Correlated equilibria of games with many players

Let G(m,n) be the class of strategic games with n players, where each player has m greater than or equal to 2 pure strategies. We are interested in the structure of the set of correlated equilibria of games in G(m,n) when n --> infinity. As the number of equilibrium constraints grows slower than the number of pure strategy profiles, it might be conjectured that the set of correlated equilibria becomes large. In this paper, we show that (1) the average relative measure of the set of correlated equilibria is smaller than 2(-n); and (2) for each 1 < c < m, the solution set contains cn correlated equilibria having disjoint supports with a probability going to 1 as n grows large. The proof of the second result hinges on the following inequality: Let c(1),...,c(l) be independent and symmetric random vectors in R-k, l greater than or equal to k. Then the probability that the convex hull of c(1),...,c(l) intersects R-+(k) is greater than of equal to 1 - 2(-l)[((l)(0)) +...+ ((l)(k-1))].