  We are interested in the convergence of the value of n-stage games as n goes
to infinity and the existence of the uniform value in stochastic games with a
general set of states and finite sets of actions where the transition is
commutative. This means that playing an action profile a 1 followed by an
action profile a 2 , leads to the same distribution on states as playing first
the action profile a 2 and then a 1. For example, absorbing games can be
reformulated as commutative stochastic games. When there is only one player and
the transition function is deterministic, we show that the existence of a
uniform value in pure strategies implies the existence of 0-optimal strategies.
In the framework of two-player stochastic games, we study a class of games
where the set of states is R m and the transition is deterministic and
1-Lipschitz for the L 1-norm, and prove that these games have a uniform value.
A similar proof shows the existence of an equilibrium in the non zero-sum case.
These results remain true if one considers a general model of finite repeated
games, where the transition is commutative and the players observe the past
actions but not the state.
