Positive value of information in games

We exhibit a general class of interactive decision situations in which all the agents benefit from more information. This class includes as a special case the classical comparison of statistical experiments `a la Blackwell. More specifically, we consider pairs consisting of a game with incomplete information G and an information structure S such that the extended game (G,S) has a unique Pareto payoff profile u. We prove that u is a Nash payoff profile of (G,S), and that for any information structure T that is coarser than S, all Nash payoff profiles of (G,S) are dominated by u. We then prove that our condition is also necessary in the following sense: Given any convex compact polyhedron of payoff profiles, whose Pareto frontier is not a singleton, there exists an extended game (G,S) with that polyhedron as the convex hull of feasible payoffs, an information structure T coarser than S and a player i who strictly prefers a Nash equilibrium in (G,S) to any Nash equilibrium in (G,S).