Repeated games with endogenous choice of information mechanisms

We consider two-player repeated games with nonobservable actions (cf. Lehrer 1989). An information mechanism for a player is a function which assigns a private signal to every action-pair of the one-shot game. In this paper, we extend the model to a situation in which both players can buy an information mechanism before playing the repeated game. Within this model, we provide a characterization of the lower equilibrium payoffs in terms of the one-shot game for the case that both players choose a nontrivial information mechanism with probability one. Moreover, we construct a lower equilibrium in a repeated game in which one of the players strictly randomizes between information mechanisms. It is shown that the corresponding payoffs cannot be induced by a lower equilibrium in which players choose a particular information mechanism with probability one.