Evolution of Commitment
in the Tullock Lottery Contest
rik van os van den abeelen
MSc. Thesis Economics University of Amsterdam October 11, 2014
I Preface
Acknowledgements
I owe my thanks to Matthijs van Veelen for his supervision of my thesis and useful comments in regards to an earlier draft. Further thanks go to Joep Sonnemans for his insights during the thesis seminar. Author Information
Written by Rik van Os van den Abeelen UvA-ID: 10670939
E-Mail: rik.vanosvandenabeelen@student.uva.nl Document Information
MSc. Thesis Economics: Behavioral Economics & Game Theory Date of submission: October 11, 2014.
II Nomenclature
xi Effort provision in an iteration of the contest by player i.
pi The probability of winning the contest by player i for a given level of personal effort, and aggregate
effort.
c Marginal cost of effort.
v Value of the prize that is contested in a given iteration of the contest game.
D Rent-dissipation: relationship between the costs of aggregate exerted effort and the value of the prize of a contest.
n Number of contestants in an iteration of the contest game.
N The total population size, from which contestants are drawn to participate in iterations of the contest game.
Γ An indirect evolutionary game.
S Set of strategies in an indirect evolutionary game.
T Type, or mutation, space of an indirect evolutionary game.
fi Evolutionary fitness; material payoffs undistorted by preferences of player i.
ui Subjective utility of player i, which may or may not be distorted by non-rational preferences.
Fi Indirect fitness of player i, maps Nash equilibrium behavior in a base game to an evolutionary
process in terms of non-rational utility preferences.
Φ The relative difference in player aggressiveness between ESP and Nash behavior.
ϕC The relative difference in player aggressiveness between ESP and Nash behavior as a result of commitment advantages.
τi Level of bias in private marginal cost perception.
θi Level of interdependency of utility.
ρ The probability that preferences of players are observable in a given interaction of the indirect evolutionary game.
Contents
Contents
I Preface ii
II Nomenclature iii
1 Introduction 1
2 Model: The Tullock Lottery Contest Game 3
2.1 Nash Equilibrium . . . 3 2.2 Evolutionarily Stable Strategy Equilibrium . . . 4
3 Indirect Evolution & Applications to Tullock Contests 6
3.1 Theory . . . 6 3.2 Previous Applications of Indirect Evolution to Contests . . . 7 3.3 Player Aggression and Commitment to Non-Rational Effort Levels . . . 9
4 Preference Evolution in the Tullock Contest with an Arbitrary Number of Players 10
4.1 Perfectly Observable Types . . . 10 4.2 Perfectly Observable Types in a Finite Population . . . 14 4.3 Unobservable Types in a Finite Population . . . 20
5 Conclusion & Discussion 24
A Supplemental Results 27
A.1 Evolution of Utility Interdependence and Marginal Cost Bias in 2-Player Contests . . . 27 A.2 Type Evolution in n-Player Contests with Occasional Observation of Preferences . . . 31
1 Introduction
Evolution of Commitment in the Tullock Lottery Contest
Abstract. I analyze the evolutionary stability of individual commitment to non-rational preferences in
the symmetric Tullock lottery contest game with an arbitrary number of players. To this end I use indirect evolution as the solution principle, where I consider biased assessments of personal marginal costs as the instigator of non-rational behavior. I show for symmetric contests with more than 2 players that the presence of advantages to overcommitment leads to more aggressive equilibrium behavior under indirect evolution than is the case for strategy evolution. I further show that the effect of commitment on behavior is eliminated in playing the field interactions or when preferences of opponents are not directly observable.
Keywords: Tullock Contest, Preference Evolution, Commitment, Non-Rational Behavior 1 Introduction
In life there exist many types of strategic interaction in which individuals devote a considerable portion of their physical or financial resources in order to get ahead of their rivals. The theory of conflict, or contest, deals with modeling and predicting strategic behavior in such situations. It follows that contest theory applies to a wide range of topics, and not surprisingly has found numerous applications which include for instance rent-seeking, political lobbying, litigation, R&D races and armed conflict.1
Formally, a contest is defined as a non-cooperative game in which players compete by making irrecover-able investments of either expenditure or effort to influence their probability of obtaining, or ‘winning’, a valuable resource; be it the acquisition of quality nutrition, monetary wealth, or political influence. A particularly useful concept in analyzing conflict is the notion of a contest success function, which uniquely maps the probability of victory of all contestants as a function of their effort level and aggregate effort, which in effect defines the strategic game. The most commonly studied specification of contest success, and also the one analyzed in the present paper, is the ratio-form from literature on rent-seeking behavior often attributed to the lottery game by Tullock (1980).
The majority of theoretical work in contest theory employs the Nash equilibrium as its solution concept. However, in experimental studies sub-optimal and overly aggressive behavior is generally observed when
1
Work that applies contest theory includes Baye et al. (1993) and Grossmann and Dietl (2012) on lobbying and rent-seeking; Farmer and Pecorino (1999), Wärneryd (2000) and Baye et al. (2005) on litigation; Szymanski (2003) and Dietl et al. (2009) on sporting competition; Baye and Hoppe (2003) on innovation; Morgan (2000) and Goeree et al. (2005) on charitable fundraising; Hirschleifer (1995) and Garfinkel and Skaperdas (1995) on military conflicts; Glazer and Gradstein (2005) and Klumpp and Polborn (2006) on political campaigns; Rosen (1986) and Bognanno (2001) on promotion contests; Schmalensee (1976) and Piga (1998) on market share competition; and, Loury (1979) and Taylor (1995) on patent race contests. Furthermore, Corchón (2007), Congleton et al. (2008a), Congleton et al. (2008b), and Konrad (2009) provide useful surveys on recent developments in contest theory.
1 Introduction
compared to the rational expected utility maximizing Nash equilibrium player (Sheremeta,2013). Recent work has applied to idea of relative payoff maximization rather than absolution payoff maximization as the appropriate yardstick for conflict behavior (Riechmann,2007; Ludwig et al.,2011). Concurrent with the idea of relative payoff maximization, it has been hypothesized that solution concepts other than the Nash equilibrium might provide a more accurate description of behavior in the context of conflict. In particular, there exists a potentially adaptive nature responsible for the observed propensity for aggressive behavior in non-cooperative games. An example of a solution concept that captures this notion of adaptiveness is the idea of an evolutionary stable strategy (ESS) from evolutionary game theory (Maynard Smith,1974). Although the standard infinite population ESS amounts to a refinement of the Nash equilibrium for the class of game the conflict game belongs to, it has been shown that the finite population of ESS by Schaffer (1988) yields behavior different from Nash equilibrium behavior (Leininger,2003), with ESS effort exertion strictly exceeding the Nash equilibrium prediction. In a concurrent study, Hehenkamp et al. (2004) embark on a complete analysis of the finite population ESS equilibrium of the Tullock contest. They confirm that equilibrium effort levels predicted by the finite population ESS of the Tullock contest strictly exceed Nash equilibrium effort, suggesting that evolutionary forces elicit spiteful behavior in conflict. I discuss this result in relation to the Nash equilibrium in more detail in section 2.
