Recursive Lexicographical Search: Finding All Markov Perfect Equilibria of Finite State Directional Dynamic Games

Abstract

We define a class of dynamic Markovian games that we call directional dynamic games (DDG) in which directionality is represented by a partial order on the state space. We propose a fast and robust state recursion algorithm that can find a Markov perfect equilibrium (MPE) via backward induction on the state space of the game. When there are multiple equilibria, this algorithm relies on an equilibrium selection rule (ESR) to pick a particular MPE.We propose a recursive lexicographic search (RLS) algorithm that systematically and efficiently cycles through all feasible ESRs and prove that the RLS algorithm finds all MPE of the overall game. We apply the algorithms to find all MPE of a dynamic duopoly model of Bertrand price competition and cost reducing investments which we show is a DDG. Even with coarse discretization of the state space we find hundreds of millions of MPE in this game.

Keywords: Dynamic games, directional dynamic games, Markov-perfect equilibrium, subgame perfect equilibrium, multiple equilibria, partial orders, directed acyclic graphs, d-subgames, generalized stage games, state recursion, recursive lexicographic search algorithm, variable-base arithmetic,successor function

JEL Classification: D92, L11, L13