Monotone Methods for Equilibrium Selection under Perfect Foresight Dynamics

This paper studies equilibrium selection in supermodular games based on perfect foresight dynamics. A normal form game is played repeatedly in a large society of rational agents. There are frictions: opportunities to revise actions follow independent Poison processes. Each agent forms his belief about the future evolution of the action distribution in the society, and takes an action that maximizes his expected discounted payoff. A perfect foresight path is defined to be a feasible path of the action distribution along which every agent with a revision opportunity takes a best response to this path itself. A Nash equilibrium is said to be absorbing if any perfect foresight path converges to this equilibrium whenever the initial distribution is suffciently close to the equilibrium; a Nash equilibrium is said to be globally accessible if for each initial distribution, there exists a perfect foresight path converging to this equilibrium. By exploiting the monotone structure of the dynamics, the unique Nash equilibrium that is absorbing and globally accessible for any small degree of friction is identified for certain classes of supermodular games. For games with monotone potentials, the selection of the monotone potential maximizer is obtained. Complete characterizations for absorption and global accessibiltiy are given for binary supermodular games. An example demonstrates that unanimity games may have multiple globally accessible equilibria for a small friction.