A Mean Field Approach for Discounted Zero-Sum Games in a Class of Systems of Interacting Objects

The paper deals with systems composed of a large number of N interacting objects (e.g., agents, particles) controlled by two players defining a stochastic zero-sum game. The objects can be classified according to a finite set of classes or categories over which they move randomly. Because N is too large, the game problem is studied following a mean field approach. That is, a zero-sum game model $$\mathcal {GM}_{N}$$ GM N , where the states are the proportions of objects in each class, is introduced. Then, letting $$N\rightarrow \infty $$ N → ∞ (the mean field limit) we obtain a new game model $$\mathcal {GM}$$ GM , independent on N, which is easier to analyze than $$\mathcal {GM}_{N}$$ GM N . Considering a discounted optimality criterion, our objective is to prove that an optimal pair of strategies in $$\mathcal {GM}$$ GM is an approximate optimal pair as $$N\rightarrow \infty $$ N → ∞ in the original game model $$\mathcal {GM}_{N}$$ GM N .