Communication complexity of Nash equilibrium in potential games

We prove communication complexity lower bounds for (possibly mixed) Nash equilibrium in potential games. In particular, we show that finding a Nash equilibrium requires poly(N) communication in two-player N × N potential games, and 2poly(n) communication in n-player two-action games. To the best of our knowledge, these are the first results to demonstrate hardness in any model of (possibly mixed) Nash equilibrium in potential games.