Repeated Games with Incomplete Information

We revisit the question of modeling incomplete information among 2 Bayesian players, following an ex-ante approach based on values of zero-sum games. K being the finite set of possible parameters, an information structure is defined as a probability distribution u with finite support over K × N × N with the interpretation that: u is publicly known by the players, (k, c, d) is selected according to u, then c (resp. d) is announced to player 1 (resp. player 2). Given a payoff structure g, composed of matrix games indexed by the state, the value of the incomplete information game defined by u and g is denoted val(u, g). We evaluate the pseudo-distance d(u, v) between 2 information structures u and v by the supremum of |val(u, g) − val(v, g)| for all g with payoffs in [−1, 1], and study the metric space Z∗ of equivalent information structures. We first provide a tractable characterization of d(u, v), as the minimal distance between 2 polytopes, and recover the characterization of Peski (2008) for u v, generalizing to 2 players Blackwell’s comparison of experiments via garblings. We then show that Z∗, endowed with a weak distance dW , is homeomorphic to the set of consistent probabilities with finite support over the universal belief space of Mertens and Zamir. Finally we show the existence of a sequence of information structures, where players acquire more and more information, and of ε > 0 such that any two elements of the sequence have distance at least ε : having more and more information may lead nowhere. As a consequence, the completion of (Z∗, d) is not compact, hence not homeomorphic to the set of consistent probabilities over the states of the world à la Mertens and Zamir. This example answers by the negative the second (and last unsolved) of the three problems posed by J.F. Mertens in his paper “Repeated Games”, ICM 1986.