Published online by Cambridge University Press: 30 May 2025
We present a Monte Carlo algorithm for efficiently finding near optimal moves and bids in the game of Bidding Hex. The algorithm is based on the recent solution of Random-Turn Hex by Peres, Schramm, Sheffield, and Wilson together with Richman’s work connecting random-turn games to bidding games.
Hex is well-known for the simplicity of its rules and the complexity of its play. Nash’s strategy-stealing argument shows that a winning strategy for the first player exists, but finding such a strategy is intractable by current methods on large boards. It is not known, