Paranoid Proof-Number Search

Over the years, an increasing number of two-player games has been added to the list of solved games. However, solving multi-player games has so far received little, if any, attention. One of the reasons is that while two-player games have a unique game-theoretical value, no unique game-theoretical value may exist in multi-player games as they can have many equilibrium points. We propose solving multi-player games under the paranoid condition. This is equivalent to find the optimal score that a player can achieve independent of the other players´ strategies. We furthermore introduce and examine an algorithm, called Paranoid Proof-Number Search (PPNS), for solving multi-player games under the paranoid condition. PPNS is tested by solving the 4×4 and 6×6 variant of the multi-player game Rolit, a multi-player generalization of Reversi (Othello). Our results show that no player can win more than the analytical minimum score in 6 × 6 Rolit while on 4 × 4 Rolit the players are able to score higher. Moreover, the experiments show that for Rolit PPNS is taking advantage of the non-uniformity of the game tree.