Following the work on evolutionary stable strategy behavior in contests a recent body of research (Leininger,2009; Noeske,2011; Boudreau and Shunda,2012; Wärneryd,2012), of which I provide a more in depth discussion in section 3, applies evolutionary methods at the preference level rather than the strategy level. The approach in this work is based on the indirect evolutionary approach pioneered in research by Güth and Yaari (1992) and Güth (1995). Indirect evolution models the evolution of player types rather than strategic behavior and assumes that individuals play rationally given the preference corresponding to their type. In the current paper I take a similar approach as the aforementioned authors where I calculate the evolutionary stability of player overoptimism through underestimation of individual marginal costs of effort provision. In effect I derive an evolutionarily stable level of effort that arises in contests through preference evolution. As reflected by previous results non-rational preferences can emerge in equilibrium as a result of relative payoff maximization in finite populations. In addition to the emergence of non-rational preferences as a result of population smallness I will argue that a player’s traits implicitly function as an unconditional strategic commitment given how preferences predict individual behavior and directly influence behavior of opponents. As such, if benefits to unconditional commitment to behavior that diverges from objective fitness maximization, or relative payoff maximization in the case of finite populations, exist in a non-cooperative game indirect evolution will capture such benefits in its prediction of equilibrium player behavior. In the context of the contest game this is relevant as benefits to overcommitment (relative to Nash equilibrium behavior) have been shown to exist in symmetric contests with more than 2 players in a general application by Dixit (1987). The main contribution of the present paper is thus the determination of an equilibrium state that reflects the presence of benefits to unconditional strategic commitments to non-rational behavior using indirect evolution as the solution principle.
The analytical results of this paper are presented as follows. In sections 4.1 and 4.2 I apply indirect evolution with perfectly observable player preferences to infinite and finite populations, respectively. I show that indirect evolution can be used to derive an equilibrium state that considers commitment advantages in symmetric contests with an arbitrary number of players. More precisely, I show that consistent with Dixit (1987)’s findings that preferences that result in overcommitment to effort are evolutionarily stable for contests with more than 2 players. Furthermore, I show that the effects of population smallness and overcommitment reinforce each other in most cases, leading to equilibrium behavior that is more
2 Model: The Tullock Lottery Contest Game
aggressive than is the case for the finite population ESS for interaction sizes greater than 2 yet smaller than full population interactions. Finally, in section 4.3 I apply indirect evolution to the contest game under the assumption of unobservable preferences. I observe that in this case the evolutionarily stable preference that results mimics finite population ESS behavior for any interaction and population size. As such, consistent with the interpretation of preferences as a means of commitment – noting that an unobservable commitment is intrinsically valueless – unobservability of preferences eliminates the strategic benefit of overcommitment to effort that arises in the indirect evolutionary equilibrium with perfectly observable preferences.
2 Model: The Tullock Lottery Contest Game
2.1 Nash Equilibrium
In the game theoretic analysis of conflict situations players are assumed to exert effort in order to increase their chances of obtaining a prize, or rent, with a particular value. The probability of success of any given individual competing in the contest is given by a contest success function which maps the vector of exerted effort levels by all contestants to a unique probability of winning for each contestant. The most common specification of contest success is the logit function as used in the Tullock lottery game (Tullock,1980). This probabilistic function is given by:
pi(x1, . . . , xn) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ xi ∑ n m=1xm if ∑ n m=1xj>0 1 n otherwise (1)
Where xi is effort exerted by player i and ∑ n
m=1xmis the sum of the effort levels of all players, that
is aggregate effort. The probability of success of any given player thus depends on their share of total effort exerted in the contest provided that aggregate effort is greater than 0. Furthermore, in the case that no effort is exerted by any player the prize is assumed to be randomly distributed. It follows that the contest success function is a probabilistic measure: the sum of the individual winning probabilities of all contestants is always equal to unity. In a contest players face the optimization problem of increasing their winning probability at the expense of costly effort. As such, players seek to maximize the following expected utility function:
fi(x1, . . . , xn) = pi(x1, . . . , xn) vi− cixi (2)
Where viis the valuation of the prize by player i, and ci is the marginal cost of effort for player i. The
most commonly studied variant, and also the one I analyze, of the Tullock contest assumes homogeneous players that share a common marginal cost of effort and common valuation of the resource at stake. That is c1, . . . , cn= cand v1, . . . , vn= v. It is a straightforward calculation to obtain the Nash equilibrium for the common marginal cost common value specification of the rent-seeking contest. Firstly, I note that the second condition of the contest success function, which occurs when no player exerts effort can never be a Nash equilibrium. If all players are currently exerting no effort, it follows that given the behavior of their opponents any player has an incentive to deviate by exerting an infinitesimal amount of effort and claiming the prize with certainty. Ignoring the second part of the contest success function, I note that the utility function of a player i amounts to:
fi(x1, . . . , xn) = v xi ∑ n m=1xm − c xi. (3)
2 Model: The Tullock Lottery Contest Game
The symmetric set of first order conditions of the contestants is given by: ∂ fi ∂xi =0 ∶ ∑ n m=1xm− xi (∑nm=1xm)2 = c v. (4)
For a symmetric solution we require that effort levels of all contestants are equal. Let x1 = ⋅ ⋅ ⋅ = xn= x, this simplifies the first order conditions as given in equation (4) to:
∂ fi ∂xi ∣ (x1=⋅⋅⋅=xn=x ) =0 ∶ (n −1) x (nx )2 = c v. (5)
Solving for x then yields the unique Nash equilibrium xN Eof the game. xN E=
(n −1) v
cn2 (6)
The general method to measure aggressiveness in the rent-seeking literature is by means of rent dissipation. Rent dissipation is defined as the ratio of the total incurred costs by players participating in the contest to the total value of the prize. As the Nash is equilibrium is symmetric it follows that rent-dissipation is equal to: DN E= cnxN E v = n − 1 n <1 (7)
As such, for Nash equilibrium behavior the rent is never completely dissipated: total effort multiplied by its cost is strictly smaller than the value of winning the contest. In the remainder of this paper I will consider Nash behavior and the corresponding level of rent dissipation as a benchmark to which I compare changes in player aggressiveness that arise from the use of evolutionary methods.
2.2 Evolutionarily Stable Strategy Equilibrium
Research by Hehenkamp et al. (2004) has provided the evolutionarily stable strategy equilibrium2of the standard rent-seeking contest. For comparison with the results of the indirect evolutionary method established below, I now briefly summarize their result for the specification of the contest as given by equation (3).3 The evolutionary stable state that is considered in Hehenkamp et al. (2004) is Schaffer (1988)’s finite population adaptation to the standard definition of an ESS by Maynard Smith (1974). In particular, this method yields the evolutionary stable strategy equilibrium of a contest for which in every iteration of the game a randomly selected subset n of the total population N participates.
Let player i be a mutant player with a strategy xMfacing a population of N − 1 players with strategy x. When entering the contest the single mutant player always faces a group of n − 1 opponents with the same strategy. As such, expected utility of player i is given by the following expression.
fi= fi(x1, . . . , xn) = v xM xM + (n −1) x − c x M (8) 2
Noting that an evolutionary stable strategy satisfies the following criterion. Let a strategy x be adopted by all players of a population. A mutant strategy xM≠ xcan invade the population if the payoff for a single individual using strategy xMis strictly greater than the payoff of a player using x. A strategy x is evolutionarily stable if it cannot be invaded by any other strategy xM.
3
In their analysis Hehenkamp et al. (2004) use a slightly different specification of the rent-seeking game. In particular, the analysis presented in their work assumes c = 1, furthermore the authors do not restrict a technology parameter that determines the efficiency of effort in the contest success function to be equal to unity as I do here. Aside from these points evolutionarily stable effort as presented here is qualitatively identical to the previous result.
