Payment schemes in infinite-horizon experimental games

We consider payment schemes in experiments that model infinite-horizon games by using random termination. We compare paying subjects cumulatively for all periods of the game; with paying subjects for the last period only; with paying for one of the periods, chosen randomly. Theoretically, assuming expected utility maximization and risk neutrality, both the cumulative and the last period payment schemes induce preferences that are equivalent to maximizing the discounted sum of utilities. The last period payment is also robust under different attitudes toward risk. In comparison, paying subjects for one of the periods chosen randomly creates a present-period bias. We further provide experimental evidence from infinitely repeated prisoners’ dilemma games that supports the above theoretical predictions.     

Essential equilibria of discontinuous games

We introduce spaces of discontinuous games in which games having essential Nash equilibria are the generic case. In order to prove the existence of essential Nash equilibria in such spaces, we provide new results on the Ky Fan minimax inequality. In the setting of potential games, we show that games with essential Nash equilibria are the generic case when their potentials satisfy a condition called weak upper pseudocontinuity that is weaker than upper semicontinuity.     

Essential equilibria in normal-form games

A Nash equilibrium x of a normal-form game G is essential if any perturbation of G has an equilibrium close to x. Using payoff perturbations, we show that for games that are generic in the set of compact, quasiconcave, and generalized payoff secure games with upper semicontinuous sum of payoffs, all equilibria are essential. Some variants of this result are also established.     

Essential equilibria of large generalized games

We characterize the essential stability of games with a continuum of players, where strategy profiles may affect objective functions and admissible strategies. Taking into account the perturbations defined by a continuous mapping from a complete metric space of parameters to the space of continuous games, we prove that essential stability is a generic property and every game has a stable subset of equilibria. These results are extended to discontinuous large generalized games assuming that only payoff functions are subject to perturbations. We apply our results in an electoral game with a continuum of Cournot-Nash equilibria, where the unique essential equilibrium is that only politically engaged players participate in the electoral process. In addition, employing our results for discontinuous games, we determine the stability properties of competitive prices in large economies.     

Discontinuous stable games and efficient Nash equilibria

In the recent paper by the author [Scalzo, V., 2010. Pareto efficient Nash equilibria in discontinuous games. Economics Letters 107, 364-365], a class of discontinuous games where efficient Nash equilibria exist has been defined. In the present paper, we complete the previous investigation and recognize a class of discontinuous games where the efficient Nash equilibria are stable with respect to perturbations of the characteristics of players.     

Learning in games with unstable equilibria

We propose a new concept for the analysis of games, the TASP, which gives a precise prediction about non-equilibrium play in games whose Nash equilibria are mixed and are unstable under fictitious play-like learning. We show that, when players learn using weighted stochastic fictitious play and so place greater weight on recent experience, the time average of play often converges in these “unstable” games, even while mixed strategies and beliefs continue to cycle. This time average, the TASP, is related to the cycle identified by Shapley [L.S. Shapley, Some topics in two person games, in: M. Dresher, et al. (Eds.), Advances in Game Theory, Princeton University Press, Princeton, 1964]. The TASP can be close to or quite distinct from Nash equilibrium.     

Selecting equilibria in common agency games

We characterize equilibrium payoffs of a delegated common agency game in a public good context where principals use smooth contribution schedules. We prove that under complete information, payoff vectors of equilibria with truthful schedules coincide with the set of smooth equilibrium payoffs, including non-truthful schedules. We next consider whether the presence of arbitrarily small amounts of asymmetric information is enough to refine this payoff set. Providing that the extensions of the equilibrium schedules beyond the equilibrium point are flatter than truthful schedules, the set of equilibrium payoffs is strictly smaller than the set of smooth (equivalently, truthful) equilibrium payoffs. Interestingly, some forms of asymmetric information do not sufficiently constrain the slopes of the extensions and fail to refine the payoff set. In the case of a uniform distribution of types and arbitrary out-of-equilibrium contributions, the refinement has no bite. If, however, one restricts out-of-equilibrium behavior in a natural way, the refinement is effective. Alternatively, we may consider an exponential distribution with unbounded support (and hence no out-of-equilibrium choices) and we find that the refinement selects a unique equilibrium payoff vector equal to Lindahl prices.As a separate contribution, equilibria with forcing contracts are also considered both under complete and asymmetric information.     

