Communication complexity of approximate Nash equilibria

For a constant ϵ, we prove a poly(N) lower bound on the (randomized) communication complexity of ϵ-Nash equilibrium in two-player N×N games. For n-player binary-action games we prove an exp⁡(n) lower bound for the (randomized) communication complexity of (ϵ,ϵ)-weak approximate Nash equilibrium, which is a profile of mixed actions such that at least (1−ϵ)-fraction of the players are ϵ-best replying.