2 Model: The Tullock Lottery Contest Game
Now consider expected utility of an opponent j ≠ i with strategy x. Under the assumption of uniform random matching in a finite population, the probability that player j will face the mutant in a contest with (n −1) opponents in a population with (N − 1) other players, is (n − 1) / (N − 1). For such a contest expected utility of the j player is given by:
fj(x1, . . . , xn∣Mutant) =
v x xM
+ (n −1) x
− c x. (9)
With the remaining probability the j player enters a contest in which the mutant player does not enter. In this case expected utility of the j player is defined as follows:
fj(x1, . . . , xn∣No Mutant) = v
n− c x. (10)
It thus follows that the player j, or any other non-mutant player with strategy x participating in the contest, has expected utility:
fj = (1 − n − 1 N − 1) fj(x1, . . . , xn∣No Mutant) + ( n − 1 N − 1) fj(x1, . . . , xn∣Mutant) = (1 − n − 1 N − 1) v n+ ( n − 1 N − 1) v x xi+ (n −1) x − c x . (11)
A strategy x is an ESS provided that no mutant strategy xM≠ xcan obtain a higher utility than strategy x when this strategy is adopted by all players (except the mutant): f (x, . . . , x) > f (xM, x, . . . , x) or equivalently f (x, . . . , x) − f (xM, x, . . . , x) > 0 ∀xM≠ x. By using this definition of the ESS condition it is possible to solve for the ESS strategy by considering the difference in payoff between a mutant and a member of a homogenous population. Consider the relative utility function of a mutant at the i-th player position Ri( fi, fj) = fi− fj. It follows that the strongest possible mutant i facing a population of
players with strategy x has a strategy xMthat maximizes Ri. As such, the strongest mutant strategy can be
obtained by solving the following maximization problem: max xM Ri( fi, fj) = v xM xM + (n −1) x − c x M − (1 − n − 1 N − 1) v n− ( n − 1 N − 1) v x xM + (n −1) x + c x. (12)
The corresponding first order condition is given by: ∂Ri
∂xM =0 ∶
N (n − 1) vx (N −1) (xM+ (n −1) x)2
− c =0. (13)
It is then possible to obtain the ESS strategy xE S Sby setting xM= x = xE S Sin the corresponding first order condition. This calculation presents the strongest mutant facing a population with some strategy x. If the mutant cannot do better than adopting the same strategy, then the condition for an ESS is automatically satisfied. Equating the effort levels in the first order condition given in (10) yields:
∂Ri ∂xM∣ (xM=x =xE S S) =0 ∶ v N (n − 1) − cn 2xE S S (N −1) (N −1) n2xE S S =0. (14)
It follows that setting the numerator in equation (14) to 0 and solving for xE S Syields the evolutionarily stable strategy of contest game with a finite population size:
xE S S= N (n − 1) v
3 Indirect Evolution & Applications to Tullock Contests
Rent dissipation for the ESS equilibrium is given by: DE S S= cnxE S S v = N (n − 1) (N −1) n ≤1. (16)
This result has two important implications. Firstly, it is immediate that the Nash and ESS equilibria coincide in an infinite population setting, i.e. N → ∞; and secondly, the finite population evolutionary equilibrium predicts effort provision and rent dissipation that are strictly larger than the Nash equilibrium of the game. It is useful to consider the relative difference in player aggressiveness between the finite population ESS (15) and the Nash equilibrium (6):
xE S S− xN E
xN E =
1
N − 1. (17)
It follows that the relative difference in aggressiveness between the finite population ESS and the Nash equilibrium is decreasing in the total population size, and does not depend on the interaction size n. As such, for a given population size N player aggressiveness exceeds the Nash aggressiveness by a constant factor for all interaction sizes. It is further worthwhile to mention that unlike the Nash equilibrium, the finite population ESS can yield an equilibrium state in which the complete rent is dissipated (i.e. DE S S= v). This occurs in the limit case in which the entire population engages in every iteration of the case, i.e. n = N. As such, in the finite population ESS equilibrium players exert effort beyond the level at which they maximize their own payoff (i.e. Nash equilibrium effort). This level of player aggressiveness is stable as players can reduce the utility of their opponents by more than the loss in their own utility that results from increasing their effort provision for effort levels that exceed Nash equilibrium effort but fall short of finite population ESS effort. By deriving the ESS of the contest game, I have established a useful benchmark to which the equilibria that arise from the indirect evolutionary method, which I turn to below, can be compared.
3 Indirect Evolution & Applications to Tullock Contests
3.1 Theory
The main object of study in the present paper are the evolutionary stable preferences associated with the Tullock contest as described previously. In an indirect evolutionary game Γ there is a population of N individuals who are repeatedly, and randomly, matched in groups of size n to play an n-player game, where 2 ≤ n ≤ N. As such, any iteration of the evolutionary game Γ is well defined for any player set {1, . . . , n} ⊆ N. Formally, the game Γ can be described by the following 3n-tuple.
Γ = ⟨⟨Si, Ti, fi⟩i={1,...,n}⟩ (18)
• Si denotes a nonempty set consisting of player i’s pure strategies si. The tuple s = (s1, . . . , sn) denotes a pure strategy vector. The set S = {(s1, . . . , sn) ∣ si ∈ Si} = ×ni=1Si is the set of all pure
strategy vectors.
• Ti is a nonempty set that indicates the mutation space of player i’s type. The mutation space
establishes the type of player i parametrically, and acts as restriction on the possible types that player i can have. The tuple t = (t1, . . . , tn)consisting of the types of all n players then denotes a type vector, where T = {(t1, . . . , tn) ∣ ti∈ Ti} = ×ni=1Tiis the set of all type vectors. Throughout the paper
3 Indirect Evolution & Applications to Tullock Contests
• fi indicates objective material payoffs, or evolutionary fitness; it is a mapping fi ∶ S →R. Note that the fitness function of a player is the result of behavior only and does not directly depend on the types of players drawn into the contest.
Let player i be an individual taking part in an iteration of the game Γ and let ui ∶ S × T → R be the individual preference of player i. A preference constitutes a subjective utility function of player i, which player i maximizes by means of its strategy si. It follows that subjective payoffs are determined by both
types and strategies. For a given type vector, the set of pure strategies and the subjective utility functions of players constitute a base game G (t1, . . . , tn)of Γ:
G (t1, . . . , tn) = ({Si, ui}i={1,...,n})∣
(t1,...,tn) (19)
Preference evolution applies the standard non-cooperative Nash equilibrium solution to the base game G (t), from which a corresponding vector of equilibrium strategies s∗
(t) = (s∗1(t) , . . . , s∗n(t)) is derived.
Indirect evolution then models the set of base games G = {G (t) ∣ t ∈ T} into an evolutionary process; as such, the vector of equilibrium strategies s∗determines not only equilibrium in a base game but also determines the objective success or evolutionary fitness of the underlying player types that instigate the strategic behavior. This gives rise to an indirect fitness function in types rather than strategies: Fi(t) = fi(s∗(t)). By using this notation for indirect fitness it is possible to simplify the expression for the indirect evolutionary game Γ as given in (18):
Γ = ⟨⟨Ti, Fi⟩i={1,...,n}⟩. (20)
The solution concept applied to the game Γ is the standard notion of ESS by Maynard Smith (1974), or in the case of a finite population the definition of ESS by Schaffer (1988). The equilibrium of the game Γ is characterized by evolutionary stable types and corresponding Nash equilibrium strategies of these types. As the evolutionarily stable strategy of this game is a player type t∗with a corresponding preference u (t∗), rather than a strategy, I follow Eaton and Eswaran (2003) and other authors in denoting the evolutionary equilibrium of the game Γ as an evolutionary stable preference (ESP). The complete solution of the indirect evolutionary game; however, is given by a vector consisting of both types and strategies, (t∗, s∗(t∗)). For which
1. t∗is an ESS (ESP) of Γ; and, 2. s∗is a Nash equilibrium of G (t∗).
