Uniform value in recursive games

We address the problem of existence of the uniform value in recursive games. We give two existence results: (i) the uniform value is shown to exist if the state space is countable, the action sets are finite and if, for some a>0, there are finitely many states in which the limsup value is less than a; (ii) for games with nonnegative payoff function, it is sufficient that the action set of player 2 is finite. The finiteness assumption can be further weakened.