Moral Hazard in Stochastic Differential Games: Beyond Markov Equilibrium

Résumé

We study n-player continuous-time repeated and stochastic games with imperfect monitoring in which the publicly observable state vector follows a jointly controlled Markov diffusion process. By extending the analysis of Sannikov (2007) to allow for more than two players and/or payoff-relevant state variables, we characterize the correspondence of (perfect public) equilibria and attainable payoffs. We introduce two types of optimality equations: an elliptic partial differential equation, and a parabolic one. Under an identifiability condition the support function of the correspondence of equilibrium payoffs solves the elliptic equation in the viscosity sense, but it may not be the unique solution. The parabolic equation has a unique viscosity solution, and we develop a numerical method to compute it. The solution of the parabolic equation converges (as the time parameter goes to infinity) to the support function of the equilibrium payoff correspondence. Relative to the two-player repeated games of Sannikov (2007), the optimal equilibria of the games that we study have new features, such as absorbing regimes that do not correspond to static Nash or Markov equilibria. Joint work with Yuliy Sannikov (Stanford GSB).