In summary, material payoffs are thus determined directly by actions and indirectly by types. However, these types and actions have different origins. Types evolve through an evolutionary process, whereas actions correspond to Nash equilibrium behavior in a base game for a given set of types. Below, I discuss the existing applications of this framework to the Tullock contest game as discussed above.
3.2 Previous Applications of Indirect Evolution to Contests
Following the pioneering work by Güth et al. (1992), and several other applications to 2 × 2-games, preference evolution has recently been applied to games with a continuum of strategies. Examples of which are the work by Eaton et al. (2003) who put forward the idea of relative payoff maximization in finite populations to indirect evolutionary methods, and the very general application by Heifetz et al. (2007) who determine the evolutionary stable preference equilibrium and sketch the properties of the
3 Indirect Evolution & Applications to Tullock Contests
associated evolutionary dynamics supplemented by an example in a Cournotesque competition game. Following the work by Eaton et al. (2003) and Heifetz et al. (2007) and the discussion of the finite population ESS of the Tullock contest by Hehenkamp et al. (2004), several authors have recently studied the ESP equilibrium in contests with finite populations. In particular, Leininger (2009) focusses on the evolutionary stability of interdependent preferences, and shows that in the unique ESP equilibrium of the contest game has players with negatively interdependent (i.e. spiteful) preferences. Three other studies present the evolutionary stability of overoptimism, either due to overvaluation of the prize (Boudreau et al.,2012), or due underestimation of the individual marginal cost of effort provision in pairwise interactions (Noeske, 2011) and full population ‘playing the field’ interactions (Wärneryd,2012). The equilibrium effort levels consistent with the equilibrium types in these papers are all equal to the (finite population) ESS earlier found by Hehenkamp et al. (2004).4The results of these authors are complementary in the sense that independent of assumed preference set they obtain an equivalent equilibrium effort prediction, which so happens to be equal to the finite population ESS of the contest game. Neither of these two equivalences, however, are coincidental.
Firstly, given the specification of the utility function of the contest game, the equivalent effort prediction between overoptimistic valuation and underestimation of marginal costs follows naturally.5However, even the observation that equilibrium effort is identical for the evolution of overconfidence and the spiteful equilibrium of Leininger (2009) is to be expected given the way the preference equilibrium is established. As it is behavior (and thus only indirectly player type) that determines fitness it follows that any type space which includes a type that exhibits ‘optimal’ behavior6will then naturally yield this type to be the evolutionary stable type.
Secondly, the equivalence between evolutionary stable preference behavior and finite population ESS behavior in 2-player contests follows from a result by Dixit (1987), who very generally shows that in symmetric two-player contests there is no advantage in effort commitment ahead in time in order to influence behavior of an opponent, and consequently no benefit in unilaterally deviating from the Nash equilibrium outcome through such commitment. If one considers the nature in which preferences affect behavior, and more importantly, the behavior of an opponent, it follows that preferences constitute a ‘commitment’ to a particular utility function and corresponding best response behavior. In the absence of advantages to commitment to non-rational behavior there naturally is nothing to be gained from preferences that commit to anything other than payoff maximization. Furthermore, as already noted by Leininger (2009), the determination of evolutionary stable preferences follows from the ESS solution,
4
Wärneryd (2012), however, does not consider equilibrium effort implied by the evolutionary stable preference and hence fails to note the equivalence between ESS and ESP effort in his result. In fact, as Eaton et al. (2003) have shown, in full population interactions the ESP and ESS effort are necessarily equal.
5
By division of the expected utility function by either (biased) costs or prize valuation one can obtain a function that is strategically equivalent (with either costs, or valuation equal to unity), but biased in only one of the two parameters. The same reasoning applies to an individual utility function that overestimates the probability of winning the contest by some constant factor.
6
Where I note that a type does not have to consist of a single bias. The idea that behavior, rather than the assumed type, is what determines stability is better illustrated when one considers a type space that considers multiple biases simultaneously. I illustrate in Appendix A that when the type space is extended to cover both interdependence of utility as well as biased marginal cost estimation that there is not a single evolutionary stable preference. Instead the subset of preferences that yield the ‘optimal’ (ESS) behavior constitutes a neutrally stable set. Consequently, I argue that without further assumptions, there is no inherent evolutionary advantage of interdependent or biased marginal cost preferences in the evolutionary game. This observation is both beneficial and problematic: although in terms of equilibrium behavior the ESP result is robust for the assumption of which type of bias drifts results away from rational behavior, it does not allow for a definitive answer to the question which preference set is the most plausible to arise in reality. In the remainder of this paper I will not touch upon this issue, arguing that experimental data is better suited than theory in answering what kind of bias is more plausible.
3 Indirect Evolution & Applications to Tullock Contests
rather than the Nash solution. Therefore, under the assumption of population smallness, rational (or absolute) profit-maximizing individuals are driven out of the population by players with preferences that mimic behavior of the finite population ESS which concerns players that maximize their relative payoffs. As the previous work on preference evolution in the two-player contest has shown, preferences that satisfy relative payoff maximization can be consistent with it spitefulness, overvaluation of the contest prize and underestimation of personal marginal costs.
3.3 Player Aggression and Commitment to Non-Rational Effort Levels
As established above, like the result for the finite population ESS it follows that the equilibria associated with indirect evolution can in some scenarios be consistent with behavior that deviates from objective expected payoff maximization. It will thus be useful to establish a consistent measure for the relative difference in player aggressiveness between ESP and Nash equilibrium behavior. To this end I denote:
Φ (n, N) =
xE SP− xN E
xN E . (21)
As both the direct and indirect evolutionary equilibria consider relative payoff maximization, rather than absolute payoff maximization, the finite population applications of both ESP and ESS feature an identical increase in player aggressiveness relative to the Nash equilibrium as a result of population smallness. This relative increase in player aggressiveness due to population smallness was shown in (17) to be equal to 1/(N − 1). It thus follows that (21) can be expressed as:
Φ (n, N) = xE SP− xE S S xN E + xE S S− xN E xN E = xE SP− xE S S xN E + 1 N − 1. (22)
As established in the previous section, in the absence of advantages for players to commit to behavior different from relative payoff maximization direct evolution and indirect evolution lead to the same equilibrium behavior. However, if advantages to such commitment are present the behavior consistent with ESS, which does not consider commitments, and behavior consistent with ESP are necessarily different. As such, commitment effects drive a wedge between ESS and ESP behavior. Although absent in 2-player contests Dixit (1987)’s result shows that advantages to overcommitment do exist for contests with more than 2 players. Hence, it is a worthwhile exercise to determine the indirect evolutionary equilibrium in contests for interactions sizes beyond the 2-player case discussed in existing applications of indirect evolution to Tullock contests. Naturally, this allows one to establish an evolutionary stable preference but more importantly also allows one to establish an equilibrium effort level that reflects the presence of advantages to overcommitment in terms of effort. To specifically quantify the effect of commitments on the evolutionarily stable level of player aggressiveness I employ the following notation:
ϕC(n, N) =
xE SP− xE S S
xN E =Φ (n, N) −
1
N − 1. (23)
If it is beneficial for players to overcommit in comparison to relative payoff maximization then ϕC >0, likewise if undercommitment is beneficial then ϕC<0. Finally, if advantages to commitment to behavior different from relative payoff maximization are absent, such that ESS and ESP behavior are equal to each other, then naturally ϕC=0.
