result, aggregate effort in this equilibrium will be lower than in a simultaneous
move Nash-equilibrium, and each player’s payoff increases, which means that, from a
welfare point of view, the equilibrium in the sequential game is a Pareto-improvement
to the equilibrium in the simultaneous game.2
1Note that, if all players have chosen the same stage before, the game of effort choice will be a
simultaneous one.
2The game we present here requires a rather complex notion of strategies. We present in the appendix,
how a strategy is defined here.
60
Concerning the relevance of sequential structures for contests in practical application,
we already refered to Morgan’s (2003) statement about the parties’ National Conventions
in U.S. presidential election campaigns. He had found that the two big parties always
chose their timing such that the party to which the current president belongs schedules
its National Convention later. If one interprets the governing party as the one with the
higher valuation (which can be supported e.g. by referring to switching costs that both
parties have to bear if the opposing party wins), this observation runs completely in line
with the results of Leininger (1993).
An example for a system with more than two (major) parties can be found in the
parliamentary election in Saxony-Anhalt 2006. The Social Democratic Party (SPD) and
the Left Party (PDS) decided upon their manifestos on the 14th and the 16th of January,
2006, respectively, while the incumbent, the Christian Democratic Union (CDU), did
not present their manifesto until the 31st of January, 2006. Final decision about this
manifesto was on 25th of February. A manifesto can be interpreted as committing oneself
to a strategy. This can be interpreted as a simultaneous decision of SPD and PDS, while
the CDU, the party providing the president, chose a timing late enough to be able to
react to the manifestoes of the SPD and the PDS. As we will show later, this timing
choice is exactly the order that our model would predict.
6.1 Introduction
In this chapter, we want to extend the analysis put forward by Leininger (1993) to a
contest in which three players decide whether they will choose their effort in an early
or in a late stage. To this aim, we analyze the equilibria of the feasible subgames,
which enables us to determine the subgame-perfect equilibrium. We will find that a
player only has an incentive to choose late if she exerts a higher effort in a simultaneous
move game than both her opponents taken together. Otherwise, all players will decide
simultaneously on the early stage. Mixed strategy equilibria can occur for a small subset
of the regarded cases when a second player also has an incentive to move late. Parameter
constellations where only two players are active are moved to the appendix.
In addition, we compare the aggregate payoff levels of the different subgames in order
to be able to see whether the sequential structures are still able to reduce dissipation.
This is not necessarily the case. Especially for rather homogenous populations, we find
that simultaneous moves are not only played on the equilibrium path, but also yield the
highest aggregate payoff among all feasible subgames.
6.2 The model
We model a rent-seeking contests with three players i = 1, 2, 3. The payoff of each
player, dependent on the effort vector, is given by (6.1). Similarly to Leininger (1993),
we examine a sequential game where every player commits herself to one of two points
in time (t = 1 or t = 2) before choosing her effort. Let us assume that, in the case
of a mixed strategy over the timing choice, the actual decision time is drawn before
61
t = 1. This justifies the assumption that players have complete information concerning
the subgame they play.
Assume throughout this paper, that V1 ≥ V2 ≥ V3. To ensure participation of all
three players in equilibrium, let us assume that
V3 ≥
V1
2
. (6.2)
For any player i with a valuation of Vi, the valid combinations of Vj and Vk are illustrated
in Figure B.1. The inequality (6.2) might be considered a restrictive assumption. How-
ever, it allows us to focus on true three-player-cases, whereas here we skip cases where
one of the players would prefer negative effort on the equilibrium path and discuss them
in Chapter A.2.
There are three generally feasible subgames: the first one, where all players move
simultaneous; the second one, where two players move early and one player moves late;
and the third one, where one player moves first while the others move later. In the
first step of our analysis, we will investigate the equilibria of these subgames one after
another. This allows us later on to derive the subgame-perfect equilibrium of the timing
game.
6.2.1 The simultaneous game
If all three players end up with the same choice of timing decision, they play a standard
rent-seeking contest with heterogenous valuations as put forward by Stein (2002). The
first-order-condition of the payoff maximization is given by:
∂pii(x)
∂xi
=
xj + xk
(xi + xj + xk)
2Vi − 1
!= 0, i, j, k = 1, 2, 3, j 6= i 6= k. (6.3)
Equating this equation for every player, we can deduce the equilibrium behavior that
brings about (xj + xk)Vi = (xi + xk)Vj = (xi + xj)Vk, which yields
xi = xj
Vi
Vj
+ xk
Vi − Vj
Vj
. (6.4)
Then
xj = xk
((Vi − Vj)Vk − ViVj)
ViVj − (Vi + Vj)Vk
. (6.5)
Inserting (6.5) in (6.4) yields
xi = xk
(
VjVk − Vi (Vj + Vk)
ViVj − (Vi + Vj)Vk
)
. (6.6)
Now inserting (6.5) and (6.6) in (6.3) gives us the combined first-order-condition:
xk = 2ViVjVk
ViVk + VjVk − ViVj
(ViVj + ViVk + VjVk)
2 ,
62
for any player k = 1, 2, 3. Now, non-negative effort requires that, for any combination of
players i, j, k = 1, 2, 3 with i 6= j 6= k, it holds that
Vi ≥
VjVk
Vj + Vk
. (6.7)
If (6.7) does not hold, the aggregate effort in this game is higher than player i’s valuation,
VjVk
Vj+Vk
, thus she has no incentive to spend positive effort. Players j and k play a standard
two-player simultaneous move game, as analyzed by Leininger (1993). It can be shown
that players j and k will also choose an aggregative effort of VjVkVj+Vk in this situation.
Yet, if (6.7) holds for all three players calculating equilibrium payoffs results in
piSi =
Vi (ViVj + ViVk − VjVk)
2
(ViVj + ViVk + VjVk)
2 , (6.8)
for any i = 1, 2, 3, where the superscript S denotes the simultaneous move game. Now
having derived the payoff a player can gain from a simultaneous move game, we need to
know the payoffs of the sequential games to derive the subgame-perfect equilibrium of
the total game. These are derived in the next two subsections.
6.2.2 Two players move early
In this subsection, we focus on the subgame where player k has chosen to move late
while the players i and j decided to move first. Then player k takes the actions of i and
j in t = 1 as given when deciding in t = 2. She maximizes
pik(x) =
xk
xi + xj + xk
Vk − xk
with the first-order condition ∂pik∂xk
!= 0 ⇔ (xi + xj)Vk = (xi + xj + xk)
2. From this, we
can conclude
xk =
√
(xi + xj)Vk − (xi + xj) . (6.9)
Note that, for player k, only the aggregate effort of her opponents matters, not the
individual choices. In t = 1, players i and j know that (6.9) characterizes the reaction
of player k and take this into consideration. This transforms the payoff function of an
early mover i to
pii(xi, xj, xk(xi, xj)) =
xi
√
(xi + xj)Vk
Vi − xi.
The first-order-condition can be transformed to
∂pii(xi, xj, xk(xi, xj))
∂xi
=
√
(xi + xj)Vk −
xiVk
2
√
(xi+xj)Vk
(xi + xj)Vk
Vi − 1
!= 0. (6.10)
Since (6.10) holds for both early movers i and j, we can conclude (see appendix for a
derivation)
xj = xi
2Vj − Vi
2Vi − Vj
. (6.11)
63
This means that xi + xj = xi
Vi+Vj
2Vi−Vj
. Inserting this in (6.10) and solving for xi yields:
xkLi =
9V 2i V
2
j (2Vi − Vj)
4 (Vi + Vj)
3 Vk
, (6.12)
where the superscript kL denotes the subgame in which player k is the only one to move
late (L). Thus, the participation constraint for an early moving player i in this subgame
is given by Vi ≥
Vj
2 .
Returning to the most interesting case of three active players, we find that player k’s
effort can be derived by inserting (6.12) for both early movers in (6.9):
xkLk =
3ViVj
4 (Vi + Vj)
2 Vk
(2Vk (Vi + Vj)− 3ViVj) . (6.13)
Player k is willing to choose a non-negative effort in this subgame, if and only if
Vk ≥
3ViVj
2 (Vi + Vj)
. (6.14)
If (6.14) does not hold, player k cannot gain a positive payoff from this subgame, and
only the players i and j will be active in the equilibrium of this subgame.
From (6.12) and (6.13), we can calculate the aggregate effort spent in this subgame.
We will need this value to derive the levels of payoff each player receives in equilibrium.
We find:
∑
h=i,j,k
xh =
3ViVj
2 (Vi + Vj)
. (6.15)
Remark that the critical value for the participation of player k in (6.14) takes the same
value. Furthermore, we see from (6.15):
Lemma 9 Consider a rent-seeking contest where two players bid in the first period and
one player bids in the second period. If all players exert positive effort, then total dissi-
pation only depends on the valuations of the two players bidding in the first stage.
Now, we can derive the equilibrium payoffs in this subgame from the above calculated
effort:
pikLi =
3V 2i Vj (2Vi − Vj)
2
4 (Vi + Vj)
3 Vk
, (6.16)
pikLj =
3ViV 2j (2Vj − Vi)
2
4 (Vi + Vj)
3 Vk
(6.17)
and
pikLk =
(2Vk (Vi + Vj)− 3ViVj)
2
4 (Vi + Vj)
2 Vk
. (6.18)
Let us now look at the question: what happens if at least one participation constraint
is binding? Note that, if a participation constraint is binding, it is always the weakest
64
V3 > 23V1 V3 ≤
2
3V1
V3 > 23V2
x1 ≥ V3 − 2V3 V2−V3V2
x1 ≤ 2V3 V1−V3V1
x1 ≥ V3 − 2V3 V2−V3V2
x1 ≤
V 23
V1
V3 ≤ 23V2 does not occur for V1 ≥ V2
x1 ≥ V3 −
V 23
V2
x1 ≤
V 23
V1
Table 6.1: Feasible equilibrium efforts of player 1, if V3 ∈
(
V1V2
V1+V2
, 32
V1V2
V1+V2
)
.
player, namely player 3, who is affected. Since (6.2) holds for player 3, her participation
is never critical if she moves early. Now presume she moves late. If her opponents choose
x1 + x2 ≥ 32
V1V2
V1+V2
in the first stage, she will choose zero effort (see (6.14)). Yet, if the
early movers play a two-player game in t = 1 without regard to player 3, the aggregate
effort takes the value of V1V2V1+V2 . Hence a plain two-player-contest is only played in t = 1
if V3 ≤ V1V2V1+V2 . Yet, this can be shown to contradict (6.2). For V3 ∈
(
V1V2
V1+V2
, 32
V1V2
V1+V2
)
,
however, a three-player-contest is played, where players 1 and 2 strategically react to
player 3’s presence in order to preempt her. Unfortunately, this game does not have a
unique equilibrium, but rather a continuum of equilibria. We present the equilibrium
effort of player 1 in these equilibria in Table 6.1. Table 6.1 gives the upper and lower
limits for the effort of player 1. For each single feasible value of x1, there exists an
equilibrium, in which player 1 chooses x1 and player 2 chooses x2 = V3 − x1 on the first
stage, while player 3 chooses x3 = 0. For a derivation of the values given in Table 6.1,
we refer to the appendix.
Now we have determined the payoff structure of a contest where two players move
early, while a third one chooses her effort after her opponents. To be able to analyze
the total game of timing decision, however, we still need to look at the situation, where
only one player decides in the early stage. In the next subsection, we take a closer look
at this subgame.
6.2.3 One player moves early
So let us now assume that only player i decides on her effort in the first round while the
players j and k move in the second round. The late movers simultaneously decide upon
their effort. So, the first-order-condition of a late mover is given by
∂pik(x)
∂xk
!= 0⇔ (xi + xj)Vk = (xi + xj + xk)
2 . (6.19)
In equilibrium, equating the first-order-condition of both late movers leads to (xi + xk)Vj =
(xi + xj)Vk, which is equivalent to xk =
Vk
Vj
(xi + xj)− xi. Inserting this in (6.19) yields:
(xi + xj)Vk =
(
(xi + xj)
Vk
Vj
+ xj
)2
(6.20)
65
As shown in the appendix, we can conclude from this that the optimal behavior of a late
moving player is given by
xiEj =
V 2j Vk − 2VjVkxi − 2V
2
k xi +
√
V 3j Vk (VjVk + 4Vkxi + 4Vjxi)
2 (Vj + Vk)
2 , (6.21)
where the superscript iE denotes the subgame in which player i is the only one who
moves early (E). This is anticipated by player i in the first period and she maximizes
pii (xi) =
xi
xi + xiEj (xi) + x
iE
k (xi)
Vi − xi.
This leads her to a first-order-condition of
(
xiEj (xi) + x
iE
k (xi)− xi
(
∂xiEj (xi)
∂xi
+
∂xiEk (xi)
∂xi
))
Vi =
(
xi + x
iE
j (xi) + x
iE
k (xi)
)2
.
(6.22)
Inserting (6.21) in (6.22) yields the maximization problem of the early mover, which is
solved by
xiEi =
V 2i (Vj + Vk)
2 − V 2j V
2
k
4VjVk (Vj + Vk)
, (6.23)
as shown in the appendix. This is a non-negative term for Vi ≥
VjVk
Vj+Vk
. As mentioned
above, this threshold value is exactly the aggregate effort that will be spent on the late
stage between players j and k, if i chooses zero effort. Now by inserting (6.23) in (6.21),
we find the effort of the players j and k (For a derivation, see appendix):
xiEj =
VjVk (2Vj + Vk)−
V 2i
Vj
(Vj + Vk)
2 + 2ViVj (Vj + Vk)
4 (Vj + Vk)
2 . (6.24)
For a late-moving player j, a non-negative effort only fulfills the first-order-condition for
Vj ≥ V˜j :=
Vi − Vk +
√
V 2i + 6ViVk + V
2
k
4
, (6.25)
with player i moving early and and player k moving late. Again, if this condition is not
fulfilled, player j cannot gain any positive payoff in this subgame.
Given (6.25), the aggregate effort in this subgame is thus given by
xiEi + x
iE
j + x
iE
k =
ViVj + ViVk + VjVk
2 (Vj + Vk)
, (6.26)
and equilibrium payoffs are given by
piiEi =
(ViVj + ViVk − VjVk)
2
4VjVk (Vj + Vk)
, (6.27)
66
piiEj =
(
2V 2j + VjVk − ViVj − ViVk
)2
4Vj (Vj + Vk)
2 and (6.28)
piiEk =
(2V 2k + VjVk − ViVj − ViVk)
2
4Vk (Vj + Vk)
2 ,
as derived in the appendix.
Let us now take a closer look at the case where (6.25) is not fulfilled. Remark that V˜j
can be shown to be strictly larger than Vi2 . For Vj ∈
(
Vi
2 ; V˜j
)
, player j quits participation.
Then the early moving player i cannot gain from increasing her effort beyond the point
where player j quits active participation, contrary to the implicit assumption behind
(6.22) that the solution is interior. Thus, the best response of player j will be to choose
an optimal reaction of zero in (6.21). The equilibrium effort of player i is given by
xi =
V 2j
Vk
. Player k will choose an effort of xk = Vj −
V 2j
Vk
. Obviously, xj = 0. Denoting
this game by ”Pr” for Preemption, we can write down the payoffs as:
piPri =
(Vi − Vj)Vj
Vk
∧ piPrj =
(Vk − Vj)2
Vk
∧ piPrk = 0.
6.3 Comparing the subgame payoffs
6.3.1 Decision Structure
So far, we have analyzed the equilibria of the subgames. This enables us to reduce the
total game to a game of a timing decision. Equilibrium behavior in the subgames is now
presumed and the normal form of the timing game is given by:
Player j
t = 1 t = 2
Pl. t = 1 piSi ,pi
S
j ,pi
S
k pi
jL
i ,pi
jL
j ,pi
jL
k
i t = 2 piiLi ,pi
iL
j ,pi
iL
k pi
kE
i ,pi
kE
j ,pi
kE
k
Player k moves in t = 1.
Player j
t = 1 t = 2
Pl. t = 1 pikLi ,pi
kL
j ,pi
kL
k pi
iE
i ,pi
iE
j ,pi
iE
k
i t = 2 pijEi ,pi
jE
j ,pi
jE
k pi
S
i ,pi
S
j ,pi
S
k
Player k moves in t = 2.
Each player can be faced with four different problems for the timing decision, two of
which can be solved equivalently, since it does not matter for the analysis whether e.g.
player i faces player j moving early and player k moving late, or the other way round.
So, we can formulate the decision situation in three problems.
• Problem I: What is a player’s best choice if her opponents decide late?
• Problem II: What is a player’s best choice if her opponents decide early?
• Problem III: What is a player’s best choice if one of her opponents decides early,
while the other one decides late?
67
Note that these problems are in a way hypothetical since they assume that the player
facing the problem decides last about his choosing time, whereas we regard choice of
timing as a simultaneous game in our analysis. Yet, solving these problems is inevitable
for finding the Nash-equilibrium of the timing game.
Problem I
Problem I can be solved by comparing the payoff player i receives from the simultaneous
game with the one she gets from moving first. If the considered player is active in both
subgames, the payoff for moving first is larger if piiEi > pi
S
i which is equivalent to
(Vi (Vj + Vk)− VjVk)
2 > 0 (6.29)
for Vi >
VjVk
Vj+Vk
, as shown in the appendix. For Vi ≤
VjVk
Vj+Vk
, player i will stay outside the
contest and thus receive zero payoff in both cases. In this case, she will be indifferent
between both timings then.
Thus, in a subgame where all players move late simultaneously, every player has an
incentive to deviate to an early move. This result is not too surprising, since it can
already be established in the two-player-case analyzed by Leininger (1993).
Problem II
Problem II can be solved by comparing the payoff player k receives from a simultaneous
game with the one in the subgame where she is the only late mover. For an interior
solution in both subgames, she will prefer moving early if piSk ≥ pi
kL
k . Comparing the
respective variants of (6.8) and (6.13), we find that piSk ≥ pi
kL
k is fulfilled with equality
for
Vk =
ψViVj
Vi + Vj
with ψ ∈
{
3−
√
57
8 ,
3+
√
57
8 , 3
}
. However, it is sufficient to concentrate on cases where the
player is willing to generate a non-negative effort as the only late mover. This happens
if Vk ≥ 32
ViVj
Vi+Vj
. Since 3+
√
57
8 <
3
2 , the only situation where pi
S
k = pi
kL
k holds and player k
chooses a positive effort is then given by Vk = 3
ViVj
Vi+Vj
. Now it is easy to show that player
k would only choose to move late if Vk > 3
ViVj
Vi+Vj
. Given assumption (6.2), this holds for
at most one player. We will present further intuition for this result in subsection 6.3.3.
Problem III
Problem III can be solved by comparing the payoff of player j when joining the one
player moving early with the payoff when joining the one player moving late. She prefers
moving late if:
pikLj > pi
iE
j . (6.30)
68
From (6.17) and (6.28) in (6.30) leads to:
3ViV 2j (2Vj − Vi)
2
4 (Vi + Vj)
3 Vk
>
(
2V 2j + VjVk − ViVj − ViVk
)2
4Vj (Vj + Vk)
2 .
For most of the constellations captured by (6.2), this can be shown to hold for all
three players. Yet, if player 1 is the player moving late, while player 3 is the player
moving early, and V3 lies only slightly above the threshold V12 , player 2 can have an
incentive to join the late mover. Table B.1 in the appendix gives an overview of the
range of constellations for which this might happen.3 It contains about 3 % of the entire
constellations captured by (6.2). Note that Table B.1 also provides an upper limit for the
valuation of a player, up to which she prefers to join the late moving opponent instead
of joining the early mover.
Let us now subsume what we learn from Table B.1. A player only has an incentive
to join the late mover if one player is (at least weakly) stronger and the other player is
strictly weaker. There is only one case where a player with highest valuation has a weak
incentive to join the late mover, and that is, if she has the same valuation as the late
mover (V1 = V2), while V3 = V12 . Note that this is a border case. The strongest player
never has a strict incentive to move late if facing this situation; the weakest player never
has any incentive at all to move late.
6.3.2 Equilibria
The previous analysis has revealed the conditions under which players prefer early com-
mitment to their effort and when they would rather wait, given their opponents’ decisions.
From these results, we can deduce which of the subgames lies on the equilibrium path
of a subgame-perfect Nash-equilibrium.
We find that, in a situation where all players choose late, there is always an incentive
to deviate and move early. Thus, all players moving late cannot be an equilibrium. In a
situation where all players move early, however, no player has an incentive to deviate to a
later move if V1 ≤ 3V2V3V2+V3 holds for the strongest player. Thus, this situation characterizes
an equilibrium. Furthermore, we find that a constellation where one of the weaker
player moves late on her own can never be equilibrium. In a similar way, there is no
equilibrium with two players moving late, since there is always at least one player who
has an incentive to deviate to moving early. The strongest, as well as the weakest, player
will always prefer an earlier move to a late move, if there is a second late mover.
Now, if V1 > 3V2V3V2+V3 , player 1 has an incentive to move late. Unless we are in a
constellation captured by Table B.1, both weaker opponents have no incentive to move
late in any situation, such that we end up in an equilibrium where the strongest player
moves late, while both the others move early. Yet, for the cases captured by Table B.1,
we find that player 2 also has an incentive to join the late mover. However, if player 2
3Note that Table B.1 also presents constellations where this problem occurs for the case of V3 >
V1
2 ,
that is discussed in Chapter A.2.
69
chooses late, player 1 no longer has an incentive to move late herself. Hence there is no
equilibrium in pure strategies, but only in mixed strategies. To derive these strategies, let
us denote the probability that player i will assign to an early move in her mixed strategy
by τi, i = 1, 2, whereas she chooses t = 2 with probability (1-τi). In a mixed strategy
equilibrium, each player must be indifferent between the alternatives she can choose.
For player 2, this means that her expected payoff from moving early equals her expected
payoff from moving late. The expected payoffs are given by E(pi2|E) = τ1piS2 +(1−τ1)pi
1L
2
for an early move and E(pi2|L) = τ1pi2L2 + (1 − τ1)pi
3E
2 for a late move. Equating these
equations yields
τ1 =
pi3E2 − pi
1L
2
pi3E2 + pi
S
2 − pi
1L
2 − pi
2L
2
.
The derivation of ρ2 runs analogously and it shows that:
τ2 =
pi2L1 − pi
3E
1
pi2L1 + pi
1L
1 − pi
S
1 − pi
3E
1
.
In the appendix, we show that the values for τi, i = 1, 2 indeed lie in the interval [0, 1]
for the relevant parameters. The case that (6.30) is violated for some player j and
V1 < 3V2V3V2+V3 never occurs. We summarize:
Theorem 9 In a game where three players decide whether they will choose effort for a
rent-seeking contest early or late, the following holds in a subgame-perfect equilibrium:
• If the three players are relatively homogenous, i.e., V1 ≤ 3V2V3V2+V3 , all players move
simultaneously in the early stage.
• If V1 > 3V2V3V2+V3 and (6.30) holds for j = 2 (with i = 3 and k = 1), player 1 moves
late while the two weaker players move early.
• If V1 > 3V2V3V2+V3 and (6.30) does not hold for j = 2 (i = 3 and k = 1), player 3 moves
in t = 1, while both her opponents choose a mixed strategy concerning their timing
decision. The probability with which player i, i = 1, 2, moves early is given by ρi.
6.3.3 Comparison to the two-player-case
For a sequential contest with two players, Dixit (1987) has shown that a player who
receives a winning probability of more than 12 in the simultaneous move game will choose
a higher effort if she moves alone early than she will in a simultaneous move game.
Furthermore, he found that a player with a winning probability lower than 12 will exert
a lower effort level moving alone early than in the simultaneous move game. Hence, as
analyzed by Baik and Shogren (1992) and Leininger (1993), the weaker player always has
an incentive to prevent the stronger player’s early move, while the strong player is better
off waiting, since the weak player behaves less aggressively as the first mover. The weaker
player will thus move first in a subgame-perfect equilibrium, while the stronger player
will move last. In our three-player-model, we find that the strongest player will only move
late if V1 ≥ 3 V2V3V2+V3 . This is the condition which induces that, in a simultaneous move
70
equilibrium, player 1’s effort is higher than the aggregate effort of both her opponents.
This directly induces a winning probability of more than 12 in the model at hand. Here
we find how the results of the two-player-case can be generalized to the three-player-case.
Corollary 8 If the effort of the strongest player in the simultaneous move game is
(strictly) higher than the aggregate effort of her opponents, i.e., her winning probability
is higher than 12 , then she has a (strict) incentive to move late. The reverse also holds.
Proof. As we can conclude from the previous analysis:
xS1 = 2V1V2V3
V1(V2 + V3)− V2V3
(V1V2 + V1V3 + V2V3)
2 ∧ x
S
2 + x
S
3 = 4
V1V 22 V
2
3
(V1V2 + V1V3 + V2V3)
2 .
Then:
xS1 > x
S
2 + x
S
3 ⇔ 2V1V2V3
V1(V2 + V3)− V2V3
(V1V2 + V1V3 + V2V3)
2 > 4
V1V 22 V
2
3
(V1V2 + V1V3 + V2V3)
2
⇔ V1(V2 + V3)− V2V3 > 2V2V3 ⇔ V1 >
3V2V3
V2 + V3
.
The analysis of this section revealed how the players behave in equilibrium when
they choose their decision time. The question of which decision time would be socially
desirable is addressed in the next section.
6.4 Welfare Aspects
As already known in contest literature, the distribution of a rent via a contest is in
general not optimal but generates dissipation. If one assumes the welfare level to be
given by
W =
∑
i=1,2,3
pii, (6.31)
then it is obvious that welfare is maximized when the total prize is allocated to the
player with highest valuation for free. Of course, this allocation cannot be achieved
in a contest. Therefore, another interesting question arises, namely, what is the least
inefficient order of moves. We now address the question which order of moves generates
the highest welfare level. In order to compare in which subgame welfare is highest, we
first calculate the level of welfare in each subgame:
• In the simultaneous move subgame, welfare is given by
W S =
Vi (Vj − Vk)
2 + Vj (Vi − Vk)
2 + Vk (Vj − Vi)
2 + ViVjVk
ViVj + ViVk + VjVk
.
71
• In the subgame where the players i and j move early, welfare is given by:
W kL =
3ViVj
(
V 2i − ViVj + V
2
j
)
+ V 2k (Vi + Vj)
2 − 3 (Vi + Vj)ViVjVk
(Vi + Vj)
2 Vk
.
• In the subgame where only player i moves early, welfare is given by:
W iE =
VjVk
(
2V 2j + VjVk + 2V
2
k
)
+ V 2i (Vj + Vk)
2 − 4ViVjVk (Vj + Vk)
2VjVk (Vj + Vk)
,
if (6.25) holds for both late players. Otherwise, if, for player j, (6.25) is violated,
the welfare level is given by4
W Pr =
Vj(Vi − Vk) + Vk(Vk − Vj)
Vk
.
We find that the simultaneous move game is indeed the subgame with the highest
welfare level, if the strongest player 1 is rather weak, i.e., V1 ≤ 3V2V3V2+V3 . Unlike in the
two-player-case, the introduction of a sequential structure cannot reduce the dissipation
level. For V1 ≥ 3V2V3V2+V3 , W
1E takes the highest value if (6.25) holds. If not, it can be
shown that the highest welfare is indeed reached by the game where the strongest player
moves early and the weakest player is preempted, as long as V1 <
V2(V 22 −V2V3+V 23 )
V 22 −V
2
3
. For
V1 >
V2(V 22 −V2V3+V 23 )
V 22 −V
2
3
, we find that the strongest player moving late on her own creates
the highest welfare. We summarize:
Proposition 5 If a contest is played between three players in two points in time, the
order of moves that generates the highest welfare level in the equilibrium of the subgame
is given:
• by all three players choosing simultaneously if the three players are relatively ho-
mogenous, i.e., V1 < 3V2V3V2+V3 ,
• by simultaneous moving or the strongest moving early on her own or the strongest
player moving late on her own, if V1 = 3V2V3V2+V3 .
• by the player with the highest valuation moving early and her opponents moving
late for V1 ∈
(
3V2V3
V2+V3
;
V2(V 22 −V2V3+V 23 )
V 22 −V
2
3
)
. Note that the weakest player will not choose
positive effort in this case.
• by the strongest player moving late, and her opponents moving early, if V1 ≥
V2(V 22 −V2V3+V 23 )
V 22 −V
2
3
.
4The case that (6.25) is violated for both players j and k cannot occur, as long as (6.2) holds.
72
Together with Theorem 9 we can conclude:
Proposition 6 For V1 ≤ 3V2V3V2+V3 or V1 ≥
V2(V 22 −V2V3+V 23 )
V 22 −V
2
3
, the order of moves that creates
the highest welfare level is the same one that the subgame-perfect equilibrium brings about.
Otherwise, welfare can be improved over the subgame-perfect equilibrium by choosing
another order of moves than the one played on the equilibrium path.
Another interesting result can be found by comparing the payoffs from the simulta-
neous game with the payoffs received when the strongest player moves late. If we make
this comparison for every single player, we find:
Lemma 10 Let V1, V2 and V3 be such that, in the equilibria of the subgames S and
1L, every player is active. Then for V1 < 3V2V3V2+V3 , the simultaneous move game pareto-
dominates subgame 1L. Yet, for V1 > 3V2V3V2+V3 , subgame 1L pareto-dominates the simulta-
neous move game.
6.5 Conclusion
We have shown that, in a rent-seeking contest where three players can decide endoge-
nously about the sequential structure, different timing strategies are only played on the
equilibrium path if the strongest player is stronger than her opponents together. We
measure this in terms of efforts chosen in the simultaneous move game. Put differently,
a player only has an incentive to move late if her winning probability exceeds 12 . Note
that, for the two-player-contest analyzed by Leininger (1993), this always holds. Yet,
this si different in a three-player-contest. If, in such a contest, the strongest player is too
weak, the three players will move simultaneously in the subgame-perfect equilibrium.
For some cases where one player is strong enough to be stronger than her opponents,
it may occur that the secodn strongest player also has an incentive to choose late. In
this situation, the timing decisions of these two players will not be in pure strategies.
Concerning welfare aspects, we found that the order of moves that evolves in a subgame-
perfect equilibrium only creates the highest aggregate payoff in a subset of all possible
constellations, e.g. if the simultaneous move game is played on the equilibrium path.
For the case of the strongest player moving late, for Leininger (1993) showed is welfare
improving in the two-player-case, we found that in the three-player-case it can be more
favorable if the strongest player moves early as the only player. In contrast, whenever
the simultaneous move game is played on the equilibrium path, the welfare can not be
increased by changing the order of moves. Compared to Leininger’s result that a welfare
improvement by sequential choice is always possible for two heterogenous players, our
result is rather bad news. We have to state that, in the presence of a third player, a
welfare improvement may not necessarily be possible. Furthermore, where it is possible,
it will not be exhausted to its full extent any more.
73
Chapter 7
Incentives in the Strategic Choice of
Decision Timings
7.1 Introduction
In the previous chapter, we found that, as soon as a third player is introduced, sequential
moves only occur in a subgame-perfect equilibrium for a rather heterogenous population.
If players are rather homogenous, a simultaneous move game is played in subgame-perfect
equilibrium. In this chapter, we take a further step to look at sequential multi-player-
contests, and relax the assumption of only three players. In order to keep the results
manageable, we have to make some other restrictions; we assume that one player has a
higher valuation than her opponents, while we assume the opponents to be equal.
As an application, one might think of a contest for a market license that grants a
monopoly rent (which can be reasonable in the case of a natural monopoly). For such a
case Harris and Vickers (1985) argue that the incumbent firm has a higher profit from
winning the rent, which we can interpret as having a higher valuation. The reason for
the higher profit is that an incumbent firm does not have to bear additional market entry
costs such as for the acquisition of machinery and personnel. For such a setup, we will
examine whether such an incumbent firm has an incentive to move earlier or later than
her opponents.
In order to keep the analysis more general, from now on, we will use a different
terminology. Following Dixit (1987) we call the firm with the higher valuation - and
thus with a stronger position - the ”favorite”, while we call those players with the lower
valuation ”underdogs”. With respect to the underdogs, we restrict our analysis to show
that, in equilibrium, a single underdog does not have an incentive to choose a late move
when the remaining underdogs choose early. This holds irrespectively of the timing
decision of the favorite.
With respect to the favorite, we will see that she will always opt for an earlier move
if allowed. If not, we can find a threshold value for her valuation, above which she will
choose to decide later than her opponents. For a lower valuation simultaneous move is
played in the subgame-perfect equilibrium. For n = 3, the threshold value should take
the same value as the one in Chapter 6 for two equal players, since the involved subgames
74
are identical. This indeed happens.
For the case where the favorite has a first-mover right, we find that she will always
exert this right. If she faces at least two other players and her valuation is sufficiently
higher than the one of her opponents, it pays for the favorite to exert such a high effort
that her opponents will not have any incentive to exert a positive effort. The welfare
analysis will show that, whenever this situation occurs, this is the best feasible sequential
order from a welfare point of view. Note that then the stronger player is the only active
player in the game. Other games where only one player is active in equilibrium have
been analyzed by Konrad and Leininger (2007) for all-pay-auction and by Schoonbeek
and Winkel (2006) for incomplete information.
When relaxing the assumption of the first-mover-right for the favorite, there is no
equilibrium in which she chooses her effort before her opponents. Yet, the results of
Leininger (1993) that the favorite will always choose to move after her opponent in
a two-player contest can again be generalized to the multi-player case in so far that
she moves late if her winning probability in the simultaneous movbe game exceeds 12 ,
similar to the results in Chapter 6. If the underdogs taken together exert a higher effort
in equilibrium, it does not pay for the favorite to separate in timing strategy and the
simultaneous game will be played.
From the view of mechanism design, we are also able to make a statement about
whether or not a first-mover-right should be granted to the player with the highest
valuation. We will provide a threshold value for the relative valuation, from which point
on the first-mover-right, and thus the game structure in which the favorite moves first,
will bring about a higher social welfare than the game where her opponents decide in
the first round.
7.2 Setup of the game
7.2.1 The model
The payoff structure of the game we analyze is described by a Tullock-contest (Tullock,
1980). We assume that the contest technology has constant returns to scale. Let the
number of players be fixed to n, denote the non-negative effort of player i by xi. The
expected payoff of player i is then determined by (6.1). We assume
V1 = θVi, θ > 1 ∧ i = 2, ..., n,
where θ represents the relative valuation of the favorite. For simplification, we write
Vi =: V , ∀i = 2, ..., n. With this, we assume the players i, i = 2, ..., n to have identical
valuations.
7.2.2 Timing decision
The aim of this chapter is to focus on the incentives for the favorite to choose her timing
decision. We will consider two variants of the meta-game:
75
1. Like the underdogs, the favorite chooses a timing t = 1 or t = 2. She cannot secure
herself an earlier choice than her opponents.
2. The favorite chooses a timing t = 0, t = 1 or t = 2. This means that the favorite
can exert the exclusive right of the first move.
For the case of a mixed strategy concerning the timing decision, let the realization of
the strategy be drawn before t = 0. Before the first effort choice is made, the subgame
played is common knowledge. The strategy set of the favorite’s timing decision consists
of two alternative timings in variant 1 and three alternatives in variant 2. The timing
decision of the favorite determines in which subgame the contest will be played. For
now, we will simply assume that the underdogs choose in t = 1. Later on, we will
point out that an underdog does not have any incentive to deviate from t = 1 to t = 2.
Thus, the equilibria presented here are subgame-perfect equilibria of a game in which
the underdogs can decide whether to choose their effort in t = 1 or t = 2.
Considering the subgames where the underdogs move simultaneously, we call the
subgame where all players choose simultaneously ”Game S”, the subgame where the
favorite moves last is called ”Game L”, and the one where she moves first is called
”Game E”. As we will see, the latter subgame will be only relevant for the analysis of
variant 2. Let us next analyze these subgames
7.3 The Subgames
7.3.1 Equilibrium Behavior in the Simultaneous Game
Consider a rent-seeking contest with constant returns to scale, where all players decide
simultaneously about their effort. Then the favorite maximizes pi1(x) = x1∑n
j=1
xj
θV −
x1,while each of the underdogs maximizes pii(x) = xi∑n
j=1
xj
V − xi, 2 ≤ i ≤ n. The first-
order-condition for the favorite is given by:
∂pi1(x)
∂x1
= 0⇔ θ
n∑
j=2
xjV =


