Risk aversion in symmetric and asymmetric contests

We analyze existence, uniqueness and properties of equilibria in incompletely discriminating Tullock contests with logistic contest success functions, when contestants are risk averse. We prove that a Nash equilibrium for such a contest exists, but give an example of a symmetric contest with both symmetric and asymmetric equilibria, showing that risk aversion may lead to multiple equilibria. Symmetric contests have unique symmetric equilibria but additional conditions are necessary for general uniqueness. We also study the effects on incumbents of additional competitors entering the contest under these conditions and examine the effects of risk aversion on rent dissipation in symmetric and asymmetric contests.     

Tournaments of Weakly Heterogeneous Players

We consider three prominent tournament formats—contests, binary elimination tournaments, and round-robin tournaments—in the case when players are heterogeneous in their abilities but the heterogeneity is, in a well-defined sense, weak. Using linear approximation, we characterize equilibrium strategies and payoffs in the three tournament games and compare them to the benchmark symmetric case of identical players. We describe small deviations from the symmetric equilibria by elasticities of a player's equilibrium effort with respect to her own ability and the abilities of her rivals. Our results only require general symmetry and smoothness assumptions but not specific functional forms for the probabilities of winning in tournaments. We show that, in equilibrium: (i) a player's effort and payoff depend on her rivals' abilities in a model-independent way, either through the average ability of the field (for static games), or through the properly discounted average ability of the field (for dynamic elimination tournaments); (ii) players respond stronger to changes in their own relative abilities than to changes in their rivals' relative abilities; (iii) aggregate effort (dissipated rent) does not change compared to the benchmark case; (iv) it is not possible to manipulate aggregate effort through seeding in binary elimination tournaments, although optimal seeding schemes for other purposes can be identified; and (v) balanced seeding and a uniform distribution of relative abilities cancel out the dependence of a player's effort on her rivals' abilities in binary elimination tournaments.     

The non-constant-sum Colonel Blotto game

The Colonel Blotto game is a two-player constant-sum game in which each player simultaneously distributes his fixed level of resources across a set of contests. In the traditional formulation of the Colonel Blotto game, the players?? resources are ??use it or lose it?? in the sense that any resources that are not allocated to one of the contests are forfeited. This article examines a non-constant-sum version of the Colonel Blotto game that relaxes this use it or lose it feature. We find that if the level of asymmetry between the players?? budgets is below a threshold, then there exists a one-to-one mapping from the unique set of equilibrium univariate marginal distribution functions in the constant-sum game to those in the non-constant-sum game. Once the asymmetry of the players?? budgets exceeds the threshold, this relationship breaks down and we construct a new equilibrium.     

On the (non-)lattice structure of the equilibrium set in games with strategic substitutes

This paper studies models where the optimal response functions under consideration are not increasing in endogenous variables, and weakly increasing in exogenous parameters. Such models include games with strategic substitutes, and include cases where additionally, some variables may be strategic complements. The main result here is that the equilibrium set in such models is a non-empty, complete lattice, if, and only if, there is a unique equilibrium. Indeed, for a given parameter value, a pair of distinct equilibria are never comparable. Therefore, with multiple equilibria, some of the established techniques for exhibiting increasing equilibria or computing equilibria that use the largest or smallest equilibrium, or that use the lattice structure of the equilibrium set do not apply to such models. Moreover, there are no ranked equilibria in such models. Additionally, the analysis here implies a new proof and a slight generalization of some existing results. It is shown that when a parameter increases, no new equilibrium is smaller than any old equilibrium. (In particular, in n-player games of strategic substitutes with real-valued action spaces, symmetric equilibria increase with the parameter.)      

Imitation with asymmetric memory

Models of learning in games based on imitation have provided fundamental insights as the relevance of risk-dominance equilibria or perfectly competitive outcomes. It has also been shown, however, that the introduction of nontrivial memory in those models fundamentally alters the results. This paper further considers the effect of asymmetric memory length in the population. We focus on two classical results and find that, while asymmetric memory crucially affects equilibrium selection in coordination games, it reinforces the stability of perfectly competitive outcomes in oligopoly games. The latter result is generalized to aggregative games and to finite population ESS in symmetric games.     

Evolution with Delay

This paper analyses evolutionary models with delays in players’ observations. I apply best response dynamics to symmetric 2 × 2 games, in particular the coordination game and the game with no symmetric pure strategy equilibrium. When the dynamics have no delay, their solutions converge to symmetric Nash equilibria monotonically. However when I introduce delays to the dynamics, although these dynamics represented by differential difference equations are seemingly simple, it shows that they have periodic solutions. JEL Classification Number: C79.     