4 Preference Evolution in the Tullock Contest with an Arbitrary Number of Players
4 Preference Evolution in the Tullock Contest with an Arbitrary Number of Players
4.1 Perfectly Observable Types
As mentioned above, existing literature on indirect evolution in rent-seeking games has restricted its attention towards pairwise interactions and full population interactions. In this section I extend these results to contests of an arbitrary interaction size without the imposition that the entire population takes part in any given contest. To obtain tractable results I restrict the mutation space of types to only include a bias in individual perception of personal marginal costs, denoted by the parameter τ, in comparable fashion to the work by Noeske (2011) and Wärneryd (2012). The parameter τi>0 (τi <0) indicates the extent to which individual i underestimates (overestimates) its marginal costs of effort. I restrict the mutation space of types to the set of types which have a corresponding preference that considers the game that is played to be a contest game. This implies that I assume that players cannot perceive their marginal cost to be 0 or negative; as such, the mutation space of any given player i is given by Ti = τi ∈ (−∞, c). The corresponding subjective utility function of a representative player i of this game is then given as follows.
ui(x1, . . . , xn∣ τi) = v xi ∑ n m=1xm − (c − τi) xi (24)
Where as before, xiis the effort of player i, v is the value of the prize of the contest which is common to all
players, and c is the marginal cost of effort which is equal for all players. Given the mutation space of types, the base game for any given iteration of the contest is given by the type vector t = (t1, . . . , tn) = (τ1, . . . , τn) and the corresponding set of preferences of the same form as (24). The objective material payoff (i.e. evolutionary fitness) of any given player i is unaffected its preferences, and is thus given by
fi(x1, . . . , xn) = v xi ∑ n m=1xm − c xi. (25)
Naturally, a rational player assesses its marginal costs correctly (i.e. τi =0) such that objective and subjective utility coincide. Given the definition of the subjective utility function equation (24) and corresponding objective material payoff of contestants (25) it is now a straightforward calculation to determine the ESP and corresponding equilibrium effort provision of this game.
Proposition 1. Consider the game described above with τ ∈ (−∞, c). For an infinite population interacting
in contests of size n the unique evolutionarily stable type is given by
τ∗
= c (n − 2)
n (n − 1)≥0. (26)
Where the inequality is strict provided that the interaction size n > 2. The corresponding ESP of a representative individual is thus defined as follows:
ui(x1, . . . , xn∣τ∗) = v xi ∑ n m=1xm − (c − τ∗) xi= v xi ∑ n m=1xm − c (1 − (n −2) n (n − 1)) xi. (27) The corresponding equilibrium effort level for each individual is given by:
xE SP = (n −1) v (c − τ∗) n2 = (n −1)2v cn (2 + (n − 2) n) (28)
4 Preference Evolution in the Tullock Contest with an Arbitrary Number of Players
Proof. To obtain the best response functions, first note the first order condition of player i: ∂ui ∂xi =0 ∶ (∑ n mxm) − xi (∑nmxm)2 v − (c − τi) =0. (29)
The first order conditions of all other players are defined analogously. Summing all partial derivatives then yields the following useful property:
n ∑ m=1 ∂um ∂xm = (n −1) ∑nmxm (∑nmxm)2 v − n ∑ m=1 (c − τm) =0. (30)
By rearranging (30) it follows that aggregate effort can be expressed in terms of subjective costs. (n −1) v ∑ n m=1xm = n ∑ m=1 (c − τm) ⇐⇒ n ∑ m=1 xm= (n −1) v ∑ n m=1(c − τm) (31)
Plugging (31) in the first order condition of player i as given in (29) and solving for xi then yields best
response effort x∗i (t) of player i in terms of exogenous variables and the subjective cost perceptions of the
participants of the contest:
x∗ i (t) =
v ((n − 1) ∑nm=1(c − τm) − (n −1)2(c − τi))
(∑nm=1(c − τm))2 . (32)
Plugging in the best responses as given in (32) in the expression for objective material payoffs (25) yields the indirect fitness function of a representative individual i as (where j ≠ i):
Fi(t) = v x∗ i (t) ∑ n mxm∗(t) v − cx∗ i (t) = v (c − τi− ∑nj τj) (c + (n −2) τi− ∑nj τj) (nc − τi− ∑nj τj) 2 (33)
As shown above, the ESS of the Tullock contest can be calculated by deriving the strongest mutant facing a homogenous population with the same strategy and equating this strategy and the mutant strategy in the corresponding first order condition. The same reasoning applies to obtaining the ESP, however, in this case the strongest mutant is a type, rather than a strategy. To obtain the ESP it is thus sufficient to calculate the strongest mutant type facing a homogeneous incumbent population and equating the mutant’s type to the type of the rest of the population in the resulting first order condition. To this end, I consider a contest in which a mutant player i with type τi faces a homogeneous opposition of n − 1 players with type
τ, where I denote this type vector of the resulting contest by ˆt. I substitute ∑nj τj = (n −1) τ in (33) to obtain the indirect fitness function of the mutant in such a contest:
Fi(ˆt) =
v (c − τi− (n −1) τ) (c + (n − 2) τi− (n −1) τ)
(nc − τi− (n −1) τ)2 . (34)
The first order condition is then given as: ∂Fi(ˆt) ∂τi =0 ∶v (n − 1) (c 2 (n −2) + c (3 − 2n) τi− c (n −1)2τ + (n − 1) τ (τi+ (n −1) τ)) (c n − τi− (n −1) τ)3 =0 (35)
4 Preference Evolution in the Tullock Contest with an Arbitrary Number of Players
To obtain the symmetric equilibrium I substitute τi = τ = τ∗in (35), this yields the following expression: ∂Fi(ˆt) ∂τi R R R R R R R R R R Rτi=τ=τ∗ =0 ∶ v (n − 1) (c (n − 2) − (n − 1) nτ ∗ ) n3(c − τ∗)2 =0. (36)
The result for the evolutionarily stable type as given in equation (26) follows immediately by solving (36) for τ∗. Given the evolutionarily stable type it is immediate that the ESP is the preference u (x1, . . . , xn∣τ∗) that corresponds to this type as given in (27). Substituting the type of all players in equation (32) by τ∗ then yields the corresponding equilibrium effort level as given in equation (28).