Rationalized subjective equilibria in repeated games

Abstract The purpose of this paper is to provide a new equilibrium concept of repeated games that involve incomplete information ( rationalized subjective equilibrium ) and a related framework ( subjective game ), and to analyse the relationships between players' subjective images of games and the realized outcomes. We demonstrate the relationships among rationalized subjective equilibria, subjective equilibria, and original Nash equilibria by providing several examples. We also present several theorems analogous to the ordinary folk theorem and a simple application using the Cournot duopoly model. JEL Classification: C70, C72
Equilibres subjectifs rationalisés dans des jeux répétés L'objectif de cet article est de proposer un nouveau concept d'équilibre dans les jeux répétés où l'information est incomplète (équilibre subjectif rationalisé) et un cadre d'analyse approprié( jeu subjectif ), ainsi que d'analyser les rapports entre les images subjectives que les acteurs ont des jeux et de leurs résultats. On montre les rapports entre équilibres subjectifs rationalisés, équilibres subjectifs, et équilibres à la Nash à l'aide de divers exemples. On présente aussi plusieurs théorèmes analogues au « ordinary folk theorem >> et une application simple utilisant le modèle de duopole de Cournot.     

Strategic market games quantal response equilibria

Summary. The paper explores a model of boundedly rational traders who act strategically. The general framework of analysis is the sell all model of Shapley and Shubik [6] whereas the behavioral rule of traders is related to the bounded rationality of McKelvey and Palfrey [5] and the subsequent of Chen Friedman and Thisse [1]. A new equilibrium concept is introduced appropriate to integrate in the basic model market participants that may choose an inferior action with positive probability. Further, assuming that these market participants are engaged in an infinite repeated game and making them able to learn playing rationally, we demonstrate that in the limit the original equilibrium of Shapley and Shubik is attained and basically emerges as a special case of our equilibrium concept.Received: 20 September 2003, Revised: 11 December 2004, I am indebted to Yanis Varoufakis and especially to Dimitrios P. Tsomocos for their helpful comments.     

Enumeration of Nash equilibria for two-player games

This paper describes algorithms for finding all Nash equilibria of a two-player game in strategic form. We present two algorithms that extend earlier work. Our presentation is self-contained, and explains the two methods in a unified framework using faces of best-response polyhedra. The first method lrsnash is based on the known vertex enumeration program lrs, for “lexicographic reverse search”. It enumerates the vertices of only one best-response polytope, and the vertices of the complementary faces that correspond to these vertices (if they are not empty) in the other polytope. The second method is a modification of the known EEE algorithm, for “enumeration of extreme equilibria”. We also describe a second, as yet not implemented, variant that is space efficient. We discuss details of implementations of lrsnash and EEE, and report on computational experiments that compare the two algorithms, which show that both have their strengths and weaknesses.     

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.     

Mixed equilibria in games of strategic complementarities

Summary. The literature on games of strategic complementarities (GSC) has focused on pure strategies. I introduce mixed strategies and show that, when strategy spaces are one-dimensional, the complementarities framework extends to mixed strategies ordered by first-order stochastic dominance. In particular, the mixed extension of a GSC is a GSC, the full set of equilibria is a complete lattice and the extremal equilibria (smallest and largest) are in pure strategies. The framework does not extend when strategy spaces are multi-dimensional. I also update learning results for GSC using stochastic fictitious play. Received: October 16, 2000; revised version: March 7, 2002 RID="*" ID="*" I am very grateful to Robert Anderson, David Blackwell, Aaron Edlin, Peter De Marzo, Ted O'Donoghue, Matthew Rabin, Ilya Segal, Chris Shannon, Clara Wang and Federico Weinschelbaum for comments and advise.     

