  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. Definability in the same structure does not
hold systematically: we provide a counterexample of a stochastic game with
semi-algebraic data yielding a non semi-algebraic but globally subanalytic
Shapley operator. Our definability results on Shapley operators are 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
highlight the key role played by the tame structure of transition functions. As
particular cases of our main results, we obtain that stochastic games with
polynomial transitions, definable games with finite actions on one side,
definable games with perfect information or switching controls have a uniform
value. Applications to nonlinear maps arising in risk sensitive control and
Perron-Frobenius theory are also given
