Private Strategy and Efficiency: Repeated Partnership Games Revisited

This paper studies repeated partnership games with only two public signals, the game studied in Radner Myerson and Maskin(1986) except that the stage game is discrete. It is well known that the public perfect equilibrium (PPE) payoff set is bounded away from the efficient frontier in this class of game. However, it has not been unanswered how this restriction to public strategies is restrictive even in this simple class of repeated games. In this paper, I construct a strongly symmetric sequential equilibrium whose payoff dominates the best symmetric PPE payoff. The strategy used to construct this equilibrium depends not only on public signals but also on player's own past actions. I call this kind of strategy private strategy and such an equilibrium a private sequential equilibrium. I also provide an example where the private sequential equilibrium approximates the efficient outcome, but the PPE payoff set is contained in an arbitrary small neighborhood of the stage game Nash equilibrium payoff. In precise, I can find a sequence of the stage game (partnership game) where the private strategy sequential equilibrium converges to the symmetric efficient outcome and the whole PPE payoff set converges to the stage game Nash equilibrium payoff. This implies that the private sequential equilibrium payoff dominates any PPE payoff. This example suggests that the difference between a PPE payoff set and a sequential equilibrium payoff set can be potentially significant.