Markov equilibria in dynamic matching and bargaining games

Rubinstein and Wolinsky [Rev. Econ. Stud. 57 (1990) 63] show that a simple homogeneous market with exogenous matching has a continuum of (non-competitive) perfect equilibria; however, the unique Markov-perfect equilibrium of this model is competitive. By contrast, in the more general case of heterogeneous markets, even the Markov property is not enough to guarantee the perfectly competitive outcome. We define a market game that allows for heterogeneous values on both sides of the market and exhibit a number of examples of (non-competitive) Markov-perfect equilibria, with and without discounting. Unlike the homogeneous case, these equilibria allow for inefficient trades and for trade at non-uniform prices. The non-competitive equilibrium may be unique.     

Anonymous sequential games: Existence and characterization of equilibria

Summary In this paper we consider Anonymous Sequential Games with Aggregate Uncertainty. We prove existence of equilibrium when there is a general state space representing aggregate uncertainty. When the economy is stationary and the underlying process governing aggregate uncertainty Markov, we provide Markov representations of the equilibria.Table of notation Agents' characteristics space ( ) - A Action space of each agent (aA) - Y Y = x A - Aggregate distribution on agents' characteristics - (X) Space of probability measures onX - C(X) Space of continuous functions onX - _X Family of Borel sets ofX - State space of aggregate uncertainty ( ) - x _t=1 aggregate uncertainty for the infinite game - = (₁,₂,...,_t,...) - ^{t t} (₁, ₂,..., _t) - L₁(^t,C ×A),v _t Normed space of measurable functions from ^t toC( x A) - 8o(^t,( x A)) Space of measurable functions from ^tto( x A) - X^t X^t= x _s=1 ^t X - _X ^t Borel field onX ^t - v Distribution on - v_t Marginal distribution of v on ^t - v(^t)((¦^t)) Conditional distribution on given ^t - v^t(^s)(v_t(¦^s)) Conditional distribution on ^t given ^s (wheres) - _t Periodt distributional strategy - Distributional strategy for all periods =(₁,₂,...,_t,...) - _t Transition process for agents' types - ( _t,^t,y)(P _t+1(, _t , ^t ,y)) Transition function associated with _t - u _t Utility function - V _t (, a, , ^t) Value function for each collection (, a, , ^t ) - W _t (, , ^t ) Value function given optimal action a - C() Consistency correspondence. Distributions consistent with and characteristics transition functions - B() Best response correspondence (which also satisfy consistency) - E Set of equilibrium distributional strategies - x _t=1 ( ^t , (x A)) - S Expanded state space for Markov construction - (, a, ) Value function for Markov construction - P( _t ^* , _t y)(P(, _t ^* , _t , _y )) Invariant characteristics transition function for Markov game We wish to acknowledge very helpful conversations with C. d'Aspremont, B. Lipman, A. McLennan and J-F. Mertens. The financial support of the SSHRCC and the ARC at Queen's University is gratefully acknowledged. This paper was begun while the first author visited CORE. The financial support of CORE and the excellent research environment is gratefully acknowledged. The usual disclaimer applies.     

Functional rational expectations equilibria in market games

The rational expectations equilibrium (REE) has been criticized as an equilibrium concept in market game environments. Such an equilibrium may not exist generically, or it may introduce unrealistic assumptions about an economic agent’s knowledge or computational ability. We define an REE as a probability measure over uncertain states of nature which exploits all available information in a market game, and which exists for almost all economies. Furthermore, if retrading is allowed, it is possible for agents to compute such a ‘functional rational expectations equilibrium’ using straightforward numerical fixed point algorithms. The approach is demonstrated in a detailed numerical example.     

Ex-post stability of Bayes-Nash equilibria of large games

We present a result on approximate ex-post stability of Bayes-Nash equilibria in semi-anonymous Bayesian games with a large finite number of players. The result allows players? action and type spaces to be general compact metric spaces, thus extending a result by Kalai (2004).     

Finding all equilibria in games of strategic complements

I present a simple and fast algorithm that finds all the pure-strategy Nash equilibria in games with strategic complementarities. This is the first non-trivial algorithm for finding all pure-strategy Nash equilibria.