We consider a general class of imperfectly discriminating contests with privately informed players. We show that findings by Athey (2001) imply the existence of a Bayesian Nash equilibrium in monotone pure strategies

Wasser, Cédric

English

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Sonderforschungsbereich/Transregio 15  www.sfbtr15.de Universität Mannheim  Freie Universität Berlin  Humboldt-Universität zu Berlin  Ludwig-Maximilians-Universität München Rheinische Friedrich-Wilhelms-Universität Bonn  Zentrum für Europäische Wirtschaftsforschung Mannheim  Speaker: Prof. Dr. Urs Schweizer.  Department of Economics  University of Bonn  D-53113 Bonn, Phone: +49(0228)739220  Fax: +49(0228)739221                       * Humboldt University of Berlin       July 2010      Financial support from the Deutsche Forschungsgemeinschaft through SFB/TR 15 is gratefully acknowledged. Discussion Paper No. 331    Existence of a pure-strategy Bayesian Nash equilibrium in imperfectly discriminating contests  * Cédric Wasser  Existence of a pure-strategy Bayesian Nashequilibrium in imperfectly discriminatingcontestsCédric Wasser∗July 13, 2010AbstractWe consider a general class of imperfectly discriminating contests with pri-vately informed players. We show that findings by Athey (2001) imply the exis-tence of a Bayesian Nash equilibrium in monotone pure strategies.JEL classification: D72, D74, D82, C72Keywords: Contest, Imperfectly discriminating, Asymmetric information, Equi-librium existence, Interdependent values1 IntroductionModels of contests have been used to study a wide range of relevant economic ques-tions.1 In many contest-like situations players are, at least to some extent, privatelyinformed about their preferences. Depending on whether the contestant who in-vested the highest effort wins with certainty or not, one can distinguish betweenperfectly and imperfectly discriminating contests. Perfectly discriminating contests∗Humboldt University of Berlin, Institut für Wirtschaftstheorie I, Spandauer Str. 1, 10178 Berlin,Germany; email: cedric.wasser@wiwi.hu-berlin.de. Financial support from the DeutscheForschungsgemeinschaft through SFB/TR 15 is gratefully acknowledged.1See, e.g., Konrad (2009) for a recent survey.1such as the all-pay auction and the war of attrition have been thoroughly stud-ied under asymmetric information (e.g., Krishna and Morgan, 1997). The literatureon imperfectly discriminating contests, however, has almost exclusively focused oncompletely informed contestants. Notable exceptions include Hurley and Shogren(1998), Malueg and Yates (2004), and Fey (2008) who study contests among two pri-vately informed players under very specific assumptions concerning the distributiontypes are drawn from and the contest success function (CSF).2The information structure and the CSF we consider in this note are much moregeneral. We show how a well-known result by Athey (2001) from the theoretical lit-erature on Bayesian games can be readily applied to imperfectly discriminating con-tests, ensuring the existence of a pure-strategy equilibrium. An important require-ment for this approach is that the CSF is everywhere continuous. Apart from thatwemake only weak assumptions, allowing for correlated signals and interdependentvaluations. We require a player’s valuation to be increasing in the signal he observesand his probability of winning to be increasing in his effort. Most notably, no as-sumptions regarding concavity and convexity, respectively, of the contest success,valuation, and cost function are needed for the existence result.We proceed by stating the assumptions and proving the main result. This is fol-lowed by a short discussion of some examples from the contest literature our resultapplies to.2 Model and Main ResultThere are n risk neutral players competing for a single prize in a contest. Before thecontest, each player i privately observes a signal θi ∈ [θ i ,θ i ] ⊂ R. It is commonknowledge among players that the vector of all signals θ := (θ1, . . . ,θn ) is drawn fromthe continuous distribution F (θ ) with bounded density f (θ ). The value of the prizeto player i can be expressed as a function vi : [θ 1,θ 1]× · · · × [θ n ,θ n ]→ [v i ,v i ] ⊂ R+.We assume that vi (θ ) is continuous in θ and nondecreasing in θi .2They consider Tullock lottery contests where a player’s probability of winning is equal to his effortdivided by the sum of all efforts. Types are either drawn from simple discrete distributions or a con-tinuous uniform distribution. Wasser (2010) extends the analysis to more than two players and moregeneral continuous distributions. Another branch of the literature considers one-sided asymmetricinformation (e.g., Wärneryd, 2003).2Players compete by simultaneously choosing the amount of effort they invest.Each player i chooses x i ∈ R+. The probability that i wins the contest (or, alter-natively, the share of the prize i obtains) depends on the vector of all efforts x :=(x1, . . . ,xn ). The winning probability is given by the CSF p i : Rn+ → [0,1], where p i (x )is continuous in x and nondecreasing in x i .Investing effort is costly. Player i ’s cost of providing effort x i can be described bya continuous function c i :R+→R+. We assume that c i (0) = 0 and that there exists anx i ∈R+ such that c i (x i )> v i for all x i > x i .Let player i ’s ex post payoff be denoted byu i (x ,θ ) := p i (x )vi (θ )− c i (x i ).Suppose each of player i ’s opponents j uses a strategy ξj : [θ j ,θ j ] → R+ and letθ−i be the vector of the signals of i ’s opponents. To simplify the notation we willuse (x i ,ξ−i (θ−i )) to refer to the vector (ξ1(θ1), . . . ,ξi−1(θi−1),x i ,ξi+1(θi+1), . . . ,ξn (θn )).Then, i ’s objective function, i.e., i ’s interim expected payoff amounts toUi (x i ,θi ) :=Ehu i (x i ,ξ−i (θ−i )),θ θii=∫Θ−ip i x i ,ξ−i (θ−i )vi (θ ) f (θ−i |θi )dθ−i − c i (x i ) (1)whereΘ−i := [θ 1,θ 