Belief-State Monte Carlo Tree Search for Phantom Go

Phantom Go is a derivative of Go with imperfect information. It is challenging in AI field due to its great uncertainty of the hidden information and high game complexity inherited from Go. To deal with this imperfect information game with large game tree complexity, a general search framework named belief-state Monte Carlo tree search (BS-MCTS) is put forward in this paper. BS-MCTS incorporates belief-states into Monte Carlo Tree Search, where belief-state is a notation derived from philosophy to represent the probability that speculation is in accordance with reality. In BS-MCTS, a belief-state tree, in which each node is a belief-state, is constructed and search proceeds in accordance with beliefs. Then, Opponent Guessing and Opponent Predicting are proposed to illuminate the learning mechanism of beliefs with heuristic information. The beliefs are learned by heuristic information during search by specific methods, and we propose Opponent Guessing and Opponent Predicting to illuminate the learning mechanism. Besides, some possible improvements of the framework are investigated, such as incremental updating and all moves as first (AMAF) heuristic. Technical details are demonstrated about applying BS-MCTS to Phantom Go, especially on inference strategy. We examine the playing strength of the BS-MCTS and AMAF-BS-MCTS in Phantom Go by varying search parameters, also testify the proposed improvements.