Leadership with commitment to mixed strategies

A basic model of commitment is to convert a game in strategic form into a “leadership game” where one player commits to a strategy to which the other player chooses a best response, with payoffs as in the original game. This paper studies subgame perfect equilibria of such leadership games for the mixed extension of a finite game, where the leader commits to a mixed strategy. In a generic two-player game, the leader payoff is unique and at least as large as any Nash payoff in the original simultaneous game. In non-generic two-player games, which are completely analyzed, the leader payoffs may form an interval, which as a set of payoffs is never worse than the Nash payoffs for the player who has the commitment power. Furthermore, the set of payoffs to the leader is also at least as good as the set of correlated equilibrium payoffs. These observations no longer hold in leadership games with three or more players. The possible payoffs to the follower are shown to be arbitrary compared to the simultaneous game or the game where the players switch their roles of leader and follower. Curiously, the follower payoff is not so arbitrary in typical

[1]  H. Stackelberg,et al.  Marktform und Gleichgewicht , 1935 .

[2]  E. Rowland Theory of Games and Economic Behavior , 1946, Nature.

[3]  A Charnes Constrained Games and Linear Programming. , 1953, Proceedings of the National Academy of Sciences of the United States of America.

[4]  T. Schelling,et al.  The Strategy of Conflict. , 1961 .

[5]  Michael Maschler A price leadership method for solving the inspector's non-constant-sum game , 1966 .

[6]  J. Cruz,et al.  On the Stackelberg strategy in nonzero-sum games , 1973 .

[7]  R. Aumann Subjectivity and Correlation in Randomized Strategies , 1974 .

[8]  James W. Friedman,et al.  Oligopoly and the theory of games , 1977 .

[9]  J. Vial,et al.  Strategically zero-sum games: The class of games whose completely mixed equilibria cannot be improved upon , 1978 .

[10]  A. Rubinstein Perfect Equilibrium in a Bargaining Model , 1982 .

[11]  T. Başar,et al.  Dynamic Noncooperative Game Theory , 1982 .

[12]  Theories of oligopoly behavior , 1989 .

[13]  Jonathan H. Hamilton,et al.  Endogenous timing in duopoly games: Stackelberg or cournot equilibria , 1990 .

[14]  Akira Okada,et al.  Inspector Leadership with Incomplete Information , 1991 .

[15]  Robert W. Rosenthal A note on robustness of equilibria with respect to commitment opportunities , 1991 .

[16]  K. Bagwell Commitment and observability in games , 1995 .

[17]  E.E.C. van Damme,et al.  Games with imperfectly observable commitment , 1997 .

[18]  B. Stengel,et al.  Team-Maxmin Equilibria☆ , 1997 .

[19]  P. Reny On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games , 1999 .

[20]  Rabah Amir,et al.  Stackelberg versus Cournot Equilibrium , 1999 .

[21]  Eric van Damme,et al.  Non-cooperative games , 2000 .

[22]  J. Morgan,et al.  Mixed Strategies for Hierarchical Zero-Sum Games , 2001 .

[23]  M. Dufwenberg,et al.  Existence and Uniqueness of Maximal Reductions Under Iterated Strict Dominance , 2002 .

[24]  Bernhard von Stengel,et al.  Chapter 51 Inspection games , 2002 .

[25]  Arthur J. Robson,et al.  Reinterpreting mixed strategy equilibria: a unification of the classical and Bayesian views , 2004, Games Econ. Behav..

[26]  Fioravante Patrone,et al.  Stackelberg Problems: Subgame Perfect Equilibria via Tikhonov Regularization , 2006 .

[27]  Dov Monderer,et al.  TO COMMIT or NOT TO COMMIT ? , .