Nonsmooth dynamics of generalized Nash games

Abstract

The generalized Nash equilibrium problem (GNEP) is an N-player noncooperative game, where each player has to solve a nonlinear optimization problem whose objective function and constraints depend on the choices of the other players. As in the case of classic Nash games, where other players' choices only impact a player's objective function, a natural question arises as to how players might evolve their strategies over time, and whether or not this evolution would allow them to reach a Nash equilibrium strategy. The approach in classical Nash games is that of introducing some form of differential equations/systems whose stable points are exactly the Nash strategies of the game. This approach leads to considering projected dynamical systems and sweeping processes. In this paper, we show that these dynamical system approaches can be extended to the case of the GNEP. We present dynamical systems that are useful in this context and discuss the new difficulties introduced by this more complex game. Finally, we show how to exploit the existence proof to build numerical methods and solve GNEP problems from the literature.