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 = { xi ∈ Ri ∣∣∣∣xiα ≥ 0,∑ α xiα = 1 } , which is the convex hull of Si, or equivalently the space of probability distributions over the finite set. For simplicity we will embed Si in ∆i such that α ∈ Si 7→ eiα ∈ Ri where eik is the kth standard unit vector in the Euclidean space Ri . We denote S = ×iSi and ∆ = ×i∆i as the pure and mixed strategy spaces respectively for the game. It is often convenient to denote the pure and mixed strategy spaces without a particular player; S−i, and ∆−i respectively. We define S−i = ×j 6=iSj, and ∆−i = ×j 6=i∆j. Elements in these sets can be interpreted many different ways. In particular S−i is a s−i = |S| si dimensional space and we would prefer to identify elements in this space with standard unit vectors in R−i as before. Unfortunately, there are s−i! ways to accomplish this. In 1 ar X iv :1 20 1. 23 84 v6 [ cs .G T ] 3 J an 2 01 3 practice, we will only use this identification when we will sum over all possible combinations of pure strategies. Using a different identification will simply result in a permutation of terms in a finite sum, which of course has no effect. k ∈ S−i is a multi-index given by (k1, . . . , ki−1, ki+1, . . . , kn). Our embedding, given by k ∈ S−i 7→ e−iβ ∈ R−i , extends to ∆−i such that x−i ∈ ∆−i = ∑ β x−iβe−iβ with x−iβ = ∏ j 6=i xjkj . If we have a single agent we will interpret S−i, ∆−i, s−i, and x−i as S, ∆, s and x respectively. We will also adopt a convention for replacement for part of a strategy profile, x ∈ ∆. We write (ti, x−i) ∈ ∆ = (x1, x2, . . . , ti, . . . , xn), where the ith component of x has been replaced by another strategy ti ∈ ∆i. Each agent will have a utility function defined over the set of all possible combinations of pure strategies S. We will denote this utility ui : S → R. These utility functions have unique n-linear extensions to ∆ given by