n∑
j=1
xi


2
, (7.1)
while the underdogs decide according to
∂pii(x)
∂xi
= 0⇔
n∑
j=1,j 6=i
xjV =


n∑
j=1
xi


2
, 2 ≤ i ≤ n. (7.2)
From Proposition 3, we can see that, from (7.2), we can conclude that xi = xj, 2 ≤ i, j ≤
n. Equating (7.1) and (7.2) and inserting xi = xj yields:
x1 = [(n− 1)θ − (n− 2)]xi. (7.3)
76
Equilibrium efforts are then given by (see appendix for derivations):
xS1 =
(n− 1)θ
(1 + (n− 1)θ)2
((n− 1)θ − (n− 2))V (7.4)
and
xSi =
(n− 1)θ
(1 + (n− 1)θ)2
V, 2 ≤ i ≤ n. (7.5)
Equilibrium payoffs are thus given by
piS1 =
((n− 1)θ − (n− 2))2
(1 + (n− 1)θ)2
θV (7.6)
and
piSi =
V
(1 + (n− 1)θ)2
, 2 ≤ i ≤ n. (7.7)
These results build on the assumption that all players move simultaneously. But
from the literature (e.g. Dixit, 1987, Baik and Shogren, 1992, Leininger, 1993) and the
previous chapter, we know that it can be a stable equilibrium, if players do not decide
at the same time but sequentially.
7.3.2 Equilibrium Behavior in Subgame L
For our analysis, let us next assume the favorite chooses to decide upon his effort level
after her opponents do. She takes the effort level of the underdogs as given. Again, we
can conclude from the results of Proposition 3 that all underdogs will choose the same
effort in the first round. Denoting this effort level by xˆ, the first-order-condition of the
favorite is given by
∂pi1(x1, [xˆ]n−1)
∂x1
=
(n− 1)xˆ
(x1 + (n− 1)xˆ)
2 θV − 1 = 0
⇔ x1 =
√
(n− 1)xˆθV − (n− 1)xˆ. (7.8)
Her opponents anticipate this in the first stage and are also able to see, in how
far this action depends on their own effort xi. As mentioned above, in equilibrium,
xi = xj, i, j = 2, ..., n. But in the decision process, note that an underdog i will
interpret ”(n− 1)xˆ” as ”(n− 2)xˆ+ xi”, taking the other underdogs’ reactions as given,
but not so her own. Each underdog maximizes her payoff pii over her own effort xi. The
reduced payoff function is given by pii(x1, [xˆ]n−1) = xiV√
θ[(n−2)xˆ+xi]V
− xi. The equilibrium
behavior of an underdog is thus characterized by
∂pii(x1, [xˆ]n−1)
∂xi
=
√
θ [(n− 2)xˆ+ xi]V − xi2
θV√
θ[(n−2)xˆ+xi]V
θ [(n− 2)xˆ+ xi]V
V − 1 = 0. (7.9)
77
Since in equilibrium all underdogs behave the same, we can set xˆ = xi and find that
(7.9) is equivalent to
xLi = xˆ =
(2n− 3)2
4(n− 1)3θ
V. (7.10)
Now inserting (7.10) in (7.8) gives us the strategy of the favorite:
xL1 =
(2n− 3) (2(n− 1)θ − (2n− 3))
4(n− 1)2θ
V. (7.11)
From the strategy, we can calculate the favorite’s level of payoff from moving last:
piL1 =
(2(n− 1)θ − 2n+ 3)2
4(n− 1)2θ
V. (7.12)
Her opponents’ payoff is given by
piLi =
(2n− 3)
4(n− 1)3θ
V. (7.13)
7.3.3 Equilibrium Behavior in Subgame E
In this section, we assume that the favorite will choose her effort x1 in the first stage
and the underdogs will follow in the second stage. Each underdog takes the effort of
the first player x1 as exogenously given and maximizes in the second stage pii(x) =
xi
x1+
∑n
j=2
xj
V − xi, 2 ≤ i ≤ n. The first-order-condition is given by
∂pii(x)
∂xi
!= 0⇔

x1 +
n∑
j=2,j 6=i
xj

V =

x1 +
n∑
j=2
xj


2
. (7.14)
For a symmetric equilibrium, which will come into place due to Proposition 3, we
can turn (7.14) into
(x1 + (n− 2)xi)V = (x1 + (n− 1)xi)
2 . (7.15)
However, we cannot directly conclude from this player i’s reaction function without
regarding the participation constraint. From Lemma 6 we know that a player i will only
choose positive effort in equilibrium, if, for her valuation Vi, it holds that Vi >
∑n
j=1 xj.
This means, that a player will only exert a positive effort if her valuation for the prize
exceeds the level of dissipation in the game. A player will only take part in the contest
as long as the total effort spent in the game does not exceed her own valuation. If we
pay attention to this restriction, we can derive the reaction function of an underdog to
the favorite’s behavior from (7.15), while taking into account the optimal behavior of
the other underdogs.
78
Lemma 11 In an intragroup symmetric equilibrium where the favorite moves first, the
optimal reaction function of an underdog i with 2 ≤ i ≤ n is given by
xi = xˆ(x1) =



