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 à la Blackwell. More specifically, we consider pairs consisting of a game with incomplete information G and an information structure script l sign such that the extended game Λ(G, script l sign) has a unique Pareto payoff profile u. We prove that u is a Nash payoff profile of Λ(G, script l sign), and that for any information structure script l sign that is coarser than script l sign, all Nash payoff profiles of Λ(G, script l sign) 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 Λ(G, script l sign) with that polyhedron as the convex hull of feasible payoffs, an information structure script l sign coarser than script l sign and a player i who strictly prefers a Nash equilibrium in Λ(G, script l sign) to any Nash equilibrium in Λ(G, script l sign).