On Norms and Coordination Games: A Rent–Seeking Approach

The aim of this paper is to extend the rent–seeking literature to the equilibrium selection problem in competitive coordination games, i.e., games in which more than one equilibrium exists, and individuals' preferences are opposed. We analyze alternative correlated equilibria: contractual agreements and legally enforced equilibria. The latter are to be understood as the outcome of rents–seeking contests in which players invest resources in order to set a norm. The contest is analyzed in its basic two–person setting and later generalized to the two–populations case. There we show that the outcomes depend on the relative payoff structure of the game, the technological properties of the contest, and the population distribution. Finally, the efficiency analysis focuses not only on the extent of the rent dissipation, but also on the comparative analysis of the inefficiencies that arise in the market (not coordinated) equilibrium.     

Information feedback and contest structure in rent-seeking games

We investigate the effects of information feedback in rent-seeking games with two different contest structures. In the share contest a contestant receives a share of the rent equal to her share of rent-seeking expenditures, while in the lottery contest a contestant wins the entire rent with probability equal to her share of rent-seeking expenditures. In share contests average expenditures converge to equilibrium levels when subjects only get feedback about own earnings, and additional feedback about rivals' choices and earnings raises average expenditures. In lottery contests information feedback has an opposite, and even stronger, effect: when subjects only get feedback on own earnings we observe high levels of rent dissipation, usually exceeding the value of the rent, and additional feedback about rivals' choices and earnings has a significant moderating influence on expenditures. In a follow-up treatment we make information feedback endogenous by allowing contestants in a lottery contest to make public or private expenditures. Subjects make the vast majority of expenditures privately and overall excess expenditures are similar to the lottery contest with own feedback.     

Sequential equilibrium in monotone games: A theory-based analysis of experimental data

A monotone game is an extensive-form game with complete information, simultaneous moves and an irreversibility structure on strategies. It captures a variety of situations in which players make partial commitments and allows us to characterize conditions under which equilibria result in socially desirable outcomes. However, since the game has many equilibrium outcomes, the theory lacks predictive power. To produce stronger predictions, one can restrict attention to the set of sequential equilibria, or Markov equilibria, or symmetric equilibria, or pure-strategy equilibria. This paper explores the relationship between equilibrium behavior in a class of monotone games, namely voluntary contribution games, and the behavior of human subjects in an experimental setting. Several key features of the symmetric Markov perfect equilibrium (SMPE) are consistent with the data. To judge how well the SMPE fits the data, we estimate a model of Quantal Response Equilibrium (QRE) [R. McKelvey, T. Palfrey, Quantal response equilibria for normal form games, Games Econ. Behav. 10 (1995) 6-38; R. McKelvey, T. Palfrey, Quantal response equilibria for extensive form games, Exp. Econ. 1 (1998) 9-41] and find that the decision rules of the QRE model are qualitatively very similar to the empirical choice probabilities.     

Effort and Performance in Public Policy Contests

Government intervention often gives rise to contests in which the possible “prizes” are determined by the status quo and some new public policy proposal. In this paper we study a general class of such two‐player public policy contests and examine the effect of a change in the proposed policy, a change that may affect the payoffs of the two contestants, on their effort and performance. Our results extend the existing comparative statics studies that focus, in symmetric contests, on the effect of a change in the value of the prize or, in asymmetric contests, on the effect of one contestant's valuation of the prize. Our results hinge on a fundamental equation that specifies the equilibrium relationship between the strategic own‐stake effect and the strategic rival's‐stake effect. This fundamental equation clarifies the role of the three possible types of ability and stake asymmetry in determining the effect of payoff variations on the efforts and performance of the contestants.     

Strategic Complements, Substitutes, and Ambiguity: The Implications for Public Goods

We examine the effect of ambiguity in symmetric games with aggregate externalities. We find that ambiguity will increase/decrease the equilibrium strategy in games with strategic complements/substitutes and positive externalities. These effects are reversed in games with negative externalities. We consider some economic applications of these results to Cournot oligopoly, bargaining, macroeconomic coordination, and voluntary donations to a public good. In particular we show that ambiguity may reduce free-riding. Comparative statics analysis shows that increases in uncertainty will increase donations, to a public good. Journal of Economic Literature C72, D81, H41.     

An indistinguishability result on rationalizability under general preferences

In this paper, we show that, in the class of games where each player??s strategy space is compact Hausdorff and each player??s payoff function is continuous and ??concave-like,?? rationalizability in a variety of general preference models yields the unique set of outcomes of iterated strict dominance. The result implies that rationalizable strategic behavior in these preference models is observationally indistinguishable from that in the subjective expected utility model, in this class of games. Our indistinguishability result can be applied not only to mixed extensions of finite games, but also to other important applications in economics, for example, the Cournot?Coligopoly model.     

Equilibria in rent-seeking contests with homogeneous success functions

Summary. We characterize pure-strategy Nash equilibria for symmetric rent-seeking contests in which the contest success function is homogeneous of degree zero. The equilibrium strategies have a simple form. We give sufficient conditions for existence of an equilibrium.Received: 11 February 2003, Revised: 27 August 2004, JEL Classification Numbers: D72. Correspondence to: Andrew J. YatesWe thank Douglas Nelson, Jac Heckelman, and two anonymous referees for helpful comments.     

