Recursive Lexicographical Search: Finding All Markov Perfect Equilibria of Finite State Directional Dynamic Games
定义了定向动态博弈（DDG）类，提出状态递归和递归字典序搜索（RLS）算法，可系统找出所有马尔可夫完美均衡（MPE），并应用于成本降低投资的伯特兰价格竞争模型，发现均衡数量随状态数指数增长。
We define a class of dynamic Markovian games, <it>directional dynamic games</it> (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 <it>any</it> Markov perfect equilibrium (MPE) of these games, much less <it>all</it> of them. We propose a fast and robust generalization of backward induction we call <it>state recursion</it> that operates on a decomposition of the overall DDG into a finite number of more tractable <it>stage games</it>, which can be solved recursively. We provide conditions under which state recursion finds at least one MPE of the overall DDG and introduce a <it>recursive lexicographic search</it> (RLS) algorithm that systematically and efficiently uses state recursion to find <it>all</it> MPE 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. We provide an <it>exact non-iterative algorithm</it> that finds all MPE of every stage game, and prove there can be only 1, 3, or 5 of them. Using the stage games as building blocks, RLS rapidly finds and enumerates all MPE of the overall game. RLS finds a unique MPE for an alternating move version of the leapfrogging game when technology improves with probability 1, but in other cases, and in any simultaneous move version of the game, it finds a huge multiplicity of MPE that explode exponentially as the number of possible cost states increases.