The Possibility of Impossible Stairways and Greener Grass

In classical game theory, players have finitely many actions and evaluate outcomes of mixed strategies using a von Neumann-Morgenstern utility function. Allowing a larger, but countable, player set introduces a host of phenomena that are impossible in finite games. Firstly, in coordination games, all players have the same preferences: switching to a weakly dominant action makes everyone at least as well off as before. Nevertheless, there are coordina- tion games where the best outcome occurs if everyone chooses a weakly dominated action, while the worst outcome occurs if everyone chooses the weakly dominant action. Secondly, the location of payoff-dominant equilibria behaves capriciously: two coordination games that look so much alike that even the consequences of unilateral deviations are the same may nevertheless have disjoint sets of payoff-dominant equilibria. Thirdly, a large class of games has no (pure or mixed) Nash equilibria. Following the proverb \the grass is always greener on the other side of the hedge", greener-grass games model constant discontent: in one part of the strategy space, players would rather switch to its complement. Once there, they'd rather switch back.coordination games;dominant strategies;payoff-dominance;nonexistence of equi- librium;tail events

Axiomatizations of the Value of Matrix Games

The function that assigns to each matrix game (i.e., the mixed extension of a finite zero-sum two-player game) its value is axiomatized by a number of intuitive properties.game theory;noncooperative games;matrix games;axiomatic methods

Learning to be Prepared

Behavioral economics provides several motivations for the common observation that agents appear somewhat unwilling to deviate from recent choices.More recent choices can be more salient than other choices, or more readily available in the agent's mind.Alternatively, agents may have formed habits, use rules of thumb, or lock in on certain modes of behavior as a result of learning by doing.This paper provides discrete-time adjustment processes for strategic games in which players display precisely such a bias towards recent choices.In addition, players choose best replies to beliefs supported by observed play in the recent past, in line with much of the literature on learning.These processes eventually settle down in the minimal prep sets of Voorneveld [Games Econ.Behav. 48 (2004) 403-414, and Games Econ.Behav. 51 (2005) 228-232].learning;adjustment;minimal prep sets;availability bias;salience;rules of thumb

Congestion, Equilibrium and Learning: The Minority Game

The minority game is a simple congestion game in which the players’ main goal is to choose among two options the one that is adopted by the smallest number of players. We characterize the set of Nash equilibria and the limiting behavior of several well-known learning processes in the minority game with an arbitrary odd number of players. Interestingly, different learning processes provide considerably different predictions.Learning;congestion games;replicator dynamic;perturbed best response dynamics;quantal response equilibria;best-reply learning

A Myopic Adjustment Process Leading to Best-Reply Matching

stochastic adjustment process;best reply;mathcing;regret equilibrium

The Cutting Power of Preparation

In a strategic game, a curb set [Basu and Weibull, Econ.Letters 36 (1991) 141-146] is a product set of pure strategies containing all best responses to every possible belief restricted to this set.Prep sets [Voorneveld, Games Econ. Behav. 48 (2004) 403-414] relax this condition by only requiring the presence of at least one best response to such a belief.The purpose of this paper is to provide sufficient conditions under which minimal prep sets give sharp predictions.These conditions are satisfied in many economically relevant classes of games, including supermodular games, potential games, and congestion games with player-specific payoffs.In these classes, minimal curb sets generically have a large cutting power as well, although it is shown that there are relevant subclasses of coordination games and congestion games where minimal curb sets have no cutting power at all and simply consist of the entire strategy space.curb sets;prep sets;supermodular games;potential games;congestion games

The Possibility of Impossible Stairways and Greener Grass

In classical game theory, players have finitely many actions and evaluate outcomes of mixed strategies using a von Neumann-Morgenstern utility function. Allowing a larger, but countable, player set introduces a host of phenomena that are impossible in finite games. Firstly, in coordination games, all players have the same preferences: switching to a weakly dominant action makes everyone at least as well off as before. Nevertheless, there are coordina- tion games where the best outcome occurs if everyone chooses a weakly dominated action, while the worst outcome occurs if everyone chooses the weakly dominant action. Secondly, the location of payoff-dominant equilibria behaves capriciously: two coordination games that look so much alike that even the consequences of unilateral deviations are the same may nevertheless have disjoint sets of payoff-dominant equilibria. Thirdly, a large class of games has no (pure or mixed) Nash equilibria. Following the proverb \the grass is always greener on the other side of the hedge", greener-grass games model constant discontent: in one part of the strategy space, players would rather switch to its complement. Once there, they'd rather switch back

Fixed points and contraction factor functions

In a complete metric space (X; d), we dene w-distance functions p : X X ! [0; 1), of which the metric d is a special case, and contraction factor functions r : X X ! [0; 1) such that if p(Tx;Ty) r(x; y)p(x; y) for all x; y 2 X, thenT : X ! X has a (unique) xed point

The Structure of the Set of Equilibria for Two Person Multicriteria Games

In this paper the structure of the set of equilibria for two person multicriteria games is analysed. It turns out that the classical result for the set of equilibria for bimatrix games, that it is a finite union of polytopes, is only valid for multicriteria games if one of the players only has two pure strategies. A full polyhedral description of these polytopes can be derived when the player with an arbitrary number of pure strategies has one criterion.game theory;equilibrium theory