Asymptotic expected number of Nash equilibria of two-player normal form games

The formula given by McLennan [The mean number of real roots of a multihomogeneous system of polynomial equations, Amer. J. Math. 124 (2002) 49–73] is applied to the mean number of Nash equilibria of random two-player normal form games in which the two players have M and N pure strategies respectively. Holding M fixed while N→∞, the expected number of Nash equilibria is approximately . Letting M=N→∞, the expected number of Nash equilibria is , where is a constant, and almost all equilibria have each player assigning positive probability to approximately 31.5915 percent of her pure strategies.     

The equilibrium set of two-player games with complementarities is a sublattice

Summary. I prove that the equilibrium set in a two-player game with complementarities, and totally ordered strategy spaces, is a sublattice of the joint strategy space. Received: May 31, 2001; revised version: October 4, 2002     

Axiomatic reference-dependence in behavior toward others and toward risk

Summary. This paper considers the applicability of the standard separability axiom for both risk and other-regarding preferences, and advances arguments why separability might fail. An alternative axiom, which is immune to these arguments, leads to a preference representation that is additively separable in a reference variable and the differences between the other variables and the reference variable. For other-regarding preferences the reference variable is the decision-maker’s own payoff, and the resulting representation coincides with the Fehr-Schmidt model. For risk preferences the reference variable is initial wealth, and the resulting representation is a generalization of prospect theory.Received: 28 April 2004, Revised: 27 April 2005, JEL Classification Numbers: D81, D64.An earlier version of this paper was titled “An Axiomatic Characterization of the Fehr-Schmidt Model of Inequity Aversion.” I am grateful to Rachel Croson, Jill Stowe, and Karl Vind for helpful comments. Financial support was provided by the Private Enterprise Research Center, the Program in the Economics of Public Policy, and the Program to Enhance Scholarly and Creative Activities.     

Weighted congestion games with separable preferences

Players in a congestion game may differ from one another in their intrinsic preferences (e.g., the benefit they get from using a specific resource), their contribution to congestion, or both. In many cases of interest, intrinsic preferences and the negative effect of congestion are (additively or multiplicatively) separable. This paper considers the implications of separability for the existence of pure-strategy Nash equilibrium and the prospects of spontaneous convergence to equilibrium. It is shown that these properties may or may not be guaranteed, depending on the exact nature of player heterogeneity.     

Uniqueness of the index for Nash equilibria of two-player games

Summary. Given a map whose roots are the Nash equilibria of a game, each component of the equilibrium set has an associated index, defined as the local degree of the map. This note shows that for a two-player game, every map with the same roots induces the same index. Moreover, this index agrees with the Shapley index constructed from the Lemke-Howson algorithm. Received: May 30, 1996; revised version June 25, 1996     

Continuous-time games of timing

We address the question of existence of equilibrium in general timing games with complete information. Under weak assumptions, any two-player timing game has a Markov subgame perfect -equilibrium, for each >0. This result is tight. For some classes of games (symmetric games, games with cumulative payoffs), stronger existence results are established.     

A group selection perspective on economic behavior, institutions and organizations

This article examines the role of group dynamics and interactions in explaining economic behavior and the evolution of institutions. Our starting point is the large literature on group selection in the biological, behavioral and social sciences. We present a range of interpretations of group selection, describe a complete set of group selection mechanisms, and discuss the empirical and experimental evidence for group selection. Unique features of cultural group selection are investigated, and opportunities for applying the latter to various areas of economic theory and economic policy are identified.     

Cournot–Nash equilibria in continuum games with non-ordered preferences

In the usual framework of continuum games with externalities, we substantially generalize Cournot–Nash existence results [Balder, A unifying approach to existence of Nash equilibria, Int. J.Game Theory 24 (1995) 79–94; On the existence of Cournot–Nash equilibria in continuum games, J. Math. Econ. 32 (1999) 207–223; A unifying pair of Cournot–Nash equilibrium existence results, J. Econ. Theory 102 (2002) 437–470] to games with possibly non-ordered preferences, providing a continuum analogue of the seminal existence results by Mas-Colell [An equilibrium existence theorem without complete or transitive preferences, J. Math. Econ. 1 (1974) 237–246], Gale and Mas-Colell [An equilibrium existence theorem for a general model without ordered preferences, J. Math. Econ. 2 (1975) 9–15], Shafer and Sonnenschein [Equilibrium in abstract economies without ordered preferences, J. Math. Econ. 2 (1975) 345–348], Borglin and Keiding [Existence of equilibrium actions and of equilibrium: a note on the “new” existence theorems, J. Math. Econ. 3 (1976) 313–316] and Yannelis and Prabhakar [Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econ. 12 (1983) 233–245].     

Lexicographic composition of simple games

Certain voting bodies can be modeled as a simple game where a coalition's winning depends on whether it wins, blocks or loses in two smaller simple games. There are essentially five such ways to combine two proper games into a proper game. The most decisive is the lexicographic rule, where a coalition must either win in G₁, or block in G₁ and win in G₂. When two isomorphic games are combined lexicographically, a given role for a player confers equal or more power when held in the first game than the second, if power is assessed by any semi-value. A game is lexicographically separable when the players of the two components partition the whole set. Games with veto players are not separable, and games of two or more players with identical roles are separable only if decisive. Some separable games are egalitarian in that they give players identical roles.     

