The Large Space Of Information Structures

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 d W , 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 worldàworldà 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.