Scale functions in equilibrium selection games

We present here an evolutionary game model, and address the issue of equilibrium selection working with the scale function of a diffusion process describing the dynamics of population processes with mutation modeled as white noise. This model is the same as the one in Foster and Young (1990) but with a different interpretation at the boundaries and with different mutation modelings. First, we justifiably assume that the boundaries of the solution of the stochastic differential equation are absorbing so that the first boundary of the interval [0,1] hit will determine the equilibrium selected. Then, working with the scale function, we obtain for 2×2 symmetric games and different mutation parameters, some new and interesting equilibrium selection results. The aim of this article is to describe another method of approach in evolutionary games with mutation which we believe will prove to be very useful in studying more general normal form games and different mutation modelings.     

Robustness of equilibria in anonymous local games

This paper studies the robustness of symmetric equilibria in anonymous local games to perturbations of prior beliefs. Two priors are strategically close on a class of games if players receive similar expected payoffs in equilibrium under the priors, for any game in that class. I show that if the structure of payoff interdependencies is sparse in a well-defined sense, the conditions for strategic proximity in anonymous local games are strictly weaker than the conditions for general Bayesian games of Kajii and Morris (1998) [11] when attention is restricted to symmetric equilibria. Hence, by exploiting the properties of anonymous local games, it is possible to obtain stronger robustness results for this class.     

Evolutionary stability of discrimination under observability

We study the evolution of preferences under perfect and almost perfect observability in symmetric 2-player games. We demonstrate that if nature can choose from a sufficiently general preference space, which includes preferences over outcomes that may depend on the opponent's preference-type, then, in most games, only discriminating preferences (treating different types of opponents differently in the same situation) can be evolutionary stable and some discriminating types are stable in a very strong sense in all games. We use these discriminating types to show that any symmetric outcome which gives players more than their minmax value in material payoffs (fitness) can be seen as equilibrium play of a player population with such strongly stable preferences.     

Are the treasures of game theory ambiguous?

Goeree and Holt (Am Econ Rev 91:1402?C1422, 2001) experimentally study a number of games. In each case, they initially find strong support for Nash equilibrium; however, by changing an apparently irrelevant parameter, they find results which contradict Nash equilibrium. In this paper, we study the five normal form games from Goeree and Holt (Am Econ Rev 91:1402?C1422, 2001). We argue that their results may be explained by the hypothesis that subjects view their opponents?? behaviour as ambiguous. Ambiguity-aversion causes players to avoid strategies, which give low out of equilibrium payoffs. Similarly, ambiguity preference can make strategies with high payoffs more attractive.     

Group efforts when performance is determined by the “best shot”

We investigate the private provision of a public good whose level is determined by the maximum effort made by a group member. Costs of effort are either commonly known or privately known. For symmetric perfect-information games, any number of players may be active and we characterize the unique (mixed-strategy) equilibrium in which active contributors use the same strategy. Increasing the number of active players leads to stochastically lower individual efforts and level of the public good. When information is private, the symmetric equilibrium is in pure strategies. Increasing the number of players yields a pointwise reduction in the equilibrium contribution strategy but an increase in equilibrium payoffs. Comparative statics with respect to costs and levels of risk aversion are derived. Finally, whether information is public or private, equilibria are inefficient—we provide mechanisms that improve efficiency.     

N-PERSON NASH BNARGAINING WITH VARIABEL THREATS*

We consider two models of n-person bargaining problems with the endogenous determination of disagreement points. In the first model, which is a direct extension of Nash's variable threat bargaining model, the disagreement point is determined as an equilibrium threat point. In the second model, the disagreement point is given as a Nash equilibrium of the underlying noncooperative game. These models are formulated as extensive games, and axiomatizations of solutions are given for both models. It is argued that for games with more than two players, the first bargaining model does not preserve some important properties valid for two-person games, e.g., the uniqueness of equilibrium payoff vector. We also show that when the number of players is large, any equilibrium threat point becomes approximately a Nash equilibrium in the underlying noncooperative game, and vice versa. This result suggests that the difference between the two models becomes less significant when the number of players is large.     

Batch queues with choice of arrivals: Equilibrium analysis and experimental study

We study the decisions agents make in two queueing games with endogenously determined arrivals and batch service. In both games, agents are asked to independently decide when to join a queue, or they may simply choose not to join it at all. The symmetric mixed-strategy equilibrium of two games in discrete time where balking is prohibited and where it is allowed are tested experimentally in a study that varies the game type (balking vs. no balking) and information structure (private vs. public information). With repeated iterations of the stage game, all four experimental conditions result in aggregate, but not individual, behavior approaching mixed-strategy equilibrium play. Individual behavior can be accounted for by relatively simple heuristics.