(n−2)V+
√
(n−2)2V 2+4(n−1)x1V
2(n−1)2 −
x1
n−1 , if x1 ≤ V
0 , if x1 ≥ V.
(7.16)
Proof. See appendix.
Like above, let us in the following denote the effort of an underdog, that is only viewed
upon in the aggregate, by xˆ. The favorite anticipates the reaction of the underdogs.
Assuming that the favorite’s effort will not exceed V , her payoff function turns to
pi1(x1, [xˆ(x1)]n−1) =
x1
x1 + (n− 1)xˆ
θV − x1
⇔ pi1(x1, [xˆ(x1)]n−1) = x1


2(n− 1)θV
(n− 2)V +
√
(n− 2)2V 2 + 4(n− 1)x1V
− 1

 . (7.17)
Since for x1 ≥ V the underdogs always react with a zero effort, it is obvious that any
choice of x1 > V leads to a lower payoff for the favorite than x1 = V , and thus cannot
maximize her payoff. So, we can restrict our analysis to x1 ≤ V . One can show that the
payoff function in (7.17) is concave, so we can conclude that the optimal strategy of the
favorite is characterized by x1 < V , if for x1 = V it holds that ∂pi1∂x1 < 0. On the other
hand, for ∂pi1(V,[xˆ(V )]n−1)∂x1 ≥ 0, we find that she will choose x1 = V . Solving this equation,
we find
Proposition 7 The favorite will choose to invest x1 = V and thus exclude her opponents
from the contest if and only if, for her relative valuation θ, holds that
θ ≥
n
n− 1
.
Her payoff will then be given by pi1 = (θ − 1)V .
Proof. See appendix.
Beyond the results of Proposition 7, let us also calculate how the players behave in
this subgame, if θ < nn−1 . In this case, the equilibrium effort of the favorite is the solution
to the first derivative of (7.17) being zero. The solution is given by
xE1 =
(n− 1)2θ2 − (n− 2)2
4(n− 1)
V. (7.18)
Inserting θ = nn−1 in (7.18) yields x
E
1 = V . Therefore, the optimal effort of the favorite
is continuous in θ. Reinserting (7.18) in (7.16) yields the effort an underdog exerts if
θ < nn−1 :
xEi =
n(n− 2) + 2(n− 1)θ − (n− 1)2θ2
4(n− 1)2
V. (7.19)
79
Equilibrium payoffs in this subgame are thus given by
piE1 =



((n−1)θ−(n−2))2
4(n−1) V, if θ <
n
n−1
(θ − 1)V, if θ ≥ nn−1 ,
(7.20)
and
piEi =



(n−(n−1)θ)2
4(n−1)2 V, if θ <
n
n−1
0, if θ ≥ nn−1 .
(7.21)
So far, this section has concentrated on the analysis of subgames that differ in the
timing decision of the favorite. Yet it is also conceivable that an underdog could have an
incentive to choose a different timing from t = 1. We address this issue by presenting:
Lemma 12 Given that all (n − 2) players j with j = 2, ..., n, j 6= i choose their effort
in t = 1, player i, i = 2, ..., n, has an incentive also to choose her effort in t = 1. This
result holds regardless of the timing decision of the favorite.
Proof. See appendix.
7.4 Subgame-perfect Equilibria
This section analyzes the subgame-perfect equilibria of the games introduced in Subsec-
tion 7.2.2. From Lemma 12, we can conclude that a combination of timing decisions,
where the underdogs move in t = 1 and the favorite does not have a strict incentive to
choose another timing choice, is the timing decision of a subgame-perfect equilibrium.
Therefore, to be able to find such equilibria it is sufficient to compare the payoffs, which
the subgames analyzed in Section 7.3 would deliver to the favorite. We continue our
analysis by determining, which subgame produces the highest payoff for the favorite
given θ. The payoffs the favorite gains in each subgame are given by equation (7.6) for
the simultaneous game, by equation (7.20) if she chooses first and by equation (7.12) if
she chooses after her opponents. Now by equating these equations pairwise, we find
Proposition 8 Let all underdogs decide about their effort in t = 1. Then
1. for any valuation θV with θ > 1, the favorite gains a higher equilibrium payoff
from moving early, i.e., t = 0, than from moving late, i.e., t = 2. This means:
piE1 > pi
L
1 , ∀θ > 1, n ≥ 3.
2. For any valuation θV with θ > 1, the favorite gains a higher equilibrium payoff
from moving early, i.e., t = 0, than from simultaneous moves, i.e., t = 1. This
means: piE1 > pi
S
1 , ∀θ > 1, n ≥ 3.
3. The equilibrium payoff of the favorite is (strictly) higher from moving late (t = 2)
than from a simultaneous move (t = 1) if and only if θ is (strictly) higher than the
threshold value θˆ = 2n−3n−1 . This means pi
L
1
>
< piS1 , iff θ
>
< θˆ, n ≥ 3.
80
Proof. See appendix.
Remark that, whenever θ < 2n−3n−1 holds, the favorite has a strict preference for moving
earlier. Now having analyzed the relation of the favorite’s payoff in different subgames,
we can take a closer look at the game in total. We comprise
Theorem 10
1. For variant 1, the subgame-perfect equilibria are characterized by
(a) the favorite choosing (7.4) and each underdog choosing (7.5), both in t = 1,
if θ < 2n−3n−1 , and by
(b) the favorite choosing (7.11) in t = 2 and each underdog choosing (7.10) in
t = 1, if θ > 2n−3n−1 .
2. For variant 2, the subgame-perfect equilibrium is given by the favorite choosing
(7.18) in t = 0 and each underdog choosing (7.19) (in t = 1), for all values of
θ > 1.
Proof. The theorem follows from Proposition 8.
For variant 1, Theorem 10 also holds for the two-player-case as one can see from
Leiniger (1993). In this case, the threshold value 2n−3n−1 reduces to 1. This means that the
favorite will always choose late. This is the same result that was shown by Leininger.
Furthermore, this result runs in line with the major result of Chapter 6. If we assume
for the analysis of Chapter 6 that both weaker players have the same valuation, then
this is the same game as the one presented here with n = 3. Both Theorems 9 and 10
state that, if the favorite’s valuation is higher than 32 of the underdog’s valuation, she
will move later, while if her valutaion is lower than 32 of the underdog’s valuation, she
will prefer a simultaneous move game.
Similar to Corollary 8, where we exposed that the favorite will prefer to move later
if her effort in the simultaneous move game is higher than the aggregate effort of her
opponents we find here that:
Corollary 9 In subgame-perfect equilibrium, the favorite will prefer a late move to a
simultaneous move if her equilibrium effort in the simultaneous move game exceeds the
aggregate effort of her opponents, i.e., if her winning probability from the simultaneous
move game exceeds 12 .
Proof. From (7.4) and (7.5), we can see that:
xS1 > (<)(n− 1)x
S
i ⇔ (n− 1)θ − (n− 2) > (<)n− 1⇔ θ > (<)
2n− 3
n− 1
.
Furthermore, we can see from Theorem 10 that whenever the favorite has the oppor-
tunity to choose earlier, she will do so. This implies directly that the equilibrium path
of the game changes depending on whether the favorite is granted a first-mover-right.
81
7.5 Welfare Implications
In this section, we analyze which type of sequential structure would maximize the social
welfare given the valuation structure. In particular, this section answers the question:
which timing a contest designer who maximizes welfare would assign to the favorite? Let
us therefore define welfare as the sum of payoffs of all players, i.e.,
W :=
n∑
j=1
pij. (7.22)
When looking at the welfare level achieved in game E, we have to split up the analysis
into two cases. For θ ≥ nn−1 , it is obvious that the welfare level is given by
WE = (θ − 1)V. (7.23)
Now looking at θ < nn−1 , we find from (7.20) and (7.21) that for this case
WE =
1 + (n− 1)2 (θ − 1)2
2(n− 1)
V. (7.24)
Now for the simultaneous game the welfare level is given by
W S =
((n− 1)θ − (n− 2))2 θ + (n− 1)
[1 + (n− 1)θ]2
V. (7.25)
Finally, in the situation where the favorite moves after her opponents, we have
WL =
2(n− 1)θ2 + (2n− 3)(1− 2θ)
2(n− 1)θ
V. (7.26)
From the levels of welfare, we can derive which sequence of moves reaches the highest
welfare level in which parameter constellation.
Proposition 9 1. Welfare is higher in the equilibrium of the subgame where the fa-
vorite moves early than in the case of simultaneous choice, i.e., WE > W S, if and
only if θ > nn−1 .
2. Welfare is higher in the equilibrium of the subgame where the favorite moves early
than when she moves late, i.e., WE > WL, if and only if θ > n+1+
√
n2−6n+13
2(n−1) .
3. The welfare level in the equilibrium of the simultaneous move game is (strictly)
higher than in the game with the favorite moving late, i.e., W S > WL if and only
if θ < 2n−3n−1 .
Proof. See appendix.
Now from Proposition 9, we can conclude that
82
Theorem 11 Let n ≥ 3. Then, subgame S brings about the highest welfare in equilib-
rium, if θ ∈
(
1, nn−1
)
and subgame E brings about the highest welfare in equilibrium, if
θ > nn−1 .
By comparing these results with Theorem 10, we find under which circumstances
equilibrium behavior leads to the subgame with highest welfare:
Corollary 10 Consider in both variants the subgame-perfect equilibrium to be played.
Then in variant 1, the subgame with the highest welfare is played if θ < nn−1 , while in
variant 2, the subgame with the highest welfare is played if θ > nn−1 .
Proof. The proof follows directly from Theorem 10 and Theorem 11.
This is especially interesting for the design of contest rules. Consider an institution
that has to constitute the rules of a rent-seeking contest with the above described set of
players. Let the appropriate social welfare function be given by (7.22) and the institution
has to decide whether the favorite should be allowed to make an early move in t = 0.
Now Corollary 10 tells us that the institution should allow the favorite to move early,
if and only if θ ≥ nn−1 . Note that in this scenario, the favorite will be the only player
choosing a poitive effort.
7.6 Conclusions
This chapter analyzes a game where a favorite faces a number of identical opponents in
a sequential rent-seeking contest. First, we focus on the incentives of the favorite given
that the underdogs choose a fixed timing. We find that the favorite will always prefer
an early move to both simultaneous and late move, if there are at least two underdogs.
She will only prefer the late move to simultaneous move, if her valuation is so high that
her effort in the simultaneous game would be higher than the sum of the effort levels
chosen by her opponents. Furthermore, we show that the underdogs do not have an
incentive to deviate from an early move. Therefore the presented results also represent
a subgame-perfect equilibrium if the underdogs are also allowed to choose their timing,
too. So far, the model generalizes parts of the results by Leininger (1993) and Chapter
6.
Furthermore, we discuss the role of a first-mover-right. We find that such a right will
be exerted by the favorite if offered. From a welfare point of view, it should be granted
that, whenever the favorite chooses such a high effort level if she moves early, she will
be the only player who chooses a strictly positive effort. This result tells us that only
one firm should actively participate in the contest. However, since this player preempts
her opponents, she does not exert zero effort (as she would do if she was the only player
at all, i.e. n = 1), but a not negligible positive effort level. This is because her passive
opponents still play an important role in the contest, since they threaten to choose a
positive effort once the favorite reduces her own effort level. In this situation, the favorite
wins with probability 1. If the valuation of the favorite is so low that it does not pay for
her to preempt her opponents, then a simultaneous move game creates a higher welfare.
For this case, it would be optimal not to grant a first-mover-right. This would result in
83
simultaneous moves on the equilibrium path, since a late move would not pay for the
favorite.
84
Chapter 8
When is the Assumption of
Simultaneous Moves in Contests
Justfiable?
8.1 Introduction
The previous chapters have analyzed the sequential structure of rent-seeking contests
with (a) three heterogenous players and (b) n players but only two different types of
players. In the next step, we will look at the incentives that a sequential structure
creates in a contest with n different types of players. Due to complexity, we will focus
on the question whether or not sequential structure will occur in a subgame-perfect
equilibrium at all. Many analyses in contest theory assume ex ante that players choose
to move simultaneously without knowing their opponent’s choice (e.g. Tullock, 1980,
Stein, 2002). But the previous analysis as well as earlier literature (e.g. by Dixit,
1987, Baik and Shogren, 1992, Leininger, 1993) show that simultaneous moves are not
necessarily played on the equilibrium path. These results especially of Baik and Shogren
and Leininger undermine the ad-hoc assumption of simultaneous moves since we now
know that, under full information, two players will always arrange themselves in an order
of moves different from the simultaneous move game. Yet, these results are restricted
to two players. The results of the Chapters 6 and 7 indicate that the simultaneous
move game can indeed be part of a subgame-perfect equilibrium without incentives to
deviate. Our analysis will now analyze under which general conditions a subgame with
simultaneous moves can be reached on the subgame-perfect equilibrium path.
8.2 When will Late Move be Preferred?
Since this chapter builds up on the previous chapters, the payoff structure of the game we
analyze here is also described by a Tullock-contest (Tullock, 1980) with constant returns
to scale. As above, let the number of active players be given by n and each player i’s
payoff be given by (6.1). Now, let every player have the opportunity to either commit to
85
their effort in an early point of time, t = 1, or to delay the decision to the late timing,
t = 2. Let us assume that every player knows who decides in t = 1 and who decides in
t = 2, when she chooses her effort.
As the aim of analysis is to determine the set of constellations where the simultaneous
move game occurs on the equilibrium path we should make clear upfront what this means:
A simultaneous move game is played on the equilibrium path if every player chooses
her effort at the same point of time and no single player has an incentive to deviate from
her strategy. This can occur in two possible ways. The first one is that every player
moves in t = 1 and no one has an incentive to deviate to t = 2. The other one is that
every player moves in t = 2 and no one has an incentive to deviate to t = 1. To be able
to identify the presence of such incentives, we first have to know the payoffs that players
can gain from the simultaneous move game. The Nash-equilibrium of this game has
already been studied by Stein (2002). By denoting the number of active players by n, we
can write the effort that an active player i spends in the equilibrium of a simultaneous
move game as:
xSi =
n− 1
∑n
j=1
1
Vj
−


n− 1
∑n
j=1
1
Vj


2
1
Vi
. (8.1)
From this, we can calculate the aggregate effort as
∑n
j=1 xj =
n−1∑n
j=1
1
Vj
and the payoff of
an individual player i as
piSi = Vi

1−
n− 1
Vi
∑n
j=1
1
Vj


2
, ∀i = 1, .., n. (8.2)
Now assume that all players would choose t = 1. If, without loss of generality, player
1 would deviate, she would be the only player to move late. In this case, player 1 will
take the aggregate effort of her opponents’ as given. We denote it by A−1 :=
∑n
i=2 xi.
Her decision will then be
∂pi1(x)
∂x1
=
A−1
(x1 + A−1)
2V1 − 1 = 0
⇔ x1 =
√
A−1V1 − A−1. (8.3)
Her opponents, of course, anticipate this in the first stage and are also able to see how
this action depends on their own effort xi. Let us now look at how a player i = 2, ..., n
will behave in this situation, reacting to the aggregate effort of the (n− 2) early movers
A−i :=
∑n
j=2,j 6=i xj. Note that A−i = A−1 − xi.
So, the payoff of player i moving early can be written as
pii(x) =
xiVi
√
(A−i + xi)V1
− xi.
This payoff function is maximized by
⇔ xi = 2A−1 − 2
√
V1
Vi
(A−1)
3
2 , (8.4)
86
as shown in the appendix.
Similarly to Stein (2002), we can now determine the aggregate effort in the early
period by summing up (8.4) for every player i = 2, ..., n. Denoting Ω =
∑n
i=2
1
Vi
we
receive:
A−1 :=
n∑
i=2
xi = 2(n− 1)A−1 − 2A
3
2
−1
√
V1Ω⇔ 2
√
A−1V1Ω = 2n− 3
⇔
√
A−1 =
2n− 3
2
√
V1Ω
.
Thus,
A−1 =
(2n− 3)2
4V1Ω2
. (8.5)
Then, the payoff of an early mover i is given by:
pi1Li =
1
Vi
2n− 3
ΩV1
(
Vi −
2n− 3
2Ω
)2
.
On the other hand, this enables us to calculate the effort of the late mover by:
x1L1 =
√
A−1V1 − A−1 =
2n− 3
2Ω
(
1−
2n− 3
2ΩV1
)
.
It is only non-negative for
V1 ≥
2n− 3
2
(∑n
j=1,j 6=i
1
Vj
) .
The payoff player 1 recieves in this situation is given by:
pi1L1 =
√
A−1V1 − A−1√
A−1V1
V1 −
√
A−1V1 + A−1
= V1 − 2
√
A−1V1 + A−1 =
(√
V1 −
√
A−1
)2
,
which holds for any aggregate effort A−1 that players spend on the early stage. Inserting
(8.5), we receive player 1’s payoff in equilibrium:
pi1L1 =


