Privately designed correlated equilibrium

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.