To establish this type as the global maximizer of fitness, I note that the second derivative of indirect fitness with respect to the type of the mutant equals:
∂2Fi(ˆt) ∂τ2 i = 2v (n − 1) (−c 2 (3 + (n − 3) n) + c (3 − 2n) τi+ τ (n −1) (τi+ (n −1) τ)) (c n − τi− (n −1) τ)4 . (37)
At the fixed point, τi = τ = τ∗with τ∗as given in equation (26), the second order condition for a maximum then follows by substituting τ∗in (37), this yields the expression:
∂2Fi(ˆt) ∂τ2 i R R R R R R R R R R Rτi=τ=τ∗ <0 ∶ − 2v (n − 1) 6 c2n (2 + (n − 2) n)3 <0. (38)
It is immediate that (38) is satisfied for any n ≥ 2 and c, v > 0. I conclude τ∗is thus a maximum of indirect fitness. As a final step I establish τ∗as the global maximizer of indirect fitness in the type space τ ∈ (−∞, c) by comparing fitness of a mutant player i at the corner of the type space τi→ cto fitness of an equilibrium player when facing a population of equilibrium players with τ∗. Expected fitness of a τ∗player facing n − 1 opponents also of type τ∗is equal to:
Fi(τ∗i, τ ∗ −i) =
v
n (2 + (n − 2) n). (39)
In comparison, a mutant at the boundary of the type space obtains: lim
τi→c
Fi(τi, τ∗−i) = − v (n − 2)
2 + (n − 2) n, (40)
which is strictly less than a player with type τ∗for any interaction size n. I conclude that τ∗is the global maximizer of the indirect fitness function Fi of the type space τ ∈ (−∞, c), and hence that this type is
indeed the evolutionarily stable type of the symmetric n-player Tullock contest.
Noting that τ∗is positive for any n > 2, it follows that given the current mutation space that it is optimal for players to underestimate their marginal costs, leading to higher effort provision than is the case for the infinite population ESS (Nash equilibrium) of the contest game.7This implies that, consistent with Dixit (1987) that in equilibrium players have a tendency to overcommit to the contest in terms of their effort provision. I note that in the absence of population smallness the deviation from objective fitness
7
I note that the derivation of the optimal type above extends to an evolutionary series of contests of varying size, for which the average interaction size can take non-integer values (larger than 2). As such I do not impose that the interaction size n has a discrete value in the analysis below.
4 Preference Evolution in the Tullock Contest with an Arbitrary Number of Players 4 6 8 10 12 14 n 0.05 0.10 0.15 Τ* Τ*
(a) Equilibrium type τ∗
for c = v = 1. 4 6 8 10 12 14 n 0.05 0.10 0.15 0.20 ΦC ΦC
(b) Relative difference in player aggressiveness as a result
of commitment effects. 2 4 6 8 10 12 14 n 0.05 0.10 0.15 0.20 0.25 0.30 x xESS xESP
(c) Comparison of Nash equilibrium effort to ESP
equi-librium effort for c = v = 1.
2 4 6 8 10 12 14 n 0.2 0.4 0.6 0.8 1.0 D DESS DESP
(d) Comparison of Nash equilibrium rent dissipation to
ESP equilibrium rent dissipation.
Figure 1: Illustrations of the optimal type and the corresponding relative difference in player aggressiveness
between the indirect evolutionary equilibrium and Nash equilibrium (infinite population ESS).
maximization that arises in the ESP equilibrium can be fully attributed to commitment effects. Using the previously established notation, it follows from (6, 28) that the difference in player aggression due to commitment effects is given by:
lim N →∞ ϕC(n, N) = lim N →∞Φ (n, N) = n − 2 2 + (n − 2) n. (41)
Where I note that ϕCcan also be also be written as τ∗
c−τ∗, revealing the relationship between cost
underesti-mation and effort provision in equilibrium. As (41) shows the presence of advantages to overcommitment yield comparatively more aggressive behavior in equilibrium when n > 2, but not when n = 2. Both these results are a direct consequence of the way in which contest success probabilities are defined.
I first note that the Tullock contest success function has players optimizing their behavior against the aggregate effort level of all their opponents. In the case of a single opponent, players directly maximize against the effort level of their opponent. As was already shown by Dixit (1987), this implies that commit-ment incentives for both players go in opposite directions. In the current specification with marginal cost preferences, this can be illustrated by analyzing the first order condition for the two player contest. For the two player case the first order condition in equation (35) reduces to:
∂Fi ∂τi ∣ (n=2) =0 ∶ − (c − τ) (τ + τi) (2c − τ − τi)3 =0 (42)
For some fixed preference of the opponent τ, this implies a fitness maximizing preference τ∗i = −τ. The
only stable solution (i.e. a symmetric solution in types) thus constitutes the case with rational preferences (τi= τ =0), and the same level of player aggression as for the Nash equilibrium. Now instead consider a
4 Preference Evolution in the Tullock Contest with an Arbitrary Number of Players
contest with more than 2 players. In this case every individual chooses an effort level to compete against the aggregate effort level by its opponents. It follows that by means of its type a player affects the best response of multiple opponents at once but only imperfectly. As a result commitment incentives are not perfectly asymmetrical as is the case in 2-player contests.8 The result is an equilibrium population of individuals which act more aggressively than the Nash equilibrium.
As is illustrated in the Figure 1c, the ESP solution also has the implication that unlike Nash equilibrium effort that individual effort exertion is increasing in n for interaction sizes in the vicinity of 2 and decreasing for larger interaction sizes, leading to the existence of an interaction size with the largest level of bias. This can be explained by noting that as n increases a player can by committing to more aggressive behavior ceteris paribus decrease optimal effort provision of an increasing number of opponents, increasing its fitness in the process. In equilibrium populations this effect thus puts an upward pressure on effort provision as the interaction size increases. However, the opponents of any given player base their decision on the aggregate effort level of all players (and thus only partly on the effort level of any individual opponent). Therefore, as the interaction size increases any player contributes an ever smaller fraction of aggregate effort and consequently has an ever smaller influence on the behavior of individual opponents. It follows that benefits to higher effort commitments decrease as the amount of players increases. In terms of effort provision I observe that the first effect dominates for interaction sizes up to n = 2 +
√
2,9and that the second effect dominates for larger interaction sizes. It follows directly that the interaction size n = 2 +
√
2 has the highest level of evolutionarily stable bias (and consequently the largest difference in player aggression compared to the Nash equilibrium).
As a final note, given that effort provision in the ESP strictly exceeds effort provision in the Nash equilibrium for any n > 2 it follows that rent dissipation consistent with ESP behavior strictly exceeds rent dissipation consistent with Nash equilibrium behavior. However, as is illustrated in figure 1d it follows that rent dissipation at the evolutionary stable preference equilibrium,
DE SP=
cnxE SP
v =
(n −1)2
n (2 + (n − 2) n), (43)
never amounts to full rent dissipation or over dissipation for any interaction size. 4.2 Perfectly Observable Types in a Finite Population
As discussed in section 3.2 results from previous applications of indirect evolution have shown that non-rational preferences are stable under the consideration of population smallness, with equilibrium behavior identical to the finite population ESS. The previous section showed that the existence of advantages to commitment in contests with more than 2 players also result in non-rational stable types. In this section I consider the interaction between both effects, I thus model an n-player Tullock contest where contestants are drawn from a finite population of N individuals. The contest game and corresponding type space are the same as in the previous section with preferences of individual players defined as in (24) and objective material payoffs as defined in (25).
Proposition 2. Consider the finite population adaptation of the game described in section 4.1 with τ ∈
(−∞, c). For a finite population of size N interacting in contests of size n the unique evolutionarily stable type is given by
τ∗
= c (N (n − 2) + n)
N (n − 1) n ≥0 ∀n, N , (44)
8
That is, when facing n − 1 opponents with the same type τ, then the strongest mutant type τ∗i ≠ −τ. 9
4 Preference Evolution in the Tullock Contest with an Arbitrary Number of Players
where the inequality is strict provided that the total population N is of finite size. The corresponding ESP of a representative individual i is then given by
ui(x1, . . . , xn∣τ∗) = v xi ∑ n m=1xm − (c − τ∗) xi = v xi ∑ n m=1xm − c (1 − (N (n −2) + n) N (n − 1) n ) xi. (45) The resulting equilibrium effort level for each individual is given by:
xE SP= (n −1) v (c − τ∗) n2 = v N (n − 1)2 cn (N (n2− 2n + 2) − n). (46)
This effort level is greater than the Nash equilibrium for any finite population size N .