√
V1 −
√
(2n− 3)2
4V1Ω2


2
= V1
(
1−
2n− 3
2V1Ω
)2
.
This enables us to compare the payoffs from the simultaneous move game to the one,
where one player moves later.
Lemma 13 Let n players participate in a rent-seeking contest with constant returns to
scale. Then, we find that player 1 moving late alone brings about a (strictly) higher
equilibrium payoff than the simultaneous move game, iff
V1 ≥ (>)
2n− 3
∑n
i=2
1
Vi
.
87
Proof. See appendix.
Note that this threshold value reduces for n = 2 to V1 ≥ V2, which underpins the
result of Leininger (1993). Yet, for n ≥ 3, we find that the threshold value is strictly
above the harmonic mean of the opponent’s valuations such that there is a substantial
set of combination, where no player exceeds the threshold value.
From this result, we are able to conclude:
Proposition 10 Let n players participate in a rent-seeking contest with simultaneous
moves. If there is no player with V1 > 2n−3∑n
j=2
1
Vj
, then no player has an incentive to deviate
to a later move.
In the previous chapters, we found that a player will only choose a late move instead
of a simultaneous move game if she spends a higher effort in the equilibrium of the
simultaneous move game than all her opponents together. We are also able to find a
corresponding result in this model:
Corollary 11 A player will prefer a late move to a simultaneous move, if her equilibrium
effort in the simultaneous move game exceeds the aggregate effort of her opponents, i.e.,
if her winning probability in the simultaneous move game exceeds 12 .
Proof. See appendix.
Now let us shed some light on the question whether a deviation to an earlier point
in time can be profitable.
8.3 When will Early Move be Preferred?
In this section, we will check whether a situation where all players move late can also
be an equilibrium. If a player would deviate from this equilibrium, we would end up in
a subgame, where one player moves early and n − 1 players move late. Now assume,
without loss of generality, that player 1 moves early on her own. Let A =
∑n
i=1 xi and
remember that Ω :=
∑n
i=2
1
Vi
and A−1 =
∑n
i=2 xi. Then the first-order-condition of a late
mover i maximizing (6.1) is:
(A− xi)Vi = A
2 ⇔ xi = A−
A2
Vi
, i = 2, ..., n.
Summing this up for i = 2, ..., n yields:
A−1 = (n− 1)A− A
2Ω.
From the definitions, it is obvious that A := A−1 + x1. Then, it follows that
A−1 =
n− 2
2Ω
− x1 +
√
x1
Ω
+
(n− 2)2
4Ω2
.
88
The early mover’s payoff function is then given by
pi1 =
x1
A−1 + x1
V1 − x1 =
2Ωx1
(n− 2) +
√
4Ωx1 + (n− 2)2
V1 − x1.
The first-order-condition is then given by ∂pi1∂x1 = 0, which can be transformed to
x1 =
Ω2V 21 − (n− 2)
2
4Ω
.
The resulting payoff is given by:
pi1E1 =
(ΩV1 − (n− 2))
2
4Ω
.
The payoff from the simultaneous move game is given by (8.2), which for player 1
can be written as: piS1 = V1
(
1− n−1ΩV1+1
)2
. One can show:
Lemma 14 Let n players participate in a rent-seeking contest. We will find that an
early move always brings about a higher equilibrium payoff than the simultaneous game.
Unless n = 2 and V1 = V2, the difference is strictly positive.
Proof.
pi1E1 > pi
S
1 ⇔
(ΩV1 − (n− 2))
2
4Ω
> V1
(ΩV1 + 1− n+ 1
ΩV1 + 1
)2
⇔ (ΩV1 + 1)
2 > 4ΩV1
⇔ Ω2V 21 + 2ΩV1 + 1 > 4ΩV1 ⇔ Ω
2V 21 − 2ΩV1 + 1 > 0⇔ (ΩV1 − 1)
2 > 0,
which is strictly fulfilled unless ΩV1 = 1. The latter is equivalent to
∑n
i=2
1
Vi
= 1V1 , which
necessarily holds for all but one player, if n ≥ 3.
We learnt that, whenever there are more than two players, there is always a player
who has a strict incentive to move early in the considered situation. Even for n = 2,
Leininger (1993) showed that simultaneous moves in t = 2 can only be equilibrium if
V1 = V2. Yet, this is not a unique equilibrium.
Theorem 12 A simultaneous move in t = 1 is only part of a subgame-perfect equilib-
rium if all players are rather homogenous, i.e. no player with V1 > 2n−3∑n
i=2
1
Vi
exists. A
simultaneous move in t = 2 is never part of a subgame-perfect equilibrium.
In line with Corollary 8 and 9, we are able to find that this result is also driven by
the strongest player exerting more than half of the effort spent in a simultaneous move
game.
Corollary 12 In a sequential rent-seeking contest, players will coordinate their timing
to a simultaneous move game if, in such an equilibrium, no player spends a higher effort
than all of her opponents together, i.e., no player has a winning probability of more than
1
2 .
Proof. See appendix.
This enables us to conclude state that, for rather homogenous populations of at least
three players, the assumption of simultaneous move can indeed be justified.
89
8.4 Conclusion
In this chapter, we looked at a sequential contest with an arbitrary number of het-
erogenous players and focused on the question: under which conditions are simultaneous
moves played on the equilibrium path of a subgame-perfect equilibrium? To this aim,
we looked at the incentives a player has to deviate from a simultaneous move game.
We found that, if deviation to an earlier point of time is possible, there is always a
player that has a strict incentive to move earlier than her opponents if the number of
players exceeds two. Even for two players, Leininger had shown, that such a situation
cannot be a prominent equilibrium, but only one of many possible equilibria with equal
payoff structure if players are homogenous.
Yet, we found that, if there is no earlier point in time simultaneous moves can indeed
be played in the subgame-perfect equilibrium if the players are sufficiently homogenous.
As we already observed in Chapters 6 and 7, we find that also in this setting a player is
only willing to choose a later timing than her opponents if her effort in the simultaneous
move game is higher than the aggregate effort of her oponents. Otherwise, the ad-hoc
assumption of simultaneous moves can indeed be justified by an endogenously chosen
order of moves.
90
Chapter 9
Conclusion
This work contributes to two different branches of literature in contest theory. As a com-
mon feature, both parts focus on rent-seeking contests with a Tullock-contest-function,
assuming that the expended effort is wasted.
In the first part, we focus on interdependent preferences. At first, we build up a
model where individuals of two homogenous groups compete for a rent. One group has
independent preferences, while the other one has negatively interdependent preferences.
The aim of this analysis is to point out under which conditions the spiteful players
attain a higher material payoff. We find that there is a broad range of situations where
spiteful preferences experience this strategic advantage, especially for two-player-contests
and contests with non-increasing returns to scale. For increasing returns to scale, we
discuss all possible forms of intragroup symmetric Nash-equilibria (concerning the order
of efforts) and point out under which conditions they can or cannot occur. Furthermore,
we discuss possible applications of this result, which can be interpreted as a strategic
advantage to negatively interdependent preferences, such as the implications that the
possibility of choosing spiteful preferences has on decisions concerning delegation.
In the next chapter, we relax the assumptions of only two different types and the
restriction to weakly spiteful preferences for a contest with linear preference functions.
We find that, among participating players, more spiteful players receive a higher material
payoff, as long as underdissipation is observed. This happens especially in the absence
of strongly spiteful players. In contrast, if the latter dominate the game, the less spite-
ful players will receive a higher payoff, although it is negative. Furthermore, we show
that, for each player with an interdependent preference, there exists a valuation (differ-
ent from the true valuation) such that a player with independent preferences and this
valuation behaves like the corresponding player with interdependent preferences. This
interdependency equivalent valuation is higher than the true valuation if the player is
spiteful and lower if the player is altruistic. Concerning the participation of players, we
find that for constant returns to scale positive effort choice cannot be taken for granted.
We show that a player quits active participation if the dissipation level is higher than
the interdependency equivalent valuation.
The last section of the first part addresses the relevance of interdependent preferences
for evolutionary questions. We find that relative payoff maximization constitutes an
91
evolutionarily stable preference for a large scale of contests, including also contests with
increasing returns to scale. Note that relative payoff maximization results from exactly
that preference that separates weakly and strongly preference. In so far, this result goes
in line with the result that, in a population with only weakly spiteful players, a more
spiteful preference yields a higher equilibrium payoff, while in a population with only
strongly spiteful preferences, the reverse holds.
The second part of the work analyzes what happens if the order of moves is endo-
genized into a contest with more than two players. We find that, for more than two
players, there is always a range of parameter constellations in the n-player-case where
players will choose to move simultaneously in the subgame-perfect equilibrium, namely
if players are rather homogenous in the sense that the valuations do not differ too much.
For the two special cases for which we provide a welfare analysis (in Chapter 6 for
three players, and in Chapter 7 for n players, where n − 1 players are equal), we find
that a simultaneous move game indeed represents the order of moves that leads to the
highest aggregate paoyff if it played on the equilibrium path.
The first analysis we present in this part focuses on a contest with three players
that can choose their timing between an early and a late move. We find that the
strongest player has an incentive to choose late if she is stronger than her opponents
together (in terms of effort chosen in a simultaneous move game). If, however, no player
dominates her opponents, all players will choose to move early in a subgame-perfect
equilibrium. For a small range of parameter constellations, we find that not only the
strongest but also the second strongest can have an incentive to move late. For this
case, an equilibrium comprises mixed strategies cocerning the timing decision. We also
compare the welfare in terms of aggregate payoff of the feasible subgames and find that,
whenever the simultaneous move game is played, this is the best subgame, while for more
heterogenous players, the subgame played on the equilibrium path need not be the one
with the highest welfare.
In the next chapter, we look at the incentive of a strong player facing (n− 1) equal
opponents. As in the case of three players, she will only move late in equilibrium if
she is stronger than her opponents together. In addition, we analyze the effect of a
first-mover-right of the stronger player. We find that the favorite will always exert
the first-mover-right if granted. If the favorite is so strong that an early move leads
to preemption of her opponents than this situation even raises the welfare level above
the level of the equilibrium without first-mover-right. In this case, the first-mover-right
should be granted to the strongest player.
Finally, in Chapter 8, we look at a game between n heterogenous players with an
endogenized order of moves. We analyze under which condition it can be an equilibrium
that all players coordinate to the same timing of effort choice, which is a standard
assumption in many models of rent-seeking contests. In line with the preceding chapters,
we come to the result that simultaneous move can only occur on the equilibrium path,
if there is no player who chooses an equilibrium effort higher than the aggregate effort
of her opponents. We provide a general condition telling us when a simultaneous game
will be played on the subgame-perfect equilibrium path. This condition generalizes both
the conditions for three players and for (n− 1) equal players. Furthermore, we find that
92
simultaneous moves can only occur on the earliest possible point of time, since there
is always an incentive for any player to deviate to an earlier timing. It is remarkable,
that for a population with only homogenous players, simultaneous moves will always be
played on the subgame-perfect equilibrium path.
In the introduction, we already alluded to some features of election campaigns, that
we regard in our contest models. For examples, we adduced that, despite the political
closeness, the Democratic candidates for the presidential election, Barack Obama and
Hillary Clinton, do not shrink back from worsening the opponent’s position. As our
theoretical analysis showed, it can indeed be reasonable for a contestant to intentionally
harm one’s opponents and not only to focus on one’s own payoff. Spiteful players, if
they are not too spiteful, will be more successful in winning contests than players with
independent preferences, and the same seems to hold for politicians. The evolutionary
analysis of Chapter 5 shows that spiteful preferences, that induce a behavior equivalent
to relative payoff maximization are an evolutionarily stable preference, if an evolutionary
process selects players according to their preferences.
Applying this to the example of presidential candidates, it is obvious that a candidate
for presidency will have to endure quite a number of contests to get into the position
where they are able to enter the contest for the candidacy of the party (e.g. to be
an esteemed senator). These contests weed out the less successful candidates, meaning
Obama and Clinton have already gone through an evolutionary process. Such a process
filters out politicians that behave like relative payoff maximizers as the most successful
ones.
For the second part of our work, the theory cannot be applied to the US-presidential
election campaign, because the pre-election campaign has a more complex structure than
our model, while in the major presidential campaign that started in August 2008 there
were only two candidates left, while our findings only regard at least three players. But,
the theory is applicable to three-party-systems. A typical example for a three-party-
system can be found in Eastern Germany. If we measure the strength of a party in a
state by its number of seats in the parliament, then CDU was the strongest party in
Saxony-Anhalt before the last parliament election with 48 seats, while the SPD and the
PDS were of equal sterngth with 25 seats. For this combination, our theory predicts,
that both smaller parties will move simultaneously while the CDU moves later. This
indeed happened. Beyond the scope of election campaigns, there are far more fields
where contest theory plays a role. So, Chapter 7 can make a contribution on whether
a market incumbent should be granted a first-mover-right if a contest for the market
license is played. Yet, the most general result of our analysis of sequential structures
is that whenever there is no player who exerts a higher effort than the aggregate effort
of her opponents, simultaneous moves will be played on the equilibrium path. This can
serve as justification for the ad-hoc-assumption of simultaneous moves that is often made
in the literature on contest theory.
The work at hand delivers an extended understanding of the role that interdependent
preferences and endogenized order of moves play in rent-seeking contests, especially if
more than two players are involved. However, we restricted our analysis of the sequential
structure to a discrete set of two points in time. As an extension, it would be conceivable
93
to extend the analysis of Chapter 6 to three timings. Furthermore, it is not clear what
will happen in equilibrium if a first point in time is not clearly defined. As an example,
one might fall back on the order of moves in election campaigns. In general, there is not
a real first point of time where an election campaign can be started earliest. Yet, it is
not clear what this means to the equilibrium behavior in our model, which says that,
for more than two players, there are always at least two players who want to choose as
early as possible.
94
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100
Chapter A
Appendix
A.1 Definition of a strategy
In our context, a strategy (t, x) is a combination of a timing decision t and an effort
choice x. The timing decision t can be viewed upon as a function t : V → T . V is
the set of vectors of valuation that are feasible. Most generally, it is given by IRn+ for a
contest with n players, but we will restrict our analysis for the main parts of our work
to the most interesting cases, where players do not differ in valuation too much, such
that all players still have an incentive to choose a positive effort. Since, in this part of
our work, we assume players to differ only with respect to their valuation, V can also be
interpreted as the set of players.
T represents the set of timings from which players can choose their actual timing
decision. In most cases, we restrict our analysis to T = {1, 2}, yet in Chapter 7, we will
also use T = {0, 1, 2}.
The effort choice x is a function that maps valuations and timing choices of all players
and already executed effort choices of opponents who decided for an earlier move to the
non-negative real numbers. Depending on the chosen timing decision tˆ, it is characterized
by
x(tˆ) : V × T × X∧t → IR≥0,
where X∧t represents the set of feasible effort vectors of players who chose t < tˆ.
A.2 Equilibria where V1 > 2V3
A.2.1 The subgames
The aim of this section is to derive the equilibria of the game where three players play
a sequential rent-seeking contest with endogenous timing and V1 > 2V3. The regarded
subgames are the following cases:
At first, let us analyze the equilibria for every single subgame. Whenever helpful, we
will fall back on wht learned from Chapter 6.
101
Player 2
t = 1 t = 2
Pl. t = 1 SE 2L
1 t = 2 1L 3E
Player 3 moves in t = 1.
Player 2
t = 1 t = 2
Pl. t = 1 3L 1E
1 t = 2 2E SL
Player 3 moves in t = 2.
The subgames SE and SL In the simultaneous move game, player 3 quits, if V3 ≤
V1V2
V1+V2
. This can be shown to be strictly smaller than V12 , if V1 > V2.
This means that inactivity of player 3 does not follow from V1 > 2V3. For V3 > V1V2V1+V2 ,
the results from Chapter 6 can be apllied, while for the case where it is indeed optimal for
her to be inactive, we find, that players 1 and 2 play a standard two-player-contest. The
aggregate effort will be x1 + x2 = V1V2V1+V2 , and thus player 3, who has a lower valuation, is
not willing to interfere in the two-player-contest.
The subgame 3L In the inner solution that fulfills the first-order-conditions for an
equilibrium in the three-player-game, player 3 is active, if and only if
V3 ≥
3
2
V1V2
V1 + V2
3
2
V1V2
V1 + V2
<
V1
2
⇔ V1 > 2V2.
This means for V1 > 2V3 player 3 will only be active if player 2 will be inactive.
This can be shown to lead to a contradiction whenever V3 ≤ V2. In this subgame, we
will not observe a game with three, but only with two, active players. Yet, this does
not mean that these two players act as if in a true two-player-contest. If players 1
and 2 ignore 3, they choose x1 + x2 = V1V2V1+V2 , as analyzed by Leininger, 1993. Yet for
V3 ∈
(
V1V2
V1+V2
, 32
V1V2
V1+V2
)
, we find that player 3 has an incentive to choose a positive effort
if the aggregate effort of players 1 and 2 is V1V2V1+V2 . As a result, players 1 and 2 increase
their effort above the level in the two-player-game in order to preempt player 3. An
aggregate effort of x1 + x2 = V3 is needed to ensure the inactivity of player 3. Yet, this
does not tell us how the aggregate effort is divided between the players. To characterize
an equilibrium, we should find for both players 1 and 2 that their payoffs decrease if
they a) increase or b) decrease their effort, given the effort of her active opponent and
the reaction function of player 3.
a) An increase of the effort would not induce a reaction from player 3. Thus, the
marginal change in payoff is given in this situation by ∂pii∂xi =
xj
(xi+xj)2
Vi− 1, which should
become non-positive in order to make an increase in effort non-profitable. This means:
∂pi1
∂x1
=
x2
(x1 + x2)2
V1 ≤ 1⇔ x2V1 ≤ (x1 + x2)
2
⇔ x2 ≥
(x1 + x2)2
V1
, (A.1)
102
and by symmetry:
x1 ≤
(x1 + x2)2
V2
. (A.2)
b) If a player decreases her effort, player 3 has an incentive to enter the game,
and to choose x3 =
√
(x1 + x2)V3 − (x1 + x2), such that the aggregate effort will be
∑3
i=1 xi =
√
(x1 + x2)V3. The marginal payoff of a change in effort by player 1 is thus
given by:
∂pi1(x1, x2, x3(x1, x2))
∂x1
=
∂
∂x1
x1
√
V3(x1 + x2)
V1 − x1 =
V1
(√
x1 + x2 − x12√x1+x2
)
√
V3(x1 + x2)
− 1.
This should be positive, such that a decreasing effort reduces the payoff of the player:
V1
(√
x1 + x2 − x12√x1+x2
)
√
V3(x1 + x2)
≥ 1⇔ V1 −
x1V1
2(x1 + x2)
≥
√
V3(x1 + x2)
⇔
x1V1
2(x1 + x2)
≤ V1 −
√
V3(x1 + x2)
⇔ x1 ≤ 2(x1 + x2)−
2(x1 + x2)
V1
√
V3(x1 + x2), (A.3)
and by symmetry:
x2 ≤ 2(x1 + x2)−
2(x1 + x2)
V2
√
V3(x1 + x2). (A.4)
Now, in an equilibrium, it has to hold that x1 + x2 = V3, such as (A.1), (A.2), (A.3)
and (A.4). One can show that, from (A.1), (A.2) and x2 = V3 − x1, it follows that
V3 −
V 23
V1
≤ x1 ≤
V 23
V2
, (A.5)
while from (A.3), (A.4) and x2 = V3 − x1, it follows that
V3 − 2V3
V2 − V3
V2
≤ x1 ≤ 2V3
V1 − V3
V1
. (A.6)
Therefore, it is necessary for the existence of an equilibrium that
V3 −
V 23
V1
≤
V 23
V2
∧ V3 − 2V3
V2 − V3
V2
≤ 2V3
V1 − V3
V1
.
⇔ V3 ≤ V
2
3
V1 + V2
V1V2
∧ V1V2 − 2V1(V2 − V3) ≤ 2V1V2 − 2V2V3
⇔ V1V2 ≤ V3V1 + V3V2 ∧ (2V1 + 2V2)V3 ≤ 3V1V2
103
⇔ V3 ≥
V1V2
V1 + V2
∧ V3 ≤
3
2
V1V2
V1 + V2
,
which is exactly the interval where the problem occurs.
Now, we have to show that both intervals defined by (A.5) and (A.6) intersect. This
holds since V3 −
V 23
V1
= V3 V1−V3V1 ≤ 2V3
V1−V3
V1
, which holds trivially, and
V3 − 2V3
V2 − V3
V2
≤
V 23
V2
⇔ V2V3 − 2V2V3 + 2V
2
3 ≤ V
2
3 ⇔ V
2
3 ≤ V3V2 ⇔ V3 ≤ V2,
which holds per definition. Note that V
2
3
V2
< 2V3 V1−V3V1 ⇔ V3 < 2
V1V2
V1+2V2
, which always holds
if V3 < V2 and V3 < V12 . Then, we can summarize that, if V3 ∈
(
V1V2
V1+V2
, 32
V1V2
V1+V2
)
, a Nash-
equilibrium has the following properties: player 1 will choose some x1 ∈
(
max
(
V3 V1−V3V1 , V3 − 2V3
V2−V3
V2
)
, V
2
3
V2
)
,
while player 2 will choose x2 = V3 − x1. Note that this implies non-uniqueness of the
Nash-equilibrium.
For V3 ≤ V1V2V1+V2 , players 1 and 2 play a standard two-player-contest.
The subgame 2L In this subgame, player 3 quits if V3 < V12 . Thus, we find that
player 3 is inactive. Player 1 and 2 play a sequential game where they need not expect
interference by player 3. Aggregate effort is V12 (see Leininger, 1993), the payoff of player
2 is given by (2V2−V1)
2
4V2
if V2 > V12 , elseif 0.
The subgame 1L For V3 > V22 every player is active in this subgame’s Nash-equilibrium
(cf. Chapter 6). For V3 ≤ V22 , player 3 quits. Player 2 and 1 play a two-player sequential
game, with an aggregate effort of V22 > V3. Hence, player 3 has no incentive to join the
contest.
The subgame 1E Player 3 quits if V3 ≤
V1−V2+
√
V 21 +6V1V2+V
2
2
4 . This holds since
V1 − V2 +
√
V 21 + 6V1V2 + V
2
2
4
>
V1
2
⇔ V1 − V2 +
√
V 21 + 6V1V2 + V
2
2 > 2V1
⇔
√
V 21 + 6V1V2 + V
2
2 > V1 + V2 ⇔ V
2
1 + 6V1V2 + V
2
2 > V
2
1 + 2V1V2 + V
2
2 .
which holds for strict positive valuations. Player 3 quits and has no incentive to interfere
in the two-player-contest equilibrium
The subgame 2E Player 3 quits active participation if:
V3 ≤
V2 − V1 +
√
V 21 + 6V1V2 + V
2
2
4
V2 − V1 +
√
V 21 + 6V1V2 + V
2
2
4
>
V1
2
⇔ V2 − V1 +
√
V 21 + 6V1V2 + V
2
2 > 2V1
104
⇔
√
V 21 + 6V1V2 + V
2
2 > 3V1−V2(⇔)V
2
1 + 6V1V2 +V
2
2 > 9V
2
1 − 6V1V2 +V
2
2 ⇔ V2 >
2
3
V1.
Thus, if V2 is rather small compared to V1, player 3 has an incentive not to quit partici-
pation.
For V2 > 23V1, if player 3 who is absent is ignored, players 1 and 2 play a stan-
dard two-player-contest where the aggregate effort will be V22 , which is smaller than
V2−V1+
√
V 21 +6V1V2+V
2
2
4 . Thus, for V3 ∈
(
V2
2 ,
V2−V1+
√
V 21 +6V1V2+V
2
2
4
)
, player 3 does not have
an incentive to choose a positive effort in the three-player-case, but to interfere in the
two-player-contest.
Thus, in equilibrium, players 1 and 2 will choose their efforts, such that the sum of
efforts will be V3. To this aim, player 2 chooses her effort, such that the reaction of her
opponents in t = 2 leads to:
x2 + x1(x2) + x3(x2) = V3,
which leads to x3(x2) = 0.
x2 + x1(x2) + x3(x2) = V3 ⇔
V 21 V3 + V1V
2
3 − 4V1V3x− 2V
2
1 x2 − 2V
2
3 x+ (V1 + V3)
√
V1V3 (V1V3 + 4(V1 + V3)x2)
2(V1 + V3)2
+ x2
=
V1V3(V1 + V3)
2(V1 + V3)2
− 2x2
V 21 + 2V1V3 + V
2
3
2(V1 + V3)2
+ x2 +
√
V1V3 (V1V3 + 4(V1 + V3)x2)
2(V1 + V3)
= V3
⇔ V1V3 +
√
V1V3 (V1V3 + 4(V1 + V3)x2) = V3 (2V1 + 2V3)
⇔
√
V1V3 (V1V3 + 4(V1 + V3)x2) = V3 (V1 + 2V3)
⇔ V 21 V
2
3 + 4V1V3x2(V1 + V3) = V
2
3
(
4V 23 + 4V3V1 + V
2
1
)
⇔ 4V1V3(V1 + V3)x2 = 4V
4
3 + 4V
3
3 V1 ⇔ x2V1(V1 + V3) = V
2
3 (V1 + V3)⇔ x2 =
V 23
V1
.
Inserting this into the reaction function of players 1 and 3 yields x3 = 0 and x1 =
V3(V1−V3)
V1
.
Now, we have to show that this is optimal from the point of view of player 2 to choose
x2 =
V 23
V1
. Her payoff is then given by pi2 =
V3(V2−V3)
V1
. For a deviation from this effort to
a lower effort, player 3 will choose positive effort, and the payoff of player 2, depending
on her own effort, and her opponent’s reaction function is given by:
pi2 =
x2 · 2(V1 + V3)
V1V3 +
√
V1V3 + 4(V1 + V3)x2
V2 − x2
∂pi2
∂x2
= 2(V1 + V3)V2
V1V3 +
√
V1V3 + 4(V1 + V3)x2 −
V1V34(V1+V3)x2
2
√
V1V3+4(V1+V3)x2
(
V1V3 +
√
V1V3 + 4(V1 + V3)x2
)2 − 1.
105
For x2 =
V 23
V1
, this should be (at least weakly) positive, such that deviation for player 2
is not profitable.
∂pi2
∂x2
(
V 23
V1
)
≥ 0
⇔ 2 (V1 + V3)V2