1]×· · ·×[θ i−1,θ i−1]×[θ i+1,θ i+1]×· · ·×[θ n ,θ n ] and f (θ−i |θi )denotesthe density of θ−i conditional on θi . We are now ready to prove the main result bymaking use of Athey (2001).Theorem 1. The contest has a pure-strategy Bayesian Nash equilibrium where eachplayer i uses a nondecreasing strategy ξi (θi ).Proof. In order to prove the result we apply Corollary 2.1 in Athey (2001). In the fol-lowing, we will verify that our model meets all the requirements of Athey’s corol-lary. First note that our assumptions are consistent with Athey’s assumption A1: f isbounded and atomless and the integral in (1) exists since u i is continuous in all of its3arguments. Because p i (x ) is nondecreasing in x i and vi (θ ) is nondecreasing in θi ,p i (xLi,x−i ) vi (θHi,θ−i )− vi (θLi,θ−i )> (≥) 0⇒ p i (xHi,x−i ) vi (θHi,θ−i )− vi (θLi,θ−i )> (≥) 0for all xHi > xLi and θHi >θLi . Consequently,Ui (xHi,θ Li)−Ui (xLi,θ Li)> (≥) 0⇒U (xHi,θHi)−U (x Li,θHi)> (≥) 0and therefore Athey’s Single Crossing Condition for games of incomplete informationis satisfied. Now, observe thatu i (x ,θ )≤ vi (θ )−c i (x i )≤ v i−c i (x i ). For x i > x i wehaveu i (x ,θ )< 0 while with x i = 0 a player can guarantee himself u i (x ,θ )≥ 0. Effort levelsx i > x i are clearly dominated. Thus, we can restrict players’ actions to x i ∈ [0,x i ] forall i – a closed interval as assumed by Athey. Moreover, u i (x ,θ ) is continuous in xfor all i . Existence of a pure-strategy Bayesian Nash equilibrium in nondecreasingstrategies hence directly follows from Corollary 2.1 in Athey (2001).3 DiscussionOur existence result applies to asymmetric information versions of many contestsconsidered in the literature. Under complete information, much attention is devotedto contests with a CSF that takes the formp i (x ) =g i (x i )∑nj=1 g j (x j )(2)where each g i (x i ) is an increasing function. Most importantly, Skaperdas (1996) pro-vides an axiomatization for the symmetric case where g i = g for all i . Under thecondition that g i (0)> 0, Theorem 1 implies existence of a pure-strategy equilibriumif such a contest is held among privately informed contestants.3 A related CSF The-orem 1 also applies to is p i (x ) = h i (x i ).1+∑nj=1h j (x j )with h i strictly increasing.3A popular specification, proposed by Tullock (1980), is g (x ) = xR for some R > 0. Theorem 1 doesnot apply because this CSF exhibits a discontinuity at x = (0, . . . ,0). However, if each player’s action isrestricted to a finite set, Theorem 1 in Athey (2001) implies existence of a pure-strategy equilibrium.4This CSF is, e.g., considered by Dasgupta and Nti (1998) and an axiomatization isprovided by Blavatskyy (2010).Under complete information contests with a CSF as in (2) are known to have onlymixed-strategy equilibria if g i is too convex (e.g., Cornes and Hartley, 2005). In con-trast, Theorem 1 shows that under asymmetric information there is always a pure-strategy equilibrium. This echoes a well-known property of the perfectly discrim-inating all-pay auction, having a pure-strategy equilibrium under asymmetric butnot under complete information. Indeed, the all-pay auction can be obtained as thelimit of a sequence of contests with increasingly convex g i . Although not neededfor equilibrium existence under asymmetric information, assuming g i to be concaveusually simplifies the analysis, especially since it allows for the equilibrium strategiesto be characterized by first-order conditions.4Hirshleifer (1989) proposes a symmetric specification of (2) where g (x i ) = e kx iwith k > 0. If n = 2, this is at the same time also a special case of the class of contestsintroduced by Lazear and Rosen (1981). They assume that the player i with the high-est x i + ǫi wins the contest, where ǫi is a randomly distributed noise term realizedonly after efforts have been chosen. Denoting the distribution function of ǫj − ǫi byL i , the corresponding CSF becomes p i (x ) = L i (x i−x j ). For continuous L i Theorem 1ensures existence of a pure-strategy equilibrium if players have private information.5Establishing the existence of a pure-strategy equilibrium, we make a first steptowards a better understanding of imperfectly discriminating contests under asym-metric information. Studying the properties of such equilibria seems to be a promis-ing direction for future research.ReferencesAMEGASHIE, J. (2006): “A contest success function with a tractable noise parameter,”Public Choice, 126(1), 135–144.4Equilibrium strategies can, in general, not be obtained in closed form. 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4Equilibrium strategies can, in general, not be obtained in closed form. For a symmetric version of (2) where g is an afﬁne function as proposed by Amegashie

5UndercompleteinformationCheandGale(2000)analyzeafamilyofpiecewiselinear Li andﬁnd that pure-strategy equilibria disappear if the contest is similar enough to the all-pay auction.

A contest success function with a tractable noise parameter,”

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Conﬂict and rent-seeking success functions: Ratio vs. difference models of relative success,”

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Contest success functions,”

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Efﬁcient Rent Seeking,” in Toward a theory of the rent-seeking society, edited by

Information in conﬂicts,”

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Rent-seeking contests with incomplete information,”

Single Crossing Properties and the Existence of Pure Strategy Equilibria in

Strategy and Dynamics in Contests,

Existence of a pure-strategy Bayesian Nash equilibrium in imperfectly discriminating contests

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Existence of a pure-strategy Bayesian Nash equilibrium in imperfectly discriminating contests

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