Stochastic Timing, Uniqueness, and Efficiency in Games

In existing game theoretic settings the timing of moves is deterministic, i.e. they occur with certainty at a pre-specified time. To add more realism we propose a framework in which, after an initial simultaneous move in time t = 0, one player gets to revise his action with positive probability at some t > 0. Since the initial action of the opponent can be observed, and payo¤s accrue over time, the setup constitutes a dynamic extension of the Stackelberg leadership concept. Allowing for an arbitrary timing distribution, and using both subgame perfection and stochastic stability, we derive the necessary and sufficient conditions under which our dynamic revision game has a unique e¢ cient outcome even if the underlying normal form game has no efficient Nash, or multiple ones. Intuitively, the fact that a player is less likely to move than the opponent may serve as a commitment device. Therefore, if the revision opportunity is expected to arrive sufficiently early then the committed player's initial cost of mis-coordination or conflict will be more than compensated by ensuring his preferred outcome after the opponent's revision. The framework allows, among other things, to address the issue of equilibrium selection in games in which traditional equilibrium selection approaches fail such as the Battle of the Sexes and the Game of Chicken. It also offers some insigths into the debate about Pareto-dominance versus risk-dominance.