V1V3 +
√
V1V3 + 4(V1 + V3)
V 23
V1
−
V1V34(V1 + V3)
V 23
V1
2
√
V1V3 + 4(V1 + V3)
V 23
V1




≥

V1V3 +
√
V1V3 + 4(V1 + V3)
V 23
V1


2
⇔ 2 (V1 + V3)V2
(
V1V3 + V3(V1 + 2V3)− 2V
2
3
V1 + V3
V1 + 2V3
)
≥ 4V 23 (V1 + V3)
2
⇔ V2
(
1−
V3
V1 + 2V3
)
≥ V3
⇔ V 23 + (V1 − V2)
V3
2
−
V1V2
2
≤ 0
⇒ V3 ≤
V2 − V1 +
√
V 21 + 6V1V2 + V
2
2
4
.
Deviation to a higher payoff cannot be profitable as well, since player 3 would not
have an influence on players 1 and 2 any more. Yet, if it was optimal despite the absence
of player 3, it should also occur in the two-player-case. This does not happen, and thus,
player 2 has no incentive to deviate from x2 =
V 23
V1
.
For V3 < V22 , a standard two-player sequential game is played by players 1 and 2,
while for V3 >
V2−V1+
√
V 21 +6V1V2+V
2
2
4 , a standard three-player sequential game is played.
The subgame 3E Player 3 chooses a positive effort if V3 > V1V2V1+V2 . According to
the main findings of subgame SE, we find here that, for this case, the three-player-case
(with three active players) is played, while for V3 ≤ V1V2V1+V2 , the players 1 and 2 play a
two-player-contest.
A.2.2 The equilibrium path
Let us now focus on which subgame is stable on the equilibrium path and in which
subgame deviation pays for at least one player. To this aim, we will check for every
subgame whether any player has an incentive to deviate.
106
Is SE stable? Subgame SE is a three-player-game, if V3 > V1V2V1+V2 . If not, only player 1
and 2 participate actively.
A deviation of player 1 would lead to subgame 1L. For the latter case, i.e., V3 ≤ V1V2V1+V2 ,
we find that, for V3 ≤ V22 this would also be a two-player-game, and we can apply
Leininger’s (1993) result. Then SE is not stable.
Now for V3 ∈
[
V2
2 ,
V1V2
V1+V2
]
, player 1 decides between a simultaneous two-player-game,
where she receives V
3
1
(V1+V2)2
and moving late in a three-player-game, receiving (2V1(V2+V3)−3V2V3)
2
4V1(V2+V3)2
=
V1
(
1− 34
V2V3
V1(V2+V3)
)
. Deviation pays if
V 31
(V1 + V2)2
< V1
(
1−
3
4
V2V3
V1(V2 + V3)
)
⇒ V 21 (V2 + V3) < V1(V1 + V2)(V2 + V3)−
3
4
V1V2V3 −
3
4
V 22 V3
⇔ 0 < V1V
2
2 + V1V2V3 −
3
4
V1V2V3 −
3
4
V 22 V3
V1V2V3
4
+ V 22
(
V1 −
3
4
V3
)
> 0,
which holds since V1 > V3.
For V3 > V1V2V1+V2 player 3 is active in subgame SE as well as in 1L. The analysis of
Chapter6 can thus be applied to show that SE is stable if V1 ≤ 3 V2V3V2+V3 .
Is 3L stable? For V3 ∈
[
V1V2
V1+V2
, 32
V1V2
V1+V2
]
, player 3 is active in SE, but not in 3L, thus
she will prefer moving early. For V3 > 32
V1V2
V1+V2
, all three players will be active in SE as
well as in 3L, but only the strongest player has an incentive to move later. Thus, 3L
cannot be stable.
For V3 < V1V2V1+V2 we find that
V2 − V1 +
√
V 21 + 6V1V2 + V
2
2
4
>
V1V2
V1 + V2
⇔ (V1 + V2)
√
V 21 + 6V1V2 + V
2
2 > V
2
1 + 4V1V − 2− V
2
2
⇔ 00(V 21 + 2V1V
2
2 + V
2
2 )(V
2
1 + 6V1V2 + V
2
2 ) > (V
2
1 + 4V1V − 2− V
2
2 )
2
= V 41 + 6V
3
1 V2 + V − 1
2V 22 + 2V
3
1 V2 + 12V
2
1 V
2
2 + 2V1V
3
2 + V
2
1 V
2
2 + 6V1V
3
2 + V
4
2
> V 41 + 8V
3
1 V2 + 14V
2
1 V
2
2 − 8V1V
3
2 + V
4
2
⇔ 8V 31 V2 + 14V
2
1 V
2
2 + 8V1V
3
2 > 8V
3
1 V2 + 14V
2
1 V
2
2 − 8V1V
3
2 .
In this case, player 3 will neither have an incentive to interact in 3L nor in 2E. Thus,
both subgames will be true two-player-games. From Leininger (1993), we know that in
this situation the stronger player will prefer late move. Thus, player 1 will deviate from
3L to game 2E. 3L will never be stable.
107
Is 2L stable? For V2 ≤ V12 , player 2 will be inactive in 2L, and thus she has an incentive
to deviate to SE. For V2 > V12 we have to distinguish for different valuations of player 3.
Let us first regard the case, where player 3 is active. Then, player 2 will prefer 2L to SE,
if (2V2−V1)
2
4V2
> (V2(V1+V3)−V1V3)
2
(V2(V1+V3)+V1V3)
2V2. Yet this never holds, if V2 ∈
[
V1
2 , V1
]
and V3 ≥ V1V2V1+V2 .
Now if player 3 is inactive in SE, then player 2 would receive
V 32
(V1+V2)
2 in SE. Deviation
to SE will thus pay if:
(2V2−V1)2
4V2
< V
3
2
(V1+V2)
2 . The latter is equivalent to
V 31 − 2V
2
1 V2 − 3V
2
1 V
2
2 + 4V
3
2 < 0,
which holds for V2 ∈
(
V1
2 , V1
)
.
Thus subgame 2L is never stable.
Is 1L stable? For V3 ≤ V22 , player 1 and 2 play a plain two-player-game, regardless of
when player 3 does (not) move. For this situation, the subgame 1L is stable!
For V3 > V22 , 1L is a three-player-game. Player 3 has no incentive to deviate, either
because her payoff is zero from deviation or from the analysis in the standard three-
player-case. Player 1 only has an incentive to deviate to subgame SE, if V3 ≥ V1V2V1+V2
(see above) and V1 ≤ 3 V2V3V2+V3 . A deviation of player 2 would lead to subgame 3E. This
pays only if pi3E2 > pi
1L
2 . Unfortunately, this can happen for some parameter ranges that
are too complex to be captured formally. Table B.1 gives an overview over the cases
where player 2 deviates from 1L to 3E, if V3 > V1V2V1+V2 . For V3 ∈
(
V2
2 ,
V1V2
V1+V2
)
, player 2
has to decide between a three-player-game where she moves early and a simultaneous
two-player-game at the late stage. This is also represented in the Table (as Maximum)
in appendix B of this work. Player 3 never has an incentive to deviate to 2E.
Is 1E stable? In subgame 1E, player 2 receives pi1E2 =
(2V2−V1)2
4V2
, if V2 ≥ V12 , and zero
elseif. The latter, of course, creates a strict incentive for her to deviate to 3L. But also
for V2 ≥ V12 , deviation to 3L would pay her:
• Regard V3 ∈ ( V1V2V1+V2 ,
3V1V2
2(V1+V2)
) first.1 Then, the minimal payoff she would get here
from a deviation, i.e., pi3L2 (x1 =
V 23
V2
, x2 = V3 −
V 23
V2
), is larger than pi1E2 :
pi3L2 (min) =
V3(V2 − V3)
V2
(V2
V3
− 1
)
=
(V2 − V3)2
V2
=
(2V2 − 2V3)2
4V2
(2V2 − 2V3)2
4V2
≥
(2V2 − V1)2
4V2
⇔ 2V2 − 2V3 ≥ 2V2 − V1 ⇔ V1 ≥ 2V3.
1 3V1V2
2(V1+V2)
is larger than V12 , if V2 is larger than
V1
2 .
108
• Now, consider V3 ≤ V1V2V1+V2 . Players 1 and 2 play a true two-player-game in 3L, where
player 2 receives: V
3
2
(V1+V2)2
. This is larger than pi1E2 =
(2V2−V1)2
4V2
for V2V1 ∈
[
1
2 , 1
]
, as
we show here:
V 32
(V1 + V2)2
≥
(2V2 − V1)2
4V2
⇔ V1
(
V 31 − 2V1V
2
2 − 3V1V
2
2 + 4V
3
2
)
≤ 0
⇔ 4V 41
(
V2
V1
+
√
17 + 1
8
)(
V2
V1
−
√
17− 1
8
)(V2
V1
− 1
)
≤ 0.
This holds for values from the regarded interval.
As a consequence, 1E is never stable.
Is 2E stable? For V3 > V22 , player 3 has a strict incentive to deviate to 1L. For the
case that player 3 is active in 2E it stems from the analysis of Chapter 6 that she will
prefer an early move. If on the other hand she chooses zero effort in 2E and V3 > V22
holds, then player 3 can receive a positive payoff by deviating and, thus, 2E cannot be
stable. For V3 ≤ V22 , she is inactive in both subgames 2E and 1L and, thus, indifferent.
Players 1 and 2 play a true two-player-game, regardless of when player 3 does (not)
move. For this situation, the subgame 2E is stable!
Is 3E stable? Assume first that V3 < V1V2V1+V2 . In this case, player 3 will be inactive
in this subgame, as well as in subgame 2L. Since player 1 has an incentive in the two-
player-game to move early, given that player 2 moves late, 3E is not stable in this area.
For larger values of V3, one can show that the payoff player 1 would gain from moving
early in two-player-game is larger than from subgame 3E with 3 active players. Subgame
3E is thus never stable.
Is SL stable? For V3 > V1V2V1+V2 we find that player 3 is active in SL as well as in 3E.
Thus, we can rely on the calculation of Chapter 6, which shows that she will deviate to
an earlier move.
For V3 ≤ V1V2V1+V2 , player 3 is inactive in SL. Now
V1V2
V1+V2
<
V2−V1+
√
V 21 +6V1V2+V
2
2
4 :
V1V2
V1 + V2
<
V2 − V1 +
√
V 21 + 6V1V2 + V
2
2
4
⇔ 4V1V2 < V
2
2 − V
2
1 + (V1 + V2)
√
V 21 + 6V1V2 + V
2
2
⇔
V 21 + 4V1V2 − V
2
2
V1 + V2
<
√
V 21 + 6V1V2 + V
2
2
V 41 + 4V
3
1 V2 − V
2
1 V
2
2 + 4V
3
1 V2 + 16V
2
1 V
2
2 − 4V1V
3
2 − V
2
1 V
2
2 − 4V1V
3
2 + V
4
2
> V 41 + 2V
3
1 V2 + V
2
1 V
2
2 + 6V
3
1 V2 + 12V
2
1 V
2
2 + 6V1V
3
2 + V
2
1 V
2
2 + 2V1V
3
2 + V
4
2
109
⇔ 8V 31 V2 + 14V
2
1 V
2
2 − 8V1V
3
2 < 8V
3
1 V2 + 14V
2
1 V
2
2 + 8V1V
3
2 .
Thus player 3 will also be inactive in subgame 2E. This means that, for this constella-
tion, the subgames 2E and SL represent two-player-contests, for which Leininger (1993)
showed that player 2 will move early. Thus player 2 will deviate, and subgame SL is
never stable.
A.2.3 Equilibria
In the subgame-perfect equilibrium, we find the following behavior:
For combinations of valuations captured by Table B.1, players 1 and 2 will choose a
mixed strategy with respect to their timing decision. Player 3 will move early.
For the remaining combinations of valuations, we find:
Whenever V3 ≥ V1V2V1+V2 ∧ V1 ≤ 3
V2V3
V2+V3
we find that a subgame-perfect equilibrium is
characterized by all players moving early. (Subgame SE) This is a unique equilibrium,
if at least one condition holds with strict inequality.
Whenever V3 ≤ V1V2V1+V2 ∨ V1 ≥ 3
V2V3
V2+V3
there is a subgame-perfect equilibrium in which the
strongest player moves late, while both weaker players move early, unless the valuations
lie in the parameter range characterized by Table B.1. (Subgame 1L) It is unique, if
V3 > V22 and at least one of the above conditions holds with strict inequality.
Whenever V3 ≤ V22 , there exists an equilibrium in which both the strongest and the
weakest player move late while the intermediate player chooses early. (Subgame 2E)
This equilibrium is never unique and player 3 will not be active.
110
Chapter B
Proofs
B.1 Comments on the Introduction
Annotation to Axiom 4 in Skaperdas (1996) While we present the Axiom 4 as
pMi (x)
pMj (x)
=
pi(x)
pj(x)
, ∀i, j ∈M ∀xi, xj ∈ S, (B.1)
Skaperdas writes:
pMi (x) =
pi(x)
∑m
j=1 pj(x)
, ∀i ∈M. (B.2)
Let us now show that both postulations are equivalent, as long as Axiom 1 holds. At
first, we can rewrite (B.1) as
pMi (x)
pi(x)
=
pMj (x)
pj(x)
,
which obviously only holds for every xi ∈ S if the function
pMi (x)
pi(x)
is a constant function,
or put differently, ∃ν ∈ IR : pMi (x) = ν · pi(x), ∀xi ∈ S. In (B.2), Skaperdas presents this
functional form, choosing ν = 1∑m
j=1
pj(x)
. This choice is due to Axiom 1, which assures
that the sum of winning probabilities yields one.
B.2 Proofs of Chapter 3
B.2.1 Preliminary Results
Proof of Lemma 3. Part (i): Fix any x = ([a]k, [b]n−m) ∈ Nsym(ΓF (m)) for some
a, b ∈ [0,∞) with a > b. It follows from the hypothesis, from negative spillovers, and
from symmetry of the material game that
pi1([a]m, [b]n−m) ≤ pin([a]m, [b]n−m) < pin(b, [a]m−1, [b]n−m) = pi1(b, [a]m−1, [b]n−m),
which contradicts a representing an optimal response of player 1 against ([a]m−1, [b]n−m).
Thus, b ≥ a.
111
Part (ii): Let m = n−1 and x = ([a]n−1, b). All we have to show is that b > a implies
pin(x) > pi1(x). Suppose, to the end of contradiction, that b > a and pin(x) ≤ pi1(x).
Observe that pin(x) ≤ pi1(x) implies ρn(x) = pin(x)(1 − 1n) −
1
n
∑
pi1(x) ≤ 0. Since b
is a best response of player n against [a]n−1 (with regard to player n’s interdependent
preferences F (pin, ρn)), it follows from ρ([a]n) = 0 ≥ ρn(x) that pin([a]n−1, b) ≥ pin([a]n).
Thus, negative spillovers and symmetry of the material game imply
pi1([a]n−1, b) < pi1([a]n) = pin([a]n) ≤ pin([a]n−1, b),
in contradiction to pin(x) ≤ pi1(x).
B.2.2 Non-increasing marginal efficiency
Proof of Theorem 2. Part 1 (Properties) Fix r ≤ 1 and m ∈ {1, . . . , n − 1} and
let xˆ = ([a]m, [b]n−m) ∈ Nsym(ΓF (m)). The following expressions will turn out useful in
establishing the proof:
ρj(x) =
[
xrj∑
h
xrh
− 1n
]
V −
[
xj − 1n
∑
h xh
]
, (B.3)
∂ρj
∂xj
(x) =
rxr−1j
∑
h 6=j
xrh
(
∑
h
xrh)
2 V − 1 + 1n , (B.4)
and
∂pih
∂xh
(x) =
rxr−1h
∑n
i=1,i 6=h
xri
(
∑
i
xri )
2 V − 1. (B.5)
These hold for any x, any j ∈ J ≡ {m+1, . . . , n}, and any h ∈ {1, . . . , n}. Notice that, for
all x, ∂ρj(x)∂xj < 0 implies
∂pij(x)
∂xj
< 0 and, inversely, ∂pij(x)∂xj > 0 implies
∂ρj(x)
∂xj
> 0. Moreover,
neither of the two cases can be part of an intragroup symmetric Nash equilibrium.
Therefore, in the former case, any status-seeker would be strictly better off with b − ε,
while, in the latter case, she could do strictly better expending effort b+ ε, where ε > 0
has to be chosen sufficiently small in each case. Therefore, in any intragroup symmetric
equilibrium xˆ, it has to be, for any status-seeker j ∈ J, that ∂ρj(x)∂xj ≥ 0 and
∂pij(xˆ)
∂xj
≤ 0.
From (B.4) and (B.5), it thus follows that 0 ≤ ∂pij∂xj (xˆ) +
1
n ≤
1
n , which is equivalent to
rbr−1[(m− 1)ar + (n−m)br]
[mar + (n−m)br]2
V = γ, (B.6)
for some γ ∈ [n−1n , 1].
Claim (i): a > 0. Suppose to the end of contradiction that a = 0. According to
Proposition 1 in Pe´rez-Castrillo and Verdier (1992), for non-increasing contest technolo-
gies (r ≤ 1) an effort level a = 0 can only maximize player 1’s material pay-off if r = 1.
Therefore, it suffices to derive a contradiction for r = 1.
On the one hand, a = 0 implies b < Vn−m since, otherwise,
∑
h xˆh = (n −m)b ≥ V
entails
∂ρn
∂xn
(xˆ) =
(n−m− 1)V
(n−m)2b
− 1 +
1
n
≤
n−m− 1
n−m
−
n− 1
n
=
−m
(n−m)n
< 0
112
and hence ∂pin(xˆ)∂xn < 0. By the argument mentioned above, player n would do better in
terms of his interdependent preferences if he expended t = b− ε with ε > 0 sufficiently
small. However, this contradicts xˆn = b being optimal for player n against xˆ−n =
([a]m, [b]n−m−1). Thus, b < Vn−m .
On the other hand, a = 0 implies b ≥ Vn−m since, otherwise, a = 0 could not
maximize player 1’s material pay-off. This holds since, for r = 1, the best response
function of player 1 is given by x1(Z) = max{0,
√
V Z−Z}, Z > 0, (where Z =
∑
h6=1 xh)
and because x1(Z) > 0 if and only if Z < V . (Notice again that b ≥ Vn−m implies∑
h xˆh = (n−m)b ≥ V .)
Taken together, b < Vn−m and b ≥
V
n−m yield a contradiction so that we have estab-
lished a > 0.
Claim (ii): b ≥ a. Suppose to the contrary that b < a. It suffices to show ∂pin(xˆ)∂xn > 0
for this implies ∂ρn(x)∂xn > 0, which yields a contradiction to xˆn = b being optimal for player
n against xˆ−n = ([a]m, [b]n−m−1). As mentioned above, in this case, player n would be
better off with t = b+ ε.
To see ∂pin(xˆ)∂xn > 0, insert the first order condition of player 1,
∂pi1
∂x1
(xˆ) =
rar−1[(m− 1)ar + (n−m)br]
[mar + (n−m)br]
V − 1 = 0,
into ∂pin(xˆ)∂xn . This results in
∂pin
∂xn
(xˆ) =
rbr−1[mar + (n−m− 1)br]
rar−1[(m− 1)ar + (n−m)br]
− 1,
which is strictly positive because of b < a,
(
b
a
)r−1
> 1, and r ≤ 1. Thus, b ≥ a.
Claim (iii): pin(xˆ) ≥ pi1(xˆ), which holds strictly if b > a. Obviously, b = a implies
pin(xˆ) = pi1(xˆ), so we only have to deal with b > a.
Let us consider the case r = 1 first. In this case, pih(xˆ) reduces to xh[ Vma+(n−m)b − 1]
and pin(xˆ) > pi1(xˆ) is equivalent to (b− a)[V −ma− (n−m)b] > 0. Because of b > a, it
suffices to show ma+ (n−m)b < V.
Suppose to the contrary that ma + (n − m)b ≥ V. If ma + (n − m)b > V, then
pih(xˆ) < 0, for all players h. However, expending x1 = 0, player 1 would be strictly better
off because of pi1(0, xˆ−1) = 0. Hence, a would not have been optimal. On the other hand,
if ma + (n − m)b = V, then pih(xˆ) = 0, for all players h. In this case, x1 = a − ε, for
ε > 0 sufficiently small, gave player 1 strictly positive pay-off, in contradiction to a being
optimal. Thus, it must be that ma+ (n−m)b < V so that the claim follows.
Now consider r < 1. Define ϕ : [a, b]→ IR by
ϕ(z) ≡
zr
mar + (n−m)br
V − z,
and notice that ϕ(a) = pi1(xˆ) and ϕ(b) = pin(xˆ). Therefore, we have to show that
ϕ(b) > ϕ(a). At first, we determine the first and second derivative of ϕ(·) as
ϕ′(z) =
rzr−1
mar + (n−m)br
V − 1 and ϕ′′(z) =
r(r − 1)zr−2
mar + (n−m)br
V < 0.
113
Since ϕ(·) is strictly concave on [a, b], it suffices to show that ϕ′(b) > 0. Then, ϕ′(z) > 0,
for all z ∈ [a, b], implies that ϕ(a) < ϕ(b).
To see ϕ′(b) > 0, we insert the first order condition for status-seekers (B.6) into ϕ′(b),
which yields
ϕ′(b) = rb
r−1
mar+(n−m)brV − 1 = γ
mar+(n−m)br
mar+(n−m−1)br − 1
≥ n−1n
mar+(n−m)br
mar+(n−m−1)br − 1,
which can be shown to be strictly positive, using b > a.
Thus, pin(xˆ) > pi1(xˆ), for any xˆ ∈ Nsym(ΓF (m)) with b > a and any r ≤ 1 .
Claim (iv): pin(xˆ) > pi1(xˆ) if F (·, ·) is differentiable. By Claim (iii), we only have to
show that in this case b 6= a, which implies b > a and hence pin(xˆ) > pi1(xˆ).
Suppose to the contrary that b = a. By symmetry of the material game, we have that
∂pin
∂xn
(xˆ) = ∂pi1∂x1 (xˆ) = 0 since player 1 maximizes his material pay-off. Moreover,
∂pin
∂xn
(xˆ) = 0
implies ∂ρn∂xn (xˆ) =
∂pin
∂xn
(xˆ) + 1n =
1
n > 0. Hence, it follows from ∂2F > 0 that
∂F (xˆ)
∂xn
= ∂1F
∂pin(xˆ)
∂xn︸ ︷︷ ︸
=0
+∂2F
∂ρn(xˆ)
∂xn
= ∂2F
1
n
> 0,
so that there exists ε > 0 such that
F (pin(xˆ−n, b+ ε), ρn(xˆ−n, b+ ε)) > F (pin(xˆ), ρn(xˆ)).
Similar to the above arguments, this yields a contradiction to xˆ being optimal for player
n.
Part 2 (Uniqueness) Fix r ≤ 1, any k ∈ {1, . . . , n− 1} and any xˆ = ([a]m, [b]n−m) ∈
Nsym(ΓF (m)). By Part 1, we have 0 < a ≤ b. Set c := ab . Obviously, c ∈ (0, 1]. For any
status-seeker, i ∈ {1, . . . , k}, positive effort a > 0 implies ∂pii∂xi (xˆ) = 0, which is equivalent
to
ar−1[(m− 1)ar + (n−m)br]rV = [mar + (n−m)br]2. (B.7)
Similarly, for any status-seeker, j = {m+ 1, . . . , n}, we have ∂pij∂xj (xˆ) ≤ 0 and
∂ρj
∂xj
(xˆ) ≥ 0,
which implies
br−1[mar + (n−m− 1)br]rV = γ[mar + (n−m)br]2, (B.8)
for some γ ∈ [n−1n , 1]. Combining equations (B.7) and (B.8) and replacing
a
b by c, it
follows that
c1−r[mcr + (n−m− 1)] = γ[(m− 1)cr + (n−m)]. (B.9)
Define ϕ : (0, 1]→ IR by c˜ 7→ ϕ(c˜) = mc˜− γ(m− 1)c˜r + (n−m− 1)c˜1−r − γ(n−m).
Obviously, c solves (B.9) if and only if ϕ(c) = 0. Therefore, it suffices to show that
ϕ(c˜) = 0 implies ϕ′(c˜) > 0, which implies that the equation ϕ(c˜) = 0 and hence equation
(B.9) has a unique solution on (0, 1].
114
Inserting ϕ(c˜) = 0 into the derivative ϕ′(c˜) yields
ϕ′(c˜) = m− γ(m− 1)rc˜r−1 + (1− r)(n−m− 1)c˜−r
= 1
c˜
[mc˜− γ(m− 1)c˜r + (n−m− 1)c˜1−r
︸ ︷︷ ︸
=γ(n−m)
+(1− r)γ(m− 1)c˜r − r(n−m− 1)c˜1−r]
= 1
c˜
[γ(n−m) + (1− r)γ(m− 1)c˜r − r(n−m− 1)c˜1−r] .
To establish ϕ′(c˜) > 0, we show
γ(n−m) > r(n−m− 1)c˜1−r. (B.10)
For m = n− 1, this is obvious. For 1 ≤ m < n− 1, observe that γ ≥ n−1n , c˜ ∈ (0, 1],
and r ≤ 1 imply
γ(n−m)
n−m− 1
≥
(n− 1)(n−m)
n(n−m− 1)
=
n2 −mn− n+m
n2 −mn− n
> 1 ≥ rc˜1−r. (B.11)
Comparing the left and the right hand side, claim (B.10) can be seen as true.
Constant marginal efficiency
Proof of (3.13). Equation (3.13) represents the first-order condition of the maximum
of (3.4). Thus, we differentiate:
∂Fj (x)
∂xj
=
∂
∂xj


