On existence of Nash equilibria of games with constraints on multistrategies

In this paper, sufficient conditions are given, which are less restrictive than those required by the Arrow-Debreu-Nash theorem, on the existence of a Nash equilibrium of an n-player game {Y-1, ..., Y-n, f(1), ..., f(n)} in n ormal form with a nonempty closed convex constraint C on the set Y = Pi (i) Y-i of multistrategies. The ith player has to minimize the function f(i) w ith respect to the ith variable. We consider two cases. In the first case, Y is a real Hilbert space and the loss function class is quadratic. In this case, the existence of a Nash equilibrium is guaranteed as a simple consequence of the projection theorem for Hilbert spaces. In t he second case, Y is a Euclidean space, the loss functions are continuous, and f(i) is convex with respect to the ith variable. In this case, the tech nique is quite particular, because the constrained game is approximated wit h a sequence of free games, each with a Nash equilibrium in an appropriate compact space X. Since X is compact, there exists a subsequence of these Na sh equilibrium points which is convergent in the norm. If the limit point i s in C and if the order of convergence is greater than one, then this is a Nash equilibrium of the constrained game.