We give new proofs of existence of the limit of the discounted values for two person zero-sum games in the following frameworks: incomplete information, absorbing, recursive. The idea of these new proofs is to use some comparison criteria.

We give an example of a zero-sum stochastic game with four states, compact action sets for each player, and continuous payoff and transition functions, such that the discounted value does not converge as the discount factor tends to 0, and the value of the n-stage game does not converge as n...