A geometric approach to the price of anarchy in nonatomic congestion games

We present a short, geometric proof for the price-of-anarchy results that have recently been established in a series of papers on selfish routing in multicommodity flow networks and on nonatomic congestion games. This novel proof also facilitates two new types of theoretical results: On the one hand, we give pseudo-approximation results that depend on the class of allowable cost functions. On the other hand, we derive stronger bounds on the inefficiency of equilibria for situations in which the equilibrium costs are within reasonable limits of the fixed costs. These tighter bounds help to explain empirical observations in vehicular traffic networks. Our analysis holds in the more general context of nonatomic congestion games, which provide the framework in which we describe this work.     

The effects of risk preferences in mixed-strategy equilibria of games

We consider the effects of risk preferences in mixed-strategy equilibria of 2×2 games, provided such equilibria exist. We identify sufficient conditions under which the expected payoff in the mixed equilibrium increases or decreases with the degree of risk aversion. We find that (at least moderate degrees of) risk aversion will frequently be beneficial in mixed equilibria.     

Two more classes of games with the continuous-time fictitious play property

Fictitious Play is the oldest and most studied learning process for games. Since the already classical result for zero-sum games, convergence of beliefs to the set of Nash equilibria has been established for several classes of games, including weighted potential games, supermodular games with diminishing returns, and 3×3 supermodular games. Extending these results, we establish convergence of Continuous-time Fictitious Play for ordinal potential games and quasi-supermodular games with diminishing returns. As a by-product we obtain convergence for 3×m and 4×4 quasi-supermodular games.     

The envelope theorem for locally differentiable Nash equilibria of finite horizon differential games

Envelope theorems are established for a ubiquitous class of finite horizon differential games. The theorems cover open-loop and feedback information patterns in which the corresponding Nash equilibria are locally differentiable with respect to the parameters of the game. Their relationship with extant envelope results is discussed and an application of them to a generalized capital accumulation game is provided. An important implication of the theorems is that, in general, the archetypal economic interpretation of the costate vector, namely, as the shadow value of the state vector along the Nash equilibrium, is valid for feedback Nash equilibria, but not for open-loop Nash equilibria.     

The structure of the Nash equilibrium sets of standard 2-player games

Summary. In this paper I study a class of two-player games, in which both players action sets are [0,1] and their payoff functions are continuous in joint actions and quasi-concave in own actions. I show that a no-improper-crossing condition is both necessary and sufficient for a finite subset A of to be the set of Nash equilibria of such a game.Received: 21 November 2002, Revised: 9 September 2004, JEL Classification Numbers: C65, C72.I am grateful to an editor of the journal and an anonymous referee for their very helpful comments. I also would like to thank the seminar participants at City University of Hong Kong, Georgia State University, Northwestern University, and Rice University.     

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.     

Revision of conjectures about the opponent’s utilities in signaling games

In this paper we apply the concept of preference conjecture equilibrium introduced in Perea (2005) to signaling games and show its relation to sequential equilibrium. We introduce the concept of minimum revision equilibrium and show how this can be interpreted as a refinement of sequential equilibrium     

Advice and behavior in intergenerational ultimatum games: An experimental approach

In the real world, when people play a game, they often receive advice from those that have played it before them. Such advice can facilitate the creation of a convention of behavior. This paper studies the impact of advice on the behavior subjects who engage in a non-overlapping generational Ultimatum game where after a subject plays he is replaced by another subject to whom he can offer advice.Our results document the fact that allowing advice fosters the creation of a convention of behavior in Ultimatum games. In addition, by reading the advice offered we conclude that arguments of fairness are rarely used to justify the offers of Senders but are relied upon to justify rejections by Receivers.     

An experimental study of commitment in Stackelberg games with observation costs

We report on experiments examining the value of commitment in Stackelberg games where the follower chooses whether to pay some cost to perfectly observe the leader's action. Várdy [Games Econ. Behav. (2004)] shows that in the unique pure-strategy subgame perfect equilibrium of this game, the value of commitment is lost completely; however, there exists a mixed-strategy subgame perfect equilibrium where the value of commitment is fully preserved. In the data, the value of commitment is largely preserved when the cost of looking is small, while it is lost when the cost is large. Nevertheless, for small observation costs, equilibrium behavior is clearly rejected. Instead, subjects persistently play non-equilibrium strategies in which the probability of the follower choosing to observe the leader's action is a decreasing function of the observation cost.     

Revisiting games of incomplete information with analogy-based expectations

This paper studies the effects of analogy-based expectations in static two-player games of incomplete information. Players are assumed to be boundedly rational in the way they forecast their opponent's state-contingent strategy: they bundle states into analogy classes and play best-responses to their opponent's average strategy in those analogy classes. We provide general properties of analogy-based expectation equilibria and apply the model to a variety of well known games. We characterize conditions on the analogy partitions for successful coordination in coordination games under incomplete information [Rubinstein, A., 1989. The electronic mail game: Strategic behavior under ‘almost common knowledge’. Amer. Econ. Rev. 79, 385–391], we show how analogy grouping of the receiver may facilitate information transmission in Crawford and Sobel's cheap talk games [Crawford, V.P., Sobel, J., 1982. Strategic information transmission. Econometrica 50, 1431–1451], and we show how analogy grouping may give rise to betting in zero-sum betting games such as those studied to illustrate the no trade theorem.