Zero-sum repeated games: counterexamples to the existence of the asymptotic value and the conjecture maxmin=lim v(n)

We provide an example of a two-player zero-sum repeated game with public signals and perfect observation of the actions, where neither the value of the lambda-discounted game nor the value of the n-stage game converges, when respectively lambda goes to 0 and n goes to infinity. It is a counterexample to two long-standing conjectures, formulated by Mertens: first, in any zero-sum repeated game, the asymptotic value exists, and secondly, when Player 1 is more informed than Player 2, Player 1 is able to guarantee the limit value of the n-stage game in the long run. The aforementioned example involves seven states, two actions and two signals for each player. Remarkably, players observe the payoffs, and play in turn (at each step the action of one player only has an effect on the payoff and the transition). Moreover, it can be adapted to fit in the class of standard stochastic games where the state is not observed.