Existence of Equilibrium Points in Non-Product Spaces

Source abstract

An equilibrium point in game theory is a situation where a unilateral change by only one of the players does not yield an increase of his profit or “payoff” function. In this paper it will be shown that an equilibrium point exists if: the payoff functions are continuous in all the variables and are concave in the variables belonging to each relevant player, the region in which the game is played is closed, bounded and convex and satisfies a weak continuity condition.The continuity condition is automatically satisfied if the region is a convex polyhedron. It is not assumed that the region in which the game is played is the product space of the subregions of each individual player. Therefore, the existence of an equilibrium point is shown for the general case of the so-called “mixed constraints”.