5. Equilibrium Manifolds and Dynamics

This chapter describes concave games, potential games, population games and Nash/Wardrop equilibria and introduces the characterization of equilibria via variational inequalities. We then study the equilibrium manifold, where the finite strategy sets are fixed and the payoff functions vary. We prove the structure theorem of Kohlberg and Mertens, then show that every game has an essential component of Nash equilibria, and that for generic payoffs, the set of mixed equilibria is finite and its cardinality is odd. We also introduce Nash vector fields and game dynamics, such as the replicator dynamics, the Smith dynamics, the best response dynamics and the Brown–von Neumann–Nash dynamics. We conclude this chapter with an analysis of evolutionary stable strategies (ESS) and their connection with replicator dynamics.