(
β1 + β2
n− 1
n
)
pij(x)−
β2
n
∑
h6=j
pih(x)

 , (B.12)
where ∂pij(x)∂xj =
V rxr−1j
(
∑n
h=1
xrh)
2
(∑n
h=1 x
r
h − x
r
j
)
−1 and ∂pii(x)∂xj = −
V rxri x
r−1
j
(
∑n
h=1
xrh)
2 . Therefore, (B.12)
becomes
(
β1 + β2
n− 1
n
)( V rxr−1j
(
∑n
h=1 x
r
h)
2
(
n∑
h=1
xrh − x
r
j
)
− 1
)
+
β2
n
∑
h6=j
V rxrhx
r−1
j
(
∑n
h=1 x
r
h)
2 .
Since we assume that r = 1, this expression reduces to
(
β1 + β2
n− 1
n
)( V
(
∑n
h=1 xh)
2
(
n∑
h=1
xh − xj
)
− 1
)
+
β2
n
∑
h6=j
V xh
(
∑n
h=1 xh)
2 .
We set ∂Fj(x)∂xj = 0 and get
V
(
∑n
h=1 x
r
h)
2 n

(β1n+ β2 (n− 1))
∑
h6=n
xh + β2
∑
h6=n
xh

 = β1n+ β2 (n− 1)
115
⇔
V
(
∑n
h=1 x
r
h)
2 (β1 + β2)
∑
h6=n
xh = β1n+ β2 (n− 1)
⇔
V (β1 + β2)
β2 (n− 1) + β1n
∑
h6=n
xh =
(
n∑
h=1
xrh
)2
.
Proof of (3.16). To show that (3.16) holds, we take (3.14) and (3.15) and equate
them. We receive:
V
n
((m− 1) a+ (m− k) b) =
V (β1 + β2)
β1 + β2 (n− 1)n
(ma+ (n−m− 1) b)
⇔ (β1 + β2 (n− 1)n) ((m− 1) a+ (n−m) b) = n (β1 + β2) (ma+ (n−m− 1) b)
⇔ nmβ2a− nβ2a−mβ2a+ β2a+ nmβ1a− nβ1a
+n2β2b− nmβ2b− nβ2b+mβ2b+ n
2β1b− nmβ1b
= nmβ2a+ nmβ1a+ n
2β2b− nmβ2b− nβ2b+ n
2β1b− nmβ1b− nβ1b
⇔ a (β2 (1− (n+m))− β1n) = b (−β2m− β1n)
⇔ a = b
β2m+ β1n
β2 (n+m− 1) + β1n
.
Proof of Proposition 2. First, recall equation (3.18), b =
√
V an−a(n−1). Then,
b > a follows from V > an⇔ V an > (an)2 ⇔
√
V an > an⇔
√
V an− a (n− 1) > a.
To see the equality in (3.19), notice that pih([a]n−1, b) = a(n−1)a+bV − a, for all h 6= j.
Using (3.18), we obtain
ρj([a]n−1, b) = n−1n pij([a]n−1, b)−
1
n
∑
h6=j pih([a]n−1, b)
= n−1n
(
b−a
(n−1)a+bV − (b− a)
)
= n−1n
(√
V an−an√
V an
V − (
√
V an− an)
)
= n−1n
(
V√
V an
− 1
) (√
V an− an
)
.
B.2.3 The case r > 1
Type I equilibria (b ≥ a > 0)
Proof of Theorem 3. To establish the strategic advantage for the case where b ≥ a > 0
and r > 1, we look upon the difference ∆ between the (material) pay-off of status-seekers
and the one of individualists. We define
∆ := pij − pii. (B.13)
116
In case of a (strict) strategic advantage, ∆ is non-negative (positive). Inserting (2.1) in
(B.13) for an intragroup symmetric equilibrium yields us:
∆ =
br − ar
mar + (n−m)br
V − b+ a. (B.14)
For b = a, this yields ∆ = 0; status-seekers experience a weak strategic advantage.
Let us now look at the less plain case where b > a. We will see that the strategic
advantage will be strict, i.e., ∆ > 0. After solving (FOCIJ), we can take c as given and,
thus, replace b by ac with c ∈ (0, 1). Inserting this in (B.14) yields:
∆ =
(
a
c
)r
− ar
mar + (n−m)
(
a
c
)rV −
a
c
+ a =
1
cr − 1
k + n−kcr
V − a
(1
c
− 1
)
. (B.15)
From (FOCI), it follows that the best response of an individualists as a function of c is
given as
a =
m+ n−mcr − 1
(
m+ n−mcr
)2 rV. (B.16)
Inserting (B.16) in (B.15) results in
∆ = V



1
cr − 1
m+ n−mcr
−
(1
c
− 1
) m+ n−mcr − 1
(
m+ n−mcr
)2 r



=
V
(
m+ n−mcr
)2
[( 1
cr
− 1
)(
m+
n−m
cr
)
−
(1
c
− 1
)(
m+
n−m
cr
)
r +
(1
c
− 1
)
r
]
=
V
(
m+ n−mcr
)2
[(n−m
cr
+m
)( 1
cr
− 1−
r
c
+ r
)
+ r
(1
c
− 1
)]
=
V
(
m+ n−mcr
)2
︸ ︷︷ ︸
>0



(n−m
cr
+m
)
︸ ︷︷ ︸
>0
[(1
c
)r
− 1− r
(1
c
− 1
)]
+ r
(1
c
− 1
)
︸ ︷︷ ︸
>0



. (B.17)
From this, we can see that
(1
c
)r
− 1− r
(1
c
− 1
)
≥ 0 (B.18)
is sufficient for a strict strategic advantage. Writing 1c := 1 + q, we can turn (B.18) into
(1 + q)r ≥ r(1 + q − 1) + 1 = 1 + rq.
117
Since q > 0 and r > 1, we can apply the Generalized Bernoulli Inequality, which tells us
that
(1 + q)r > 1 + rq, ∀q > −1, r > 1.
Applying this knowledge to (B.17) implies
∆ > 0,
that is, higher effort b > a results in a strict strategic advantage of interdependent
preferences.
Type II equilibria (a > b > 0)
Proof of Lemma 4. Fix r > 1 and m ∈ {1, . . . , n − 1} and let xˆ = ([a]m, [b]n−m) ∈
Nsym(ΓF (m)) such that a, b > 0. Recall Z = mar + (n−m− 1)br, B(Z, λ) = [b(Z, λ)]
r =
br(Z, λ), and that b(Z, λ) solves
[Z + br]2 − rbr−1ZλV = 0. (B.19)
Then, (B.19) is equivalent to
B
1
r [Z +B]2 − rBZλV = 0. (B.20)
Part (i): We implicitly differentiate (B.20) to obtain
∂B
∂Z
= −
2B
1
r [Z +B]− rBλV
1
rB
1
r−1 [Z +B]2 + 2B
1
r [Z +B]− rZλV
= −
2B
1
r [Z +B]−B
1
r [Z +B]2 Z−1
1
rB
1
r−1 [Z +B]2 + 2B
1
r [Z +B]−B
1
r [Z +B]2B−1
, (B.21)
where the equality follows from (B.20). The numerator is non-negative if and only if
Z ≥ B. Similarly, the denominator is positive if and only if
1
r
[Z +B] + 2B − [Z +B] > 0
⇔
r + 1
r − 1
B > Z.
Now, consider Z ≤ B first. Since this implies Z < r+1r−1B, we have
∂B
∂Z ≥ 0 for all Z ≤ B.
On the other hand, if B < Z < r+1r−1B, then we have
∂B
∂Z < 0. Since both the numerator
and the denominator of (B.21) are positive, ∂B∂Z ≥ −1 is equivalent to
Numerator (B.21) ≤ Denominator (B.21)
⇔ −B
1
r [Z +B]2 Z−1 ≤
1
r
B
1
r−1 [Z +B]2 −B
1
r [Z +B]2B−1
⇔ −B ≤
1− r
r
Z
⇔
r
r − 1
B ≥ Z.
118
Since Z ≤ rr−1B implies Z <
r+1
r−1B, this completes the proof of part (i).
Part (ii): Notice first that ∂B∂λ (Z, λ) = r[b(Z, λ)]
r−1 ∂b
∂λ(Z, λ) and that Z > 0 implies
b(Z, λ) > 0. Therefore, it is sufficient to show that ∂B∂λ (Z, λ) > 0 if and only if Z <
r+1
r−1B.
Implicitly differentiating (B.20), we obtain
∂B
∂λ
= −
−rBZV
1
rB
1
r−1 [Z +B]2 +B
1
r 2 [Z +B]− rZλV
.
Obviously, ∂B∂λ is strictly positive if and only if the denominator is strictly positive, which
by part (i) is equivalent to r+1r−1B > Z.
Part (iii): Suppose a > b and Z ≤ rr−1B. Then, part (ii) implies that
∂B
∂λ (Z
′, λ) > 0,
for all 0 < Z ′ ≤ Z. Derive a condition from the first order condition of an individualist
that is similar to (B.20), namely,
C
1
r [Y + C]2 − rCY V = 0, (B.22)
where C = ar and Y = (m− 1)ar + (n−m)br. Comparing (B.22) with (B.20), it follows
that C(Y ) = B(Y, 1). Since a > b implies Y < Z, we can thus apply part (ii) and λ > 1
to conclude that
C(Y ) = B(Y, 1) < B(Y, λ). (B.23)
Moreover, it follows from part (i) that ∂B∂Z (Z
′, λ) ≥ −1, for all Z ′ ≤ rr−1B, and hence
B(Z, λ)−B(Y, λ)
Z − Y
≥ −1
⇔ B(Y, λ)−B(Z, λ) ≤ Z − Y. (B.24)
Combining the two inequalities (B.23) and (B.24), we obtain
C(Y )−B(Z, λ) < B(Y, λ)−B(Z, λ)
≤ Z − Y = ar − br = C −B,
which yields a contradiction because xˆ = ([a]m, [b]n−m) ∈ Nsym(ΓF (k)) implies C(Y ) = C
and B(Z, λ) = B. Thus, a ≤ b.
B.3 Proofs of Chapter 4
Proof of Lemma 5. The first order condition of the maximization problem of player
i is given by ∂Fi(pii(x),ρi(x))∂xi
!= 0, which is equivalent to
rxr−1i
∑
j 6=i
xrj
n
n+ αi
V =
(
n∑
h=1
xrh
)2
. (B.25)
Now assume, on the other hand, that player i maximizes the utility function (2.1) in
a rent-seeking contest with a fixed prize of V˜ := λiV . His first-order-condition is then
given by
∂pii(x)
∂xi
=
rxr−1i
∑
j 6=i x
r
j
(
∑n
h=1 x
r
h)
2 V˜ − 1
!= 0
119
⇔ rxr−1i
∑
j 6=i
xrjλiV =
(
n∑
h=1
xrh
)2
. (B.26)
Obviously, since we assumed λi := nn+αi , the decision situation in (B.25) and (B.26)
is the same. Since we know from Fang (2002) that maximization of the material payoff
(2.1) for a fixed prize V has a unique solution, we can apply this result also to (B.26).
We find that there is one unique solution for the optimization of the material payoff
competing for λiV , and from (B.25) and (B.26), we can conclude that this is also the
unique solution in maximizing (4.1) when competing for a prize of V .
Proof of Lemma 6. The first-order-condition of a player i is given by
∑
j 6=i
xjλiV −


