Anti-Coordination Games and Dynamic stability

We introduce the class of anti-coordination games, including the hawk-dove game as a special case. A symmetric two-player game is said to have the anti-coordination property if any worst response to a mixed strategy is in the support of that mixed strategy. Every anti-coordination game has a unique interior Nash equilibrium. We investigate stability of the static equilibrium under several dynamics with one-population setting. Specifically we focus on the best response dynamic (BRD), where agents in a large population take myopic best responses, and the perfect foresight dynamic (PFD), where agents take best responses to the time average of the action distributions from the present to the future. For any anti-coordination game we show (i) that, for any initial distribution, BRD has a unique solution, which reaches the static equilibrium in a finite time, (ii) that the same path is one of the solutions in PFD, and (iii) that no path escapes from the static equilibrium in PFD once the path reaches the equilibrium. Moreover, in some subclasses of anti-coordination games, we show that any solution from any initial state converges to the static equilibrium in PFD. All the results for PFD hold for any discount rate.