Proof. Noting that individual preferences and evolutionary fitness are identical to the previous section, it follows that for a given type vector the best response of any given player x∗i (t) and indirect fitness Fi(t) are still given by (32) and (33), respectively. Let player i be a single mutant with a type τithat arises in a
homogeneous population of N − 1 identical players with preference τ. It follows that when the mutant enters the contest it will always face a group of n − 1 players of with type τ, as before I denote this type vector by ˆt. Expected indirect fitness of the mutant is thus identical to expression (34). However, expected indirect fitness of non-mutant players in any given contest depends on whether the mutant player takes part in the contest.
Consider a player j to be representative of the non-mutant population. For an unspecified type vector t expected indirect fitness of the j player is analogous to the expression an i player as given in (33):
Fj(t) = v x∗ j (t) ∑ n mxm∗ (t) v − cx∗ i (t) = v (c − ∑nmτm) (c + (n −1) τj− ∑nmτm) (nc − ∑nmτm)2 . (47)
Under the assumption of uniform random matching the probability that player j will face the mutant (player i) from the remaining (N − 1) players in the population in a contest with (n − 1) participants is equal to (n − 1) / (N − 1). If this event occurs then the j player faces an opposition of n − 2 other non-mutant players of type τ and the single mutant with type τi. Substituting ∑
n
mτm= τi+ (n −1) τ and τj= τin (47) yields indirect fitness of the non-mutant player for the type vector ˆt:
Fj(ˆt) =
v (c − τi− (n −1) τ) (c − τi) (nc − τi− (n −1) τ)2
(48)
With the remaining probability a non-mutant player enters a contest with n − 1 other non-mutants. I denote the corresponding type vector by tN M. By substituting ∑
n
mτm= nτand τj = τin (47) I obtain expected indirect fitness of the j player of a contest without the mutant player:
Fj(tN M) = v n−
(n −1) vc
n2(c − τ). (49)
To derive the finite population ESP equilibrium given these expressions I apply the finite population evolutionary equilibrium as presented in Schaffer (1988) to types. Given the expected indirect fitness of the mutant (34) and the non-mutant players (48, 49) I note that the strongest mutant type τiwhen facing
4 Preference Evolution in the Tullock Contest with an Arbitrary Number of Players
a population of N − 1 non-mutants with type τ is the type that maximizes relative expected indirect fitness Ri = Fi− Fj, that is: maxτ i Ri(Fi, Fj) = Fi(ˆt) − n − 1 N − 1 Fj(ˆt) − (1 − n − 1 N − 1) Fj(tN M) = v (c − τi− (n −1) τ) (c + (n − 2) τi− (n −1) τ) (nc − τi− (n −1) τ)2 − v (n − 1) (c − τ i) (c − τi− (n −1) τ) (N −1) (nc − τi− (n −1) τ)2 − (1 − n − 1 N − 1) ( v c − (n −1) vc n2(c − τ) ) . (50)
The maximization problem in (50) has the following first order condition: ∂Ri ∂τi =0 ∶ vc (n − 1) (nc − τi− (n −1) τ) (N −1) (nc − τi− (n −1) τ)3 + v N (n − 1) (c 2 (n −2) − c (n − 1)2τ + c (3 − 2n) τi+ (n −1) τ ((n − 1) τ + τi)) (N −1) (nc − τi− (n −1) τ)3 =0 (51) A strategy (type) is an ESS (ESP) if the strongest mutant cannot do better than the strategy (type) adopted by an equilibrium population. By setting τi = τ = τ∗, the first order condition above simplifies to the following expression: ∂Ri ∂τi∣ (τi=τ=τ∗) =0 ∶ v (n − 1) (cn + N (c (n − 2) − (n − 1) nτ ∗ )) (N −1) n3(c − τ∗)2 =0. (52)
Solving (52) for τ∗yields the evolutionarily stable type and corresponding ESP as presented in equations (44) and (45), respectively. The corresponding equilibrium effort level xE SPfollows by substituting the types of all players by τ∗in expression (32).
To establish this interior solution as the unique maximizer of relative fitness, I first note that the second derivative of relative fitness with respect to the type of the mutant is given as follows:
∂2Ri ∂τ2 i = − 2vc (n − 1) ((n − 1) τ + τ i− nc) (N −1) (nc − τi− (n −1) τ)4 − 2vN (n − 1) (c 2 (n −2) − c (n − 1)2τ + c (3 − 2n) τi+ (n −1) τ ((n − 1) τ + τi)) (N −1) (nc − τi− (n −1) τ)4 (53) The second order condition for a maximum at τ∗as defined in (44) is given by:
∂2Ri ∂τ2 i ∣ (τi=τ=τ∗) <0 ∶ − 2N 4(n − 1)6v (N −1) c2n (−n + N (2 + (n − 2) n)) <0 (54)
Which holds for any c, v > 0 and N ≥ n ≥ 2. Finally, comparing relative fitness of the interior solution Ri(τ∗i, τ
∗
−i) =0 and the corner solution
lim τi→c Ri(τi, τ∗−i) = − N2(n −1)2v (N −1) n (N (2 + (n − 2) n) − n) <0 (55)
4 Preference Evolution in the Tullock Contest with an Arbitrary Number of Players 2 4 6 8 10 12 14 n 0.05 0.10 0.15 0.20 0.25 0.30 x xNE xESS xESP
(a) Comparison of equilibrium effort levels.
2 4 6 8 10 12 14 n 0.2 0.4 0.6 0.8 1.0 D DNE DESS DESP
(b) Comparison of equilibrium rent dissipation. Figure 2: Illustration of equilibrium individual effort and rent dissipation for different interaction sizes in a
finite population N = 15 with c = v = 1.
shows that τ∗as given in the proposition is indeed the unique global maximizer of relative fitness of the type space τ ∈ (∞, c) for any N ≥ n ≥ 2 and c, v > 0. As such I conclude that the type τ∗and corresponding preference u (x1, . . . , xn∣τ∗)constitute the finite population ESP of the n-player Tullock contest.
Consistent with earlier findings by Noeske (2011) and Wärneryd (2012), which are special cases of the current specification with n = 2 and n = N respectively, the result here shows that when the mutation space of types consists solely of the subjective interpretation of marginal costs of effort it is evolutionarily optimal in finite populations for individuals to underestimate their marginal costs. Figure 2 illustrates individual effort and rent dissipation levels in the Nash and finite population ESS equilibrium, as derived in section 2, and the finite population ESP equilibrium for a population N = 15. It follows that player aggressiveness generally differs between the three solution concepts, with equilibrium effort levels that result from evolutionary methods strictly exceeding Nash equilibrium effort. Using the notation from section 3.3 I note that the relative difference in player aggressiveness between the ESP and the Nash equilibrium Φ for an unspecified population size is given by:
Φ (n, N) = n + N (n − 2)
N (2 + (n − 2) n) − n. (56)
It follows that Φ > 0 for any valid interaction and population size, i.e. N ≥ n ≥ 2, provided that N is finite. Expression (56) shows that for a given interaction size a smaller population always results in more aggressive player behavior in the indirect evolutionary equilibrium.10As was the case for the finite population ESS the finite population ESP only results in complete rent dissipation for full population interactions. Furthermore, consistent with the section 4.1, I note that benefits to overcommitment lead to the result that ESP effort is greater than ESS effort for all but two special cases for any given population size. The relative difference in player aggressiveness due to commitment effects can be expressed in general for all interaction and population sizes:
ϕC(n, N) = Φ (n, N) − 1 N − 1 = N (N − n) (n − 2) (N −1) (N (2 + (n − 2) n) − n). (57) 10 Noting that: ∂Φ ∂N = − (n −1) n2 (N (2 + (n − 2) n) − n)2 <0, for all N ≥ n ≥ 2.
