On the existence of general equilibrium in finite games and general game dynamics

A notion of incentive for agents is introduced which leads to a very general notion of an equilibrium for a finite game. Sufficient conditions for the existence of these equilibria are given. Known existence theorems are shown to be corollaries to the main theorem of this paper. Furthermore, conditions for the existence of equilibria in certain symmetric regions for games are also given. From the notion of general equilibrium, a general family of game dynamics are derived. This family incorporates all canonical examples of game dynamics. A proof is given for the full generality of this system. 1 Notation and Definitions We shall denote the finite set of agents by N = {1, 2,..., n} for some n ∈ N. Each agent i is endowed with a finite set of pure strategies, which will be denoted Si = {1, 2,..., si}, with si ∈ N as well. To allow the agents to mix their strategies, they may choose strategies from the simplex on si vertices, ∆i