∑
j 6=i
xj + xi


2
= 0. (B.27)
A choice of xi = 0 is endogenously optimal if
∑
j 6=i
xjλiV =


∑
j 6=i
xj


2
⇔ λiV =
n∑
j=1
xj.
Note that
∑n
j=1 xj and
∑
j 6=i xj can be set equal, since xi = 0. Applying the implicit
function theorem to equation (B.27) results in
∂xi
∂λi
= −
∑
j 6=i xj
−2
∑n
j=1 xj
> 0, ∀xi > −
∑
j 6=1
xj,
telling us that a player i that is characterized by λiV <
∑n
j=1 xj would then prefer
negative effort. Thus, she stays outside the contest.
Proof of Proposition 3. At first we show how (4.3) turns into the ratio of the
preference parameters:
(4.3)⇔ xr−1i
∑
h6=i
xrhλi = x
r−1
j
∑
h6=j
xrhλj
⇔
λi
λj
=
A− xrj
A− xri
(
xi
xj
)1−r
, (B.28)
where A :=
∑
i = 1nxri .
Part (i): Assume λi > λj. This means that the left-hand-side of equation (B.28) is
larger than 1. For equating (B.28), at least one of the factors on the right-hand-side also
has to be larger than 1. This means:
A− xrj
A− xri
> 1 ∨
(
xi
xj
)1−r
> 1
⇒ xi > xj ∨ (xi > xj ∧ r < 1) .
120
Thus from λi > λj follows xi > xj. Due to reciprocity, we can also conclude that
λi < λj ⇒ xi < xj
Part (ii): Now if λi = λj, the left-hand-side of (B.28) equates 1. Then for the two
factors on the right-hand-side, it has to hold that either one has to be smaller and
one larger than 1, or both have to equate 1. One can show that the first of these
possibilities induces a contradiction. Thus the latter must hold, which means that xixj =
1 or equivalently xi = xj has to hold. Since these results cover the whole range of
possibilities for the ratio of xi and xj, we can conclude that the equivalency has to hold,
too.
Proof of Theorem 7. We will restrict the proof to the case, where λi ≤ nn−1 , ∀i. The
proof for λi ≥ nn−1 , ∀i runs analogously. Assume, like in Chapter 3, the aggregate to be
spent and, in consequence, A to be fixed. Then, we can define a function ϕ(z) : IR+ → IR,
such that:
ϕ(z) =
zr
A
V − x. (B.29)
Note that, applying Proposition 3, it is sufficient to show that a higher effort induces
higher material payoff. For this purpose, we show that the first derivative of (B.29),
ϕ′(z) = rz
r−1
A V − 1, is positive over the whole range of efforts chosen in the contest.
Denoting the highest effort chosen in equilibrium by x, we do this by showing, that ϕ(x)
is concave for any effort level and ϕ′ (x) > 0, from which we can conclude that material
payoff is strictly increasing in λi, whenever effort is strictly increasing.
The second derivative of (B.29) is given by ϕ′′(z) = r(r−1)z
r−2
A V ≤ 0, r ≤ 1, which
already proves concavity. Now let us look ϕ′ (x).
ϕ′ (x) =
rxr−1
A
V − 1 (B.30)
Denoting λi of the player i choosing x by λ, we can transform (4.2) to:
xr−1 =
A2
r (A− xr)λV
. (B.31)
Inserting (B.31) in (B.30) yields:
ϕ′ (x) =
A
(A− xr)λ
− 1 ≥
n− 1
n
A
A− xr
− 1 =
n−1
n A− (A− x
r)
A− xr
=
xr − An
A− xr
> 0.
The last expression is larger than zero since xr is per definition the largest summand of
A. Since we assumed a heterogenous population, the largest summand of A is larger than
the average summand, An . The numerator is strictly positive. Since A is a sum of several
positive summands, where xr is only one of them, the denominator is also strictly positive,
and thus the player with higher preference parameter realizes the higher material payoff.
Proof of Corollary 7. We have to show here that, for any player i with λi ≤
1, an effort level of xi > 0 cannot be the best response to an aggregate effort where
∑
j 6=i xj > V . To this aim, remember that λi ≤ 1 implies αi ≥ 0. Denote with xi0 =
121
(x1, ..., xi−1, 0, xi+1, ..., xN). Then, if i remains passive, her objective function will take
the value
F (xi0) =
αi
N
∑
j 6=i
pij (xi0) =
αi
N
∑
j 6=i
xj
V −
∑
j 6=i xj
∑
j 6=i xj
.
Now suppose to the contrary that she would choose xi > 0 as best response to some
∑
j 6=i xj > V . Writing xi = (x1, ..., xi−1, xi > 0, xi+1, ..., xN) we have:
F (xi) =
(
1 +
αi
N
)( xi
∑
j 6=i xj + xi
V − xi
)
+
αi
N
∑
j 6=i
(
xj
∑
k 6=i xk + xi
V − xj
)
=
(
1 +
αi
N
)
xi
V −
∑
j 6=i xj − xi
∑
j 6=i xj + xi
︸ ︷︷ ︸
<0
+
αi
N
∑
j 6=i
xj
V −
∑
j 6=i xj − xi
∑
j 6=i xj + xi
︸ ︷︷ ︸
<
V−
∑
j 6=i
xj
∑
j 6=i
xj
< F (xi0) .
B.4 Proofs of Chapter 5
Proof of Lemma 8. As we said, q1 and q2 are positive. Furthermore, define q3 :=
2(n− 1)x2ryr. This turns ω′′(y) > 0 to
−q1 ((r − 1)q2 − q3r) > 0,
with q1, q2, q3 > 0. From this, we can conclude to
r < 1 +
q3
q2 − q3
=: r∗ > 1, if q2 > q3.
If q2 ≤ q3, we end up in q2 > r(q2 − q3), which is then always fulfilled.
B.5 Proofs of Chapter 6
Derivation of (6.11): Equating (6.10) for player i and player j yields:
√
(xi + xj)Vk −
xiVk
2
√
(xi+xj)Vk
(xi + xj)Vk
Vi =
√
(xi + xj)Vk −
xjVk
2
√
(xi+xj)Vk
(xi + xj)Vk
Vj
⇔
√
(xi + xj)VkVi −
xiVk
2
√
(xi + xj)Vk
Vi =
√
(xi + xj)VkVj −
xjVk
2
√
(xi + xj)Vk
Vj
⇔ 2 (xi + xj)ViVk − xiViVk = 2 (xi + xj)VjVk − xjVjVk
⇔ 2Vixj + Vixi = 2Vjxi + Vjxj ⇔ xj = xi
2Vj − Vi
2Vi − Vj
.
122
Derivation of Table 6.1:
We look at a rent-seeking contest, where player 3 decides on her effort in t = 2, while
players 1 and 2 choose their effort levels in t = 1. Assume that V1V2V1+V2 < V3 <
3
2
V1V2
V1+V2
.
The participation constraint of player 3 says, that she will be inactive in the standard
three-player-model. The standard model implicitly assumes that a player who is strictly
worse off from participating chooses a negative effort, resulting in a negative winning
probability. Since this is not feasible by assumption, the reaction functions of players 1
and 2 cannot be adapted in this case. Let us now focus on what will actually be the
equilibrium effort of player 1 and 2.
Presume, first, that players 1 and 2 completely ignore the presence of player 3 and
play a mere two-player-game. According to Leininger (1993), their efforts would be
x1 =
V 21 V2
(V1+V2)2
∧ x2 =
V1V 22
(V1+V2)2
, which means that x1 + x2 = V1V2V1+V2 . The latter is smaller
than V3 per assumption. Yet, (6.9) shows that for x1 + x2 < V3, player 3 will choose
a positive effort. Hence, if players 1 and 2 would behave as if in a two-player-contest,
player 3 would choose a positive effort. This cannot constitute an equilibrium.
From the standard analysis of the three-player-case, we can conclude that x3 > 0
cannot be part of an equilibrium in this setting since this equilibrium would also be
feasible without the restriction on x3 ≥ 0.
In order to keep player 3 out of the contest, the aggregate effort of players 1 and 2
has to be at least V3. Furthermore it has to hold that each active player has a positive
return of the last marginal effort. Hence, the first derivative of the payoff must be weakly
positive for the case that x1 + x2 = V3. We illustrate this for player 1:
∂pi1 (x1, x2, x∗3(x1, x2))
∂x1
≥ 0|x1+x2=V3
⇔
∂
∂x1
x1
√
(x1 + x2)V3
V1 − x1 ≥ 0|x1+x2=V3
⇔
√
(x1 + x2)V3 − x1V3
2
√
(x1+x2)V3
(x1 + x2)V3
V1 ≥ 1|x1+x2=V3
⇔ V1V3 −
1
2
x1V1 ≥ V
2
3 ⇔ x1 ≤ 2V3
V1 − V3
V1
.
For player 2, it analogously has to hold that
x2 ≤ 2V3
V2 − V3
V2
.
The upper limit for aggregate effort x1 + x2 = 4V3 − 2V1+V2V1V2 V
2
3 for these calculations
should lie above V3. This holds, if and only if V3 < 32
V1V2
V1+V2
.
Furthermore, we have to check whether a player has an incentive to deviate to a higher
effort. If one of the players does so, she only has to face one opponent, since player 3
123
was already preempted without an increase of her opponents efforts. An aggregate effort
of V3 is thus stable, if
∂pin=21 (x1, x2)
∂x1
≤ 0|x1+x2=V3 ⇔
∂
∂x1
( x1
x1 + x2
V1 − x1
)
≤ 0|x1+x2=V3
⇔
x2
(x1 + x2)2
V1 − 1 ≤ 0|x1+x2=V3 ⇔ x2 ≤
V 23
V1
,
and respectively player 2 has no incentive to increase her effort if x1 ≤
V 23
V2
.
Here, we can see that this analysis provides an additional upper limit for effort
vectors (x1, x2) that is given by
V 23
V1
+ V
2
3
V2
= V1+V2V1V2 V
2
3 . This is larger than V3 if and only if
V3 > V1V2V1+V2 .
Derivation of (6.21): From (6.20), we can conclude
(xi + xj)Vk =
(
(xi + xj)
Vk
Vj
+ xj
)2
⇔ x2j + xjV
2
j
2
(
Vk
Vj
)2
xi + 2
Vk
Vj
xi − Vk
(Vk + Vj)
2 +
Vkxi
(Vk + Vj)
2
(
Vkxi − V
2
j
)
= 0
⇒ xj =
VkV 2j − 2V
2
k xi − 2VkVjxi
2 (Vk + Vj)
2 ±
√
√
√
√
√
(
VkV 2j − 2V
2
k xi − 2VkVjxi
)2
4 (Vk + Vj)
4 +
Vkxi
(
V 2j − Vkxi
)
4 (Vk + Vj)
4 4 (Vk + Vj)
2
=
VkV 2j − 2V
2
k xi − 2VkVjxi
2 (Vk + Vj)
2 ±
1
2 (Vk + Vj)
2 · V
3
j Vk (VjVk + 4xi (Vk + Vj))
1
2 .
Derivation of (6.23):
Let us first take a look at the aggregate effort the late moving players will exert in
reaction to player i’s choice:
xEj + x
E
k =
VjVk +
√
VjVk (VjVk + 4 (Vj + Vk)xi)
2 (Vj + Vk)
− xi. (B.32)
Furthermore, the first-order-condition of player i depends on the derivation
(
∂x∗j (xi)
∂xi
+ ∂x
∗
k(xi)
∂xi
)
,
which equals the derivation of (B.32) for xi. This is given by
∂
(
x∗j (xi) + x
∗
k (xi)
)
∂xi
=
√
VjVk
VjVk + 4 (Vj + Vk)xi
− 1. (B.33)
Now, inserting (B.32) and (B.33) into (6.22) yields
124

VjVk +
√
VjVk (VjVk + 4 (Vj + Vk)xi)
2 (Vj + Vk)
−
√
VjVk
VjVk + 4 (Vj + Vk)xi
xi

Vi
=


VjVk +
√
VjVk (VjVk + 4 (Vj + Vk)xi)
2 (Vj + Vk)


2
.
One can show that this equation is solved by
xi =
V 2i (Vj + Vk)
2 − V 2j V
2
k
4VjVk (Vj + Vk)
. (B.34)
Derivation of (6.24):
Inserting (6.23) in (6.21) yields:
xj =
V 2j Vk − 2Vk (Vj + Vk)
V 2i (Vj+Vk)
2−V 2j V
2
k
4VjVk(Vj+Vk)
2 (Vj + Vk)
2
+
Vj
√
VjVk
(
VjVk + 4 (Vj + Vk)
V 2i (Vj+Vk)
2−V 2j V
2
k
4VjVk(Vj+Vk)
)
2 (Vj + Vk)
2
⇔ x∗j =
VjVk (2Vj + Vk)−
V 2i
Vj
(Vj + Vk)
2 + 2ViVj (Vj + Vk)
4 (Vj + Vk)
2 .
Derivation of (6.26):
From (6.24), we can conclude:
x∗j + x
∗
k =
VjVk (2Vk + Vj + 2Vj + Vk)− (Vj + Vk)
2
(
V 2i
Vj
+ V
2
i
Vk
)
+ 2Vi (Vj + Vk)
2
4 (Vj + Vk)
2
=
3V 2j V
2
k + 2ViVjVk (Vj + Vk)− V
2
i (Vj + Vk)
2
4VjVk (Vj + Vk)
.
Then
x∗i + x
∗
j + x
∗
k =
V 2i (Vj + Vk)
2 − V 2j V
2
k
4VjVk (Vj + Vk)
+
3V 2j V
2
k + 2ViVjVk (Vj + Vk)− V
2
i (Vj + Vk)
2
4VjVk (Vj + Vk)
=
ViVj + ViVk + VjVk
2 (Vj + Vk)
.
125
Derivation of (6.27) and (6.28):
pii =
V 2i (Vj + Vk)
2 − V 2j V
2
k
4VjVk (Vj + Vk)
·
2ViVj + 2ViVk − ViVj − ViVk − VjVk
ViVj + ViVk + VjVk
=
(ViVj + ViVk − VjVk)
2
4VjVk (Vj + Vk)
.
pij =
VjVk (2Vj + Vk)−
V 2i
Vj
(Vj + Vk)
2 + 2ViVj (Vj + Vk)
4 (Vj + Vk)
2
·
2V 2j + 2VjVk − ViVj − ViVk − VjVk
ViVj + ViVk + VjVk
=
(
2V 2j + VjVk − ViVj − ViVk
)2
4Vj (Vj + Vk)
2 .
pik can be derived analogously.
Derivation of (6.29):
By inserting (6.8) and (6.27), we get:
piiEi > pi
S
i ⇔
(ViVj + ViVk − VjVk)
2
4VjVk (Vj + Vk)
>
Vi (ViVj + ViVk − VjVk)
2
(ViVj + ViVk + VjVk)
2
⇔ (ViVj + ViVk + VjVk)
2 > 4ViVjVk(VjVk)
⇔ V 2i (Vj + Vk)
2 + 2ViVjVk(VjVk) + V
2
j V
2
k > 4ViVjVk(VjVk)
⇔ V 2i (Vj + Vk)
2 − 2ViVjVk(VjVk) + V
2
j V
2
k > 0
⇔ (Vi (Vj + Vk)− VjVk)
2 > 0.
Annotations to τi:
τ1 ∈ [0, 1]⇔ pi3E2 ≥ pi
1L
2 ∧ pi
S
2 > pi
2L
2 .
pi3E2 ≥ pi
1L
2 specifies the case we are interested in, since this has to hold in order for
player 2 to have an incentive to deviate from joining the weak player moving early to
joining the strong player moving late.
piS2 > pi
2L
2 : From the analysis of problem II, we know that this is equivalent to
V2 ≤ 3V1V3V1+V3 . Let us now show that this has to hold if V3 ≥
V1
2 , as assumed in (6.2). Note
that 3V1V3V1+V3 increases in V3. Then, we find:
3V1V3
V1 + V3
≥
3V1 V12
V1 + V12
= V1.
Yet, V2 ≤ V1 holds by assumption, and thus V2 ≤ 3V1V3V1+V3 .
τ2 ∈ [0, 1]⇔ pi2L1 ≥ pi
3E
1 ∧ pi
L
1 > pi
S
1
pi2L1 ≥ pi
3E
1 always holds, as we can see from the analysis of Problem III.
Since we only regard mixed strategies for the case V1 ≥ 3V2V3 V2 + V3, pi
L
1 > pi
S
1 holds,
as we have shown when analyzing Problem II.
126
B.6 Proofs of Chapter 7
Derivations of Equilibrium Strategies and Payoffs
Inserting xi = xj and (7.3) in (7.1) yields:
θ(n− 1)xiV = [((n− 1) + (n− 1)θ − (n− 2))xi]
2
⇔ θ(n− 1)V xi = (1 + (n− 1)θ)
2 x2i
⇔ xi =
(n− 1)θ
(1 + (n− 1)θ)2
V (∨xi = 0). (B.35)
Remark that xi = 0 will not be chosen by any player i, since we know from Stein (2002)
that at least two players will choose positive effort in the simultaneous game, while
Chapter 4 tells us that players with identical valuation choose the same effort level.
(7.4) is derived by inserting (B.35) in (7.3). It follows:
pi1 =
(n− 1)θ
(1 + (n− 1)θ)2
((n− 1)θ − (n− 2))V
(
θV
∑n
j=1 xj
− 1
)
=
(n− 1) ((n− 1)θ − (n− 2)) θ
(1 + (n− 1)θ)2
V ·



θV
(n−1)θ
(1+(n−1)θ)2
[((n− 1)θ − (n− 2))V + (n− 1)V ]
− 1



=
(n− 1) ((n− 1)θ − (n− 2)) θ
(1 + (n− 1)θ)2
(1 + (n− 1)θ)2 − (n− 1) ((n− 1)θ + 1)
(n− 1) ((n− 1)θ + 1)
V
=
((n− 1)θ − (n− 2))2
(1 + (n− 1)θ)2
θV
pii =
(n− 1)θ
(1 + (n− 1)θ)2
V
(
(1 + (n− 1)θ)V
(n− 1)θV
− 1
)
=
V
(1 + (n− 1)θ)2
.
Proof of Lemma 11. The optimal reaction function on all opponents is given by
xi =



√
V
(∑
j 6=i xj
)
−
∑
j 6=i xj, if
∑
j 6=i xj ≤ V
0 , if
∑
j 6=i xj ≥ V,
which generalizes the equilibrium in the sequential rent-seeking game introduced by
Leininger (1993). Now since all underdogs behave the same, this yields
xi =
√
(x1 + (n− 2)xi)V − x1 − (n− 2)xi, for x1 + (n− 2)xi ≤ V.
Due to symmetric behavior of the underdogs, the non-activity of one underdog in equi-
librium implies non-activity of all underdogs, and thus xi = 0, if
∑
j 6=i xj ≥ V .
127
On the second stage, the game between the underdogs behaves like a simultaneous
game, resulting in:
(n− 1)xi =
√
(x1 + (n− 2)xi)V − x1
⇔ (n− 1)2x2i + (2(n− 1)x1 − (n− 2)V )xi + x1(x1 − V ) = 0.
After applying the quadratic formula, we get a reaction function that describes each
underdog’s optimal reaction to the favorite’s and the other underdogs’ actions. It is
given by:
xi =
(n− 2)V − 2(n− 1)x1
2(n− 1)2
+
√
√
√
√(n− 2)
2V 2 − 4(n− 1)(n− 2)x1V + 4(n− 1)2x21
4(n− 1)4
+
x1(V − x1)
(n− 1)2
=
1
2(n− 1)2
[
(n− 2)V +
√
(n− 2)2V 2 + 4(n− 1)x1V
]
−
x1
n− 1
, (B.36)
which is the optimal behavior of an underdog if she actively takes part in the game.
Note that, for x1 = V , (B.36) delivers an optimal interior solution of xi = 0, such that
the reaction function of the underdogs is continuous.
Proof of Proposition 7. Denoting T := (n− 2)2V 2 + 4(n− 1)x1V we find:
∂pi1
∂x1
=
2(n− 1)θV
(n− 2)V +
√
T
− 1− x1
2(n− 1)θV 1√
T
[
(n− 2)V +
√
T
]2
1
2
4(n− 1)V
=
2(n− 1)θV
(n− 2)V +
√
T
·

