Infinite player noncooperative games and the continuity of the Nash equilibrium correspondence

The usual definition of a noncooperative game is extended in two different ways: first, by replacing the finite player set with a measure space, and second, by eliminating the player set and considering a distribution of the players’ characteristics. Feasible strategy profiles and Nash equilibria obtained from the two approaches are compared. The feasible strategy profile correspondence is shown to be continuous. The Nash equilibrium correspondence is shown to be upper hemicontinuous and nearly lower hemicontinuous on the class of convex and equicontinuous games. These results show when it is reasonable to use an infinite player game as an approximation of a large, but finite, player game.