A Reformulation-Based Simplicial Homotopy Method for Approximating Perfect Equilibria

As a strict refinement of Nash equilibrium, the concept of perfect equilibrium was formulated by Selten (Int J Game Theory 4(1):25–55, 1975). A well-known application of this concept is that every perfect equilibrium of the agent normal form game of an extensive form game with perfect recall yields a trembling-hand perfect equilibrium (consequently a sequential equilibrium). To compute a perfect equilibrium, this paper extends Kohlberg and Mertens’s equivalent reformulation of Nash equilibrium to a perturbed game. This extension naturally leads to a homotopy mapping on the Euclidean space. With this homotopy mapping and a triangulation, we develop a simplicial homotopy method for approximating perfect equilibria. It is proved that every limit point of the simplicial path yields a perfect equilibrium. Numerical results further confirm the effectiveness of the method.