We consider a setting where each player of a simultaneous-move game

privately designs an information structure before playing the

game. One of these information structures is chosen at random to determine the

distribution of the private messages that players receive. These

messages allow players to correlate their actions; however, their private

design implies a push from correlated to Nash

equilibria. Indeed, the sequential equilibrium payoffs of the

extensive-form game with privately designed information structures are correlated

equilibrium payoffs of the underlying simultaneous-move game, but not all

correlated equilibrium payoffs are sequential equilibrium

payoffs. In generic 2-player games, the latter are specific convex

combinations of two Nash equilibrium payoffs.