From Nash Equilibria to Chain Recurrent Sets: Solution Concepts and Topology

Nash's universal existence theorem for his notion of equilibria was essentially an ingenious application of fixed point theorems, the most sophisticated result in his era's topology --- in fact, recent algorithmic work has established that Nash equilibria are in fact computationally equivalent to fixed points. Here, we shift focus to universal non-equilibrium solution concepts that arise from an important theorem in the topology of dynamical systems that was unavailable to Nash. This approach takes as input both a game and a learning dynamic, defined over mixed strategies. Nash equilibria are guaranteed to be fixed points of such dynamics; however, the system behavior is captured by a more general object that is known in dynamical systems theory as chain recurrent set. Informally, once we focus on this solution concept, every game behaves like a potential game with the dynamic converging to these states. We characterize this solution for simple benchmark games under replicator dynamics, arguably the best known evolutionary dynamic in game theory. For potential games it coincides with the notion of equilibrium; however, in simple zero sum games, it can cover the whole state space. We discuss numerous novel computational as well as structural, combinatorial questions that chain recurrence raises.