4 Preference Evolution in the Tullock Contest with an Arbitrary Number of Players
(a) Plot of ϕCfor population sizes up to 15 players and all valid interaction
sizes given the population size.
(b) Contour plot associated with Figure 3a.
Figure 3: Relative difference in player aggressiveness in terms of individual effort in equilibrium between the
4 Preference Evolution in the Tullock Contest with an Arbitrary Number of Players
It follows that ϕC =0 for either n = 2 or N = n, but that ϕC >0 for all N > n > 2. Figure 3 presents an accompanying illustration of (57). With the exception of the special cases discussed by Noeske (2011) and Wärneryd (2012), I observe that as was the case for the application to infinite population sizes in the previous section that benefits to overcommitment drive ESP effort beyond finite population ESS equilibrium effort. Type evolution thus results in the stability of a type that overcommits to effort in comparison to relative fitness maximization for N > n > 2.11
The main insight of the current application of preference evolution with finite populations is that reductions in the overall population size decrease the scope for overcommitment for a given interaction size as given in (57); which in the limit case n = N results in benefits to overcommitment being eliminated completely. The underlying nature of this effect has been formally analyzed by Eaton et al. (2003), who have shown that for any game with full population interactions the ESP and ESS coincide in terms of equilibrium behavior. To interpret this finding first note that commitment to an action by means of a non-rational player type can be valuable, both in absolute and relative fitness terms, due to the direct effect on the strategic behavior of opponents in a given interaction. Contrary to commitment, which as mentioned affects behavior of opponents directly, relative fitness maximizing strategic behavior does not consider the opponents in a given contest, but rather a player’s relative performance for a given population state.
As an illustration, consider a population of incumbents with an identical strategy. When a mutant enters such a population relative fitness maximizing incumbents adjust their strategy to maximize their payoff in interactions facing the mutant and interactions without the mutant weighted for the probability of occurrence of either contest. As such, the presence of a mutant only indirectly influences the strategic behavior of the incumbents by its effect on the population state.12It then follows that in full population interactions any given member of the non-mutant population faces the mutant with certainty in every iteration of the evolutionary game. Hence, players of the incumbent population adjust to the mutant in every iteration of the evolutionary contest in the same manner as an incumbent player in an indirect evolutionary game that directly considers types (and commitment through types) would when placed in an iteration of the evolutionary game which includes the mutant. The player aggressiveness that results from strategy evolution (or in the case of indirect evolution, the evolution of the share of base games with a type vector that includes the mutant player) thus in equilibrium eliminates the benefit of further commitment in the base game that corresponds to the equilibrium indirect evolutionary game. As such, for playing the field interactions the evolutionary stable preference naturally coincides with the type of player in the type space that mimics finite population ESS behavior.
11
Furthermore, as was the case for populations of infinite size the relative difference in equilibrium player aggressiveness due to commitment effects depends on the interaction size n. Increases in the interaction size can coincide with either a positive or a negative impact of additional effort exertion on fitness, such that for any given population size N there is an interaction size with an evolutionarily stable type with the largest degree of marginal cost bias for that population size:
∂ϕC ∂n =0 ⇐⇒ nM a x ∣N= 2N + √ 2 √ N2− N 1 + N .
It follows from the expression above that given N ∈ [2, ∞) that the maximum level of bias evolves for n ∈ [2, 2 + √
2). The strongest effects of commitment on player aggression thus occur for populations of infinite size. Noting that for infinite populations, the positive effect of additional effort dominates the negative effect in the largest subdomain of n, resulting not just in the largest value of nM a x ∣N, but also in the largest difference between ESS and ESP effort exertion for any interaction
size n > 2.
12
Where I note that in an infinite population this effect is negligible, which results in the equivalence of behavior observed for the absolute payoff maximizing Nash equilibrium player and the and the relative payoff maximizing ESS player, as illustrated by the results presented in (6-7, 15-17).
4 Preference Evolution in the Tullock Contest with an Arbitrary Number of Players
In the context the standard Tullock contest as discussed here, in addition to 2-player contests it implicitly follows that overcommitment decreases in value as the portion of the rent dissipated as a result of relative payoff maximization increases. As discussed above, in the limit case when the rent is fully dissipated as a result of strategy evolution, i.e. the case in playing the field interactions, the benefits to overcommitment in base games of the indirect evolutionary game are eliminated completely. I thus conclude that the general increase in aggressiveness in the population due to population smallness effects reduces the scope to which players benefit from directly influencing their opponents by committing to more aggressive behavior than relative fitness maximizing behavior.
4.3 Unobservable Types in a Finite Population
The equilibrium type and corresponding behavior derived in the previous sections was based on the underlying assumption that individuals are capable of assessing their opponents’ type and corresponding subjective cost level in every iteration of the contest. Under these assumptions with the exception of the two special cases discussed in the previous sections equilibrium behavior was shown more aggressive than the ESS, both in an infinite as well as a finite population setting. I argued that types with overcommitment in terms effort are stable due to the direct impact of preferences on behavior of opponents.
In the following, to illustrate the nature of preferences as a commitment device, I will analyze preference evolution when preferences are not observed directly. Rather, I assume that players are only aware of the current average type in the population and condition their behavior on this type rather than directly on the types of their opponents in any given interaction. In similar vein to applications by Heifetz et al. (2007) and Dekel et al. (2007) I assume that players play a Bayesian equilibrium with a common prior regarding the distribution of types and average type in the population.
Let Ω denote the current state of the population, and let ω be the average type of this population state. It follows that for a given population state Ω the best estimation of the type preference of any given opponent is the average type present in the population. The expected type of any given player is thus given by E [τ ∣ Ω] = ω. Given these assumptions, I note that expected subjective utility of a representative individual i is defined as follows:
E [ui(x1, . . . , xn∣ τi) ∣Ω] =
v xi xi+ (n −1) xω
− (c − τi) xi (58)
Where xω=E [x ∣ Ω], the expected effort level of a player given the current average type in the population ω. Objective material payoffs do not directly depend on types and expectations regarding behavior of opponents, and as such are still given as in the previous sections, i.e. as in (25).
Proposition 3. Consider the game described above with τ ∈ (−∞, c). For a finite population interacting in
contests of size n the unique evolutionarily stable type is given by τ∗
= c
N >0, ∀N ≥ n ≥ 2. (59)
The corresponding ESP for a representative individual i is given by:
E [ui(x1, . . . , xn∣ τ∗) ∣Ω] = v xi xi+ (n −1) xω − (c − τ∗) xi= v xi xi+ (n −1) xω −c (N − 1) N xi. (60)
In an equilibrium population Ω∗, all players are of type τ∗such that ω = τ∗, the corresponding equilibrium
individual effort level of all players is then defined by:
xE SP= v N (n − 1) cn2(N −
