We show that every N-player K 1 × ... × K N game possesses a correlated equilibrium with at least $$\prod_{i=1}^{N} K_i -1 - \sum_{i=1}^{N} K_i (K_i -1)$$ zero entries. In particular, the largest N-player K × ... × K games with unique fully supported correlated equilibrium are two-player games.