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

We define a class of dynamic Markovian games, directional dynamic games (DDG), where directionality is represented by a strategy-independent partial order on the state space. We show that many games are DDGs, yet none of the existing algorithms are guaranteed to find anyMarkov perfect equilibrium (MPE) of these games, much less all of them. We propose a fast and robust generalization of backward induction we call state recursion that operates on a decomposition of the overall DDG into a finite number of more tractable stage games, which can be solved recursively. We provide conditions under which state recursion finds at least one MPE of the overall DDG in a finite number of steps. We introduce a recursive lexicographic search (RLS) algorithm that systematically and efficiently uses state recursion to find allMPE of the overall game in a finite number of steps. We apply RLS to find all MPE of a dynamic model of Bertrand price competition with cost reducing investments which we show is a DDG. RLS rapidly finds and enumerates each one of the hundreds of millions of MPE of this game.