On Monte-Carlo tree search for deterministic games with alternate moves and complete information^★

¹ Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Université Paris Diderot, Case courrier 7012, 75205 Paris Cedex 13, France.
² Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Université Pierre-et-Marie Curie, Case 188, 4 place Jussieu, 75252 Paris Cedex 5, France.

^* Corresponding author: sylvain.delattre@univ-paris-diderot.fr

Received: 2 June 2017
Accepted: 15 January 2018

Abstract

We consider a deterministic game with alternate moves and complete information, of which the issue is always the victory of one of the two opponents. We assume that this game is the realization of a random model enjoying some independence properties. We consider algorithms in the spirit of Monte-Carlo Tree Search, to estimate at best the minimax value of a given position: it consists in simulating, successively, n well-chosen matches, starting from this position. We build an algorithm, which is optimal, step by step, in some sense: once the n first matches are simulated, the algorithm decides from the statistics furnished by the n first matches (and the a priori we have on the game) how to simulate the (n + 1)th match in such a way that the increase of information concerning the minimax value of the position under study is maximal. This algorithm is remarkably quick. We prove that our step by step optimal algorithm is not globally optimal and that it always converges in a finite number of steps, even if the a priori we have on the game is completely irrelevant. We finally test our algorithm, against MCTS, on Pearl’s game [Pearl, Artif. Intell. 14 (1980) 113–138] and, with a very simple and universal a priori, on the game Connect Four and some variants. The numerical results are rather disappointing. We however exhibit some situations in which our algorithm seems efficient.