On values of repeated games with signals.

Summary We study the existence of different notions of values in two-person zero-sum repeated games where the state evolves and players receive signals. We provide some examples showing that the limsup value and the uniform value may not exist in general. Then, we show the existence of the value for any Borel payoff function if the players observe a public signal including the actions played. We prove also two other positive results without assumptions on the signaling structure: the existence of the $\sup$-value and the existence of the uniform value in recursive games with non-negative payoffs.