Curb Sets Research Articles

On finding curb sets in extensive games

We characterize strategy sets that are closed under rational behavior (curb) in extensive games of perfect information and finite horizon. It is shown that any such game possesses only one minimal curb set, which necessarily includes all its subgame perfect Nash equilibria. Applications of this result are twofold. First, it lessens computational burden while computing minimal curb sets. Second, it implies that the profile of subgame perfect equilibrium strategies is always stochastically stable in a certain class of games.

Clever agents in adaptive learning

Saez-Marti and Weibull (J. Econom. Theory 86 (1999) 268) investigate the consequences of letting some agents play a myopic best reply to the myopic best reply in Young's (J. Econom. Theory 59 (1993) 145) bargaining model, which is how they introduce “cleverness” of players. I analyze such clever agents in general finite two-player games and show Young's (Individual Strategy and Social Structure, Princeton University Press, Princeton, NJ, 1998) prediction to be robust: adaptive learning with clever agents does select the same minimal curb set as in the absence of clever agents, if their population share is less than one. However, the long-run strategy distribution in such a curb set may vary with the share of clever agents.

A myopic adjustment process leading to best-reply matching

We analyze a myopic strategy adjustment process in strategic-form games. It is shown that the steady states of the continuous time limit, which is constructed assuming frequent play and slow adjustment of strategies, are exactly the best-reply matching equilibria, as discussed by Droste, Kosfeld, and Voorneveld (2000. Mimeo, Tilburg University). In a best-reply matching equilibrium every player ‘matches’ the probability of playing a pure strategy to the probability that this pure strategy is a best reply to the pure-strategy profile played by his opponents. We derive stability results for the steady states of the continuous time limit in 2×2 bimatrix games and coordination games. Analyzing the asymptotic behavior of the stochastic adjustment process in discrete time shows convergence to minimal curb sets of the game. Moreover, absorbing states of the process correspond to best-reply matching equilibria of the game.

Communication, Risk, and Efficiency in Games

This paper uses curb sets to study the evolution of effective pre-play communication in games where a single communication round precedes a simultaneous-move, complete-information game. It is shown that the effectiveness of one-sided pre-play communication is inversely related to risk in the underlying game, and to the size of the message space. If messages have somea prioriinformation content, then multi-sided communication is more effective than one-sided communication; i.e., risk and the size of the message space play no role.Journal of Economic LiteratureClassification Number: C72.

Learning by Forgetful Players

A product set of pure strategies is said to be closed under best replies if all best replies against all possible mixtures of these strategies are contained in the set. Minimal sets with this property are called minimal curb sets. This paper presents a dynamic learning process that has two main characteristics: Players have a bounded memory and they play best replies against beliefs, formed on the basis of strategies used in the recent past. It is shown that this learning process leads the players to playing strategies from a minimal curb set. Several variations of the process are considered. Journal of Economic Literature Classification Numbers: C70, C72.

Strategy subsets closed under rational behavior

A set of strategy profiles is here said to be closed under rational behavior (curb) of it contains all its best replies. Each curb set contains the support of at least one Nash equilibrium in mixed ...