Subgame-Perfect Equilibria for Stochastic Games

For an n-person stochastic game with Borel state space S and compact metric action sets A¹, A²,…, Aⁿ, sufficient conditions are given for the existence of subgame-perfect equilibria. One result is that such equilibria exist if the law of motion q(⋯∣ s, a) is, for fixed s, continuous in a = (a¹,a²,…,aⁿ) for the total variation norm and the payoff functions f¹, f²,…,fⁿ are bounded, Borel measurable functions of the sequence of states (s₁, s₂,…) ∈ S^ℕ and, in addition, are continuous when S^ℕ is given the product of discrete topologies on S.