A Social Equilibrium Existence Theorem

In a wide class of social systems each agent has a range of actions among which he selects one. His choice is not, however, entirely free and the actions of all the other agents determine the subset to which his selection is restricted. Once the action of every agent is given, the outcome of the social activity is known. The preferences of each agent yield his complete ordering of the outcomes and each one of them tries by choosing his action in his restricting subset to bring about the best outcome according to his own preferences. The existence theorem presented here gives general conditions under which there is for such a social system an equilibrium, i.e., a situation where the action of every agent belongs to his restricting subset and no agent has incentive to choose another action.

This theorem has been used by Arrow and Debreu² to prove the existence of an equilibrium for a classical competitive economic system, it contains the existence of an equilibrium point for an N-person game (see Nash⁸ and Section 4) and, naturally, as a still more particular case the existence of a solution for a zero-sum two-person game (see von Neumann and Morgenstern, Ref. 11, Section 17.6).

In Section 1 the topological concepts to be used are defined. In Section 2 an abstract definition of equilibrium is presented with a proof of the theorem. In Section 3 saddle points are presented as particular cases of equilibrium points and in connection with the closely related MinMax operator. Section 4 concludes with a short historical survey of results about saddle points, fixed points for multi-valued transformations and equilibrium points.

Only subsets of finite Euclidean spaces will be considered here.