Definable Zero-Sum Stochastic Games

Definable zero-sum stochastic games involve a finite number of states and action sets, reward and transition functions that are definable in an o-minimal structure. Prominent examples of such games are finite, semi-algebraic or globally subanalytic stochastic games. We prove that the Shapley operator of any definable stochastic game with separable transition and reward functions is definable in the same structure. This result is used to prove that any separable definable game has a uniform value; in the case of polynomially bounded structures we also provide convergence rates. Using an approximation procedure, we actually establish that general zero-sum games with separable definable transition functions have a uniform value. These results cover the particular case of stochastic games with polynomial transitions, definable games with finite actions on one side, definable games with perfect information or switching controls, as well as nonlinear maps arising in risk sensitive control and Perron-Frobenius theory