1−
2x1(n− 1)V
(
(n− 2)V +
√
T
)√
T

− 1.
We will now insert x1 = V into ∂pi1∂x1 to show that
∂pi1(x1=V )
∂x1
≥ 0 is equivalent to θ ≥ nn−1 .
Beforehand, let us mention that T transforms to
T = (n2 − 4n+ 4)V 2 + (4n− 4)V 2 = n2V 2.
The condition for the favorite excluding her opponents is thus given by
2(n− 1)θV
(n− 2)V + nV
·
[
1−
2(n− 1)V 2
((n− 2)V + nV )nV
]
≥ 1
⇔ θ ≥
n
n− 1
.
Proof of Lemma 12.
To prove Lemma 12, we have to show that an underdog (at least weakly) prefers
moving in t = 1 to moving in t = 2 if all the other underdogs move in t = 1 and
1. the favorite also moves in t = 1
128
2. the favorite moves later, i.e., in t = 2
3. the favorite moves earlier, i.e., in t = 0.
We will now allude to these three different cases. However, we will only give an
extensive analysis to only the first case:
1. Let us first assume that the favorite and (n− 2) underdogs move early, while one
underdog denoted by n chooses late. Player n will then maximize her payoff with
the first-order-condition
∂pin(x)
∂xn
= 0⇔
n−1∑
h=1
xhV =
(
n−1∑
h=1
xh
)2
.
The reaction function of player n is thus given by
xn =
√
√
√
√
(
n−1∑
h=1
xh
)
V −
n−1∑
h=1
xh.
Each player i, i = 1, , n− 1, anticipates this in the early stage, and thus maximizes
pii(x−n, xn(x−n)) =
xi
√(∑n−1
j=1 xj
)
V
θiV − xi,
where θi = θ for i = 1 and θi = 1 for i 6= 1. The first-order-condition of payoff-
maximization by player i is given by
∂pii(x−n, xn(x−n))
∂xi
=
√(∑n−1
h=1 xh
)
V − 12xi
√
V∑n−1
h=1
xh
(∑n−1
h=1 xh
)
V
θiV − 1
!= 0
⇔
√
√
√
√
(
n−1∑
h=1
xh
)
V −
1
2
xi
√
V
∑n−1
h=1 xh
=
∑n−1
h=1 xh
θi
⇔ xi = 2
n−1∑
h=1
xh − 2
(
n−1∑
h=1
xh
) 3
2 1
√
V θi
.
Summing this up for all early moving players results in:
n−1∑
h=1
xh = 2(n− 1)
n−1∑
h=1
xh − 2
(
n−1∑
h=1
xh
) 3
2 1
√
V
·
n−1∑
h=1
1
θh
⇔
n−1∑
h=1
xh = 2(n− 1)
n−1∑
h=1
xh − 2
(
n−1∑
h=1
xh
) 3
2 1
V
·
(n− 2)θ + 1
θ
129
⇔ 2
(n− 2)θ + 1
θ
√
V
√
√
√
√
n−1∑
h=1
xh = 2n− 3
⇔
n−1∑
h=1
xh =
(2n− 3)2θ2
4((n− 2)θ + 1)2
V.
From this, we are able to conclude that
xn =
√
√
√
√
n−1∑
h=1
xhV −
n−1∑
h=1
xh =
(2n− 3)θ
2((n− 2)θ + 1)
V −
(2n− 3)2θ2
4((n− 2)θ + 1)2
V
=
(2n− 3)θ (2(n− 2)θ + 2− (2n− 3)θ)
4((n− 2)θ + 1)2
V
⇔ xn =
(2n− 3)(2− θ)θ
4((n− 2)θ + 1)2
V. (B.37)
The aggregate effort of all players, including player n, is then given by:
n∑
h=1
xh =
(2n− 3)2θ2
4((n− 2)θ + 1)2
V +
(2n− 3)(2− θ)θ
4((n− 2)θ + 1)2
V
=
(2n− 3)θV
4((n− 2)θ + 1)2
((2n− 3)θ + 2− θ)
=
2(n− 2)θ + 2
(2(n− 2)θ + 2)2
(2n− 3)θV
=
(2n− 3)θV
2(n− 2)θ + 2
.
Then, the payoff of player n is given by
pin(x) = xn
V −
∑n
h=1 xh
∑n
h=1 xh
=
(2n− 3)(2− θ)θ
4((n− 2)θ + 1)2
V ·
(
V −
(2n− 3)θ
2(n− 2)θ + 2
V
)
2(n− 2)θ + 2
(2n− 3)θV
=
(2− θ)
2(n− 2)θ + 2
·
(2n− 4)θ + 2− (2n− 3)θ
2(n− 2)θ + 2
V
⇔ pinLn =
(2− θ)2
4((n− 2)θ + 1)2
V (B.38)
If we now consider under which parameter constellation this payoff is larger than
the payoff from the simultaneous move game, we find that the payoff function in
(7.7) is larger than in (B.38), if θ ∈
[
4n−7−
√
16n2−40n+33
2(n−1) , 0
]
∪
[
1
n−1 ,
4n−7+
√
16n2−40n+33
2(n−1)
]
.
We regard only values of θ ≥ 1 and it can be be seen from (B.37) that the late
moving player will only choose a positive effort, if θ < 2. For n ≥ 2, it holds that
1 ≥ 1n−1 and 2 ≤
4n−7+
√
16n2−40n+33
2(n−1) := φ(n).
130
The latter can be shown by inserting n = 2 into φ(n) (which yields θ ≈ 2.56155)
and then showing that φ increases in n. To this aim, we can show that ∂φ∂n > 0
is equivalent to (n − 1)2 > 0, for n 6= 1. Therefore, the relevant values of θ lie in
[
1
n−1 ,
4n−7+
√
16n2−40n+33
2(n−1)
]
, and we can conclude that an underdog has always an at
least weakly higher payoff from moving in t = 1 than from moving in t = 2, if all
of her opponents move in t = 1.
2. Next, we show that an underdog will stick to her choice of t = 1, if the favorite
moves in t = 2 (at least for θ > 2n−3n−1 ). To this aim, we consider that (n−2) players
stick to t = 1, while the players 1 and n move in t = 2. In analogy to the analysis
behind (B.34), one can show that the aggregate effort by the (n− 2) early movers
is given by
n−1∑
h=2
xh =




θ(n− 3) + (2n− 5)
4
√
θ(θ + 1)(n− 2)
+
√
θ2(n− 1)2 + 4θ(n− 2)2 + (2n− 5)2 − 2θ(n− 3)
4
√
θ(θ + 1)(n− 2)


2
−
θ
4(θ + 1)


V.
Following the analysis from Chapter 6 for the case of late movers, we can compute
the equilibrium effort of both late moving players and their respective payoffs.
This enables us to calculate player n’s payoff from deviating to a late move, if
the favorite moves late, while the remaining players stick to t = 1. We skip the
formula here, since it is too complex, but simulations show that it never pays for
an underdog to join the late moving favorite.
3. Now, consider the case where we allow the favorite to move earlier than the un-
derdogs, i.e., in t = 0. Note that, in this case, in subgame-perfect equilibrium the
underdogs choose an effort of xi = 0 if θ ≥ nn−1 . Since all underdogs have the same
valuations, they will make the same decision concerning the question whether or
not to choose a strictly positive effort in any equilibrium. This means that, for
these levels of θ, it is optimal for all underdogs to choose xi = 0, regardless of
whether they choose in t = 1 or t = 2. Since the function of the favorite’s reaction
to the aggregate effort of the underdogs in t = 0 does not depend on whether the
underdogs move simultaneously or sequentially, it is given by (7.16). This allows
us to conclude from the analysis of Section 7.3.3 that the underdogs are inactive, if
and only if θ ≥ nn−1 . For lower values of θ, the comparison becomes rather complex,
but simulations show that underdogs will prefer an earlier move to a later move.
Proof of Proposition 8. (i) We will compare the payoffs given by the equations
(7.20) and (7.12) and find that the latter is smaller for every θ > 1 and every n ≥ 3.
The case θ < nn−1 :
piE1 > pi
L
1
⇔
((n− 1)θ − (n− 2))2
4(n− 1)
V >
(2(n− 1)θ − 2n+ 3)2
4(n− 1)2θ
V
131
⇔ Ξ(θ) := (n− 1)3θ3 − 2n(n− 1)2θ2 +
(
n2 + 4n− 8
)
(n− 1)θ − (2n− 3)2 > 0.
To prove Ξ(θ) being positive for θ > 1, one can show that Ξ(1) > 0 and Ξ′(θ) > 0 for
θ > 1 and n ≥ 3.
The case θ ≥ nn−1 :
We now similarly insert:
(θ − 1)V >
(2(n− 1)θ − 2n+ 3)2
4(n− 1)2θ
V
⇔ 4(n− 1)2θ2 − 4(n− 1)2θ > 4(n− 1)2θ2 − 4(n− 1)(2n− 3)θ + (2n− 3)2
⇔ 4(n− 1)θ(2n− 3− n+ 1) > (2n− 3)2.
Since we deal with the case θ ≥ nn−1 , we can write
4(n− 1)(n− 2)θ ≥ 4n2 − 8n > 4n2 − 12n+ 9⇔ 4n > 9.
The latter holds for any n ≥ 3. So we can conclude that piE1 > pi
L
1 for every θ > 1 and
n ≥ 3.
(ii) The case θ < nn−1 : To this aim, we compare the equations (7.20) and (7.6):
piE1 > pi
S
1 ⇔
((n− 1)θ − (n− 2))2
4(n− 1)
V >
((n− 1)θ − (n− 2))2
((n− 1)θ + 1)2
θV
⇔ (n− 1)2θ2 + 2(n− 1)θ + 1 > 4(n− 1)θ ⇔ ((n− 1)θ − 1)2 > 0.
This is obviously fulfilled for every θ > 1 and n ≥ 3.
The case θ ≥ nn−1 :
piE1 > pi
S
1
⇔ (θ − 1)V >
((n− 1)θ − (n− 2))2
(1 + (n− 1)θ)2
θV
⇔ (n− 1)2θ(θ − 1) > 1.
For θ ≥ nn−1 , we can write
(n− 1)2θ(θ − 1) ≥ (n− 1)2θ(
n
n− 1
−
n− 1
n− 1
) = (n− 1)θ > 1.
(iii) Let us now take a look at the condition, under which the payoffs of moving
simultaneously and moving later are equal for player 1, i.e., piS1 = pi
L
1 . This is equivalent
to
((n− 1)θ − (n− 2))2
(1 + (n− 1)θ)2
θV −
(2(n− 1)θ − 2n+ 3)2
4(n− 1)2θ
V = 0. (B.39)
Calculating the roots of (B.39) yields:
θ =
2n− 3
n− 1
∨ θ =
n
2 −
9
8 +
√
16n2−40n+33
8
n− 1
∨ θ =
n
2 −
9
8 −
√
16n2−40n+33
8
n− 1
.
132
For n ≥ 3, the last two roots can be shown to be smaller than 1, such that we find player
1 indifferent between the outcome of subgame L and that of subgame S, only if θ = 2n−3n−1 .
Due to continuity, it is sufficient to notice that, for θ = 1, it holds that piL1 < pi
S
1 and, for
θ = 2 on the other hand, it holds that piL1 > pi
S
1 , which proves the result of Proposition
8 (iii).
Proof of Proposition 9.
(i) Let us first look at the case where θ < nn−1 . Then W
E > W S implies
1 + (n− 1)2(θ − 1)2
2(n− 1)
>
((n− 1)θ − (n− 2))2 θ + (n− 1)
(1 + (n− 1)θ)2
,
which can be shown to be equivalent to
⇔ g(θ) = (n− 1)4θ4− 2(n− 1)4θ3 + (n− 1)4θ2− 2(n− 1)2θ2 + 2(n− 1)2θ−n(n− 2) > 0.
Note that g(1) < 0, while g
(
n
n−1
)
= 0. Next, let us show that θ = nn−1 is the only
the root of g(θ) for θ ∈
]
1, nn−1
]
. To this aim, we analyze the first derivative of h, given
by
g′(θ) = 4(n− 1)4θ3 − 6(n− 1)4θ2 + 2(n− 1)4θ − 4(n− 1)2θ + 2(n− 1)2
= (n− 1)4
(
4θ3 − 6θ2 + 2θ
)
+ (n− 1)2(2− 4θ).
Since g(θ) is a polynomial of the third degree, we can conclude that it has, at most,
one local minimum and one local maximum. From g(1) < 0 and g
(
n
n−1
)
= 0, we can
conclude that there can only be a root in
]
1, nn−1
[
, if the only local maximum lies in this
interval. But since g′(0) = 2(n − 1)2 > 0 and g′(1) = −2(n − 1)2 < 0, we can conclude
that the local maximum is given by some θ ∈]0, 1[. Thus, there is no root in
]
1, nn−1
[
, or
put differently, both subgames only reach the same level of welfare if and only if θ = nn−1 .
Since g(1) < 0, it is obvious that, for 1 < θ < nn−1 , it holds that W
E < W S.
Now looking at θ ≥ nn−1 we have that
WE > W S ⇔ (θ − 1) >
(n− 1)θ − (n− 2))2 θ + (n− 1)
(1 + (n− 1)θ)2
,
which can be transformed to
⇔ θ2 − θ −
n
(n− 1)2
> 0.
Applying the quadratic formula, we find that the latter condition is fulfilled whenever
θ > 1 exceeds the threshold value
θ1 =
1
2
+
√
1
4
+
n
(n− 1)2
=
1
2
+
√
√
√
√
n2 + 2n+ 1
4(n− 1)2
=
(n− 1) + (n+ 1)
2(n− 1)
=
n
n− 1
.
(ii) As above, we have to split the proof. For θ < nn−1 , we find
WE > WL
133
⇔
1 + (n− 1)2(θ − 1)2
2(n− 1)
V >
2(n− 1)θ2 + (2n− 3)(1− 2θ)
2(n− 1)θ
V
⇔ (n− 1)2θ3 − 2n(n− 1)θ2 +
(
n2 + 2n− 4
)
θ − (2n− 3) > 0.
Since the left-hand-side of the last inequality is a polynomial of the third degree, it
has at most three roots. These can be shown to be given by θ = n+1−
√
n2−6n+13
2(n−1) , θ = 1
and θ = n+1+
√
n2−6n+13
2(n−1) . Only for the latter root does it hold that θ > 1. From the sign of
the first coefficient, we can conclude that WE > WL holds for θ ∈
]
n+1+
√
n2−6n+13
2(n−1) ,
n
n−1
]
,
while for θ < n+1+
√
n2−6n+13
2(n−1) we find W
E < WL.
Now for θ ≥ nn−1 we have:
θ − 1 >
2(n− 1)θ2 + (2n− 3)(1− 2θ)
2(n− 1)θ
⇔ 2(2n− 3− n+ 1)θ − 2n+ 3 > 0
⇔ 2(n− 2)θ − (2n− 3) > 0.
Since we assumed that θ > nn−1 , we can write
2(n− 2)θ − (2n− 3) > 2
n2 − 2n
n− 1
− (2n− 3) =
n− 3
n− 1
≥ 0.
Thus, we can conclude that, for θ > nn−1 , it always holds that W
E > WL.
(iii)
W S > WL
⇔
((n− 1)θ − (n− 2))2 θ + (n− 1)
(1 + (n− 1)θ)2
>
2(n− 1)θ2 + (2n− 3)(1− 2θ)
2(n− 1)θ
⇔ −2(n− 1)2θ3 + (n− 1)(5n− 9)θ2 − 2
(
n2 − 5n+ 5
)
θ − (2n− 3) > 0.
The only root with θ > 1 to the function on the left-hand-side of the last inequality is
given by θ = 2n−3n−1 . Because the left-hand-side of that inequality is positive for θ = 1,
we can conclude that W S > WL for θ < 2n−3n−1 , while, for θ >
2n−3
n−1 , it has to hold that
W S < WL.
B.7 Proofs of Chapter 8
Derivation of (8.4)
√
(A−i + xi)V1Vi − xiViV1
2
√
(A−i+xi)V1
(A−i + xi)V1
− 1 = 0
134
⇔√
A−1V1Vi − xiViV1
2
√
A−1V1
A−1V1
= 1
(A−1V1)
3
2 = A−1V1Vi −
1
2
xiViV1
⇔ A
3
2
−1
√
V1 − A−1Vi = −
1
2
xiVi
⇔ xi = 2A−1 − 2
√
V1
Vi
(A−1)
3
2 .
Proof of Lemma 13. Player i would prefer moving late if
pi1L1 ≥ pi
S
1 ⇔ V1
(
1−
2n− 3
2V1Ω
)2
≥ V1

1−
n− 1
V1
(
1
V1
+ Ω
)


2
(
V1 −
2n− 3
2Ω
)2
≥

V1 −
n− 1
(
1
V1
+ Ω
)


2
⇔ V 21 − (2n− 3)
V1
Ω
+
(2n− 3)2
4Ω2
≥ V 21 −
2(n− 1)V1
(
1
V1
+ Ω
) +
(n− 1)2
(
1
V1
+ Ω
)2
⇔ −(2n− 3)
V1
Ω
+
(2n− 3)2
4Ω2
≥ −
(2n− 2)V1
1
V1
+ Ω
+
(n− 1)2
(
1
V1
+ Ω
)2
⇔ −(2n− 3)V1Ω
( 1
V1
+ Ω
)2
+
(2n− 3)2
4
( 1
V1
+ Ω
)2
≥ −(2n− 2)V1Ω
2
( 1
V1
+ Ω
)
+ (n− 1)Ω2
⇔ −(2n− 3)V1Ω− 2(2n− 3)V
2
1 Ω
2 − (2n− 3)V 31 Ω
3
+
(2n− 3)2
4
+
(2n− 3)2
2
V1Ω +
(2n− 3)2
4
V 21 Ω
2
≥ −(2n− 2)V 31 Ω
3 + (n2 − 2n+ 1− 2n+ 2)V 2i Ω
2
⇔ V 31 Ω
3 + V 21 Ω
2
(21
4
− 3n
)
+ V1Ω
(
2n2 − 3n+
15
2
)
+
(
n2 − 3n+
9
4
)
≥ 0
⇔ (V1Ω− (2n− 3))
(
V1Ω +
9
8
−
n
2
+
√
16n2 − 40n+ 33
8
)
·
(
V1Ω +
9
8
−
n
2
−
√
16n2 − 40n+ 33
8
)
≥ 0.
Now it is easy to see, for which values of V1Ω this fulfills with equality. For the roots of
this inequality, we find that
n
2
−
9 +
√
16n2 − 40n+ 33
8
< 0 <
n
2
−
9−
√
16n2 − 40n+ 33
8
<
2n− 3
2
< 2n− 3,
135
whenever n ≥ 2. For positive values of V1, player 1 will thus be willing to move late,
choosing a non-negative effort if
V1 ≥
2n− 3
∑n
i=2
1
Vi
.
Proof of Corollary 11. We want to show that:
xS1 ≥
n∑
h=2
xSi ⇔ V1 ≥
2n− 3
∑n
i=2
1
Vi
.
To this aim, we start by inserting the respective values for each player in (8.1), and
receive:
xS1 ≥
n∑
h=2
xSi ⇔
n− 1
∑n
i=1
1
Vi
−
(n− 1)2
(∑n
i=1
1
Vi
)2
1
V1
≥
(n− 1)2
∑n
i=1
1
Vi
−
(n− 1)2
(∑n
i=1
1
Vi
)2 Ω
⇔ 1−
n− 1
∑n
h=1 x
S
i
1
V1
≥ (n− 1)−
n− 1
∑n
i=1
1
Vi
Ω
1
V1
≤
n∑
i=2
1
Vi
−
n− 2
n− 1
n∑
i=1
1
Vi
=
(
1−
n− 2
n− 1
) n∑
i=2
1
Vi
−
n− 2
n− 1
1
V1
⇔
2n− 3
n− 1
1
V1
≤
1
n− 1
n∑
i=2
1
Vi
⇔ V1 ≥
2n− 3
∑n
i=2
1
Vi
.
Proof. For the proof, we show that, in the simultaneous move game, xi < A−1 ⇔
V1 < 2n−3∑n
i=2
1
Vi
:
n− 1
Ω + 1V1
−
(
n− 1
Ω + 1V1
)2
1
V1
<
n∑
i=2
xSi =
(n− 1)2
Ω + 1V1
−
(
n− 1
Ω + 1V1
)2
Ω
⇔
(
n− 1
Ω + 1V1
)2 (
Ω−
1
V1
)
< (n− 2)
n− 1
Ω + 1V1
⇔ (n− 1)
(
Ω−
1
V1
)
< (n− 2)
(
Ω +
1
V1
)
⇔ V1 <
2n− 3
Ω
.
136
B.8 Tables
Early Mover Minimum Maximum
V3/V1 V2/V1 V2/V1
0.164431 0.196790 0.196790
0.17 0.203812 0.207567
0.18 0.216466 0.227668
0.19 0.229172 0.248724
0.2 0.241934 0.270212
0.21 0.254755 0.293591
0.22 0.267638 0.317308
0.23 0.280589 0.341793
0.24 0.293613 0.366962
0.25 0.306715 0.392717
0.26 0.319902 0.418952
0.27 0.333182 0.445553
0.28 0.346563 0.472406
0.29 0.360054 0.499396
0.3 0.373667 0.526418
0.31 0.387412 0.553371
0.32 0.401303 0.580170
0.33 0.415356 0.606740
0.34 0.429588 0.633020
Early Mover Minimum Maximum
V3/V1 V2/V1 V2/V1
0.35 0.444018 0.658962
0.36 0.458668 0.684530
0.37 0.473566 0.709698
0.38 0.488741 0.734449
0.39 0.504232 0.758775
0.4 0.520081 0.782673
0.41 0.536343 0.806146
0.42 0.553083 0.829201
0.43 0.570385 0.851847
0.44 0.588357 0.874095
0.45 0.607143 0.895959
0.46 0.626942 0.917452
0.47 0.648037 0.938588
0.48 0.670871 0.959382
0.49 0.696183 0.987984
0.5 0.72541 1.00000
0.51 0.76215 0.95630
0.52 0.82810 0.88522
0.52099 0.85650 0.85650
Table B.1: Range when joining the late mover 1 is preferable for 2
137
B.9 Figures
-
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Figure B.1: The figure illustrates the constellations of valuations that are allowed by
(6.2). Other constellations than the ones in the hatched area are only considered in the
appendix.
Note that (6.2) can be split up into Vi ≥
Vj
2 and Vi ≥
Vk
2 , which can be transformed
for player 1, 2 and 3, respectively, to:
Vi
2
≤ Vj ≤ 2Vi,
Vi
2
≤ Vk ≤ 2Vi and
Vk
2
≤ Vj ≤ 2Vk.
138