Subgame-Perfect Equilibria in Discrete and Continuous Games: Does Discretization Matter?

Abstract

This chapter discusses the impact of discretization of a continuous game on the set of subgame-perfect equilibria. The transition from discrete to continuous games is continuous in the observables and discontinuous in the unobservables. The limit path of any sequence of equilibrium path of discrete games that converge to a continuous game is a subgame-perfect equilibrium path of the continuous game. However, the limits—if they exist at all—of the associated equilibrium strategies in general do not form an equilibrium strategy combination of the continuous game. The chapter provides a constructive proof of the first assertion, which gives a clear explanation for this often encountered phenomenon and is used to compute the equilibrium strategies that sustain the limit path in the continuous game. It presents as a by-product, a new proof for the existence of subgame-perfect equilibria in continuous extensive games.