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 duopoly games; this is the topic of a companion paper (von Stengel (2003)). ∗Support by a STICERD Distinguished Visitorship for Shmuel Zamir at the London School of Economics is gratefully acknowledged.