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

The paper [Harry Buhrman, Michal Koucký, Nikolay Vereshchagin. Randomized Individual Communication Complexity. IEEE Conference on Computational Complexity 2008: 321-331] considered communication complexity of the following problem. Alice has a binary string x and Bob a binary string y, both of length n, and they want to compute or approximate Kolmogorov complexity C(x|y) of x conditional to y. ...