Existence of Nash Equilibria on Convex Sets

We analyze a non-cooperative game in which the set of feasible strategy profiles is compact and convex but possibly non-rectangular. Thus, a player's feasible strategies may depend on the strategies used by others, as in Debreu's (1952, 1982) generalized games. In contrast to the model of Debreu, we do not require preferences to be defined over infeasible strategy profiles, and we do not require a player's feasible strategy correspondence to have non-empty values. We prove existence of Nash equilibria under a lower hemicontinuity condition, and we give examples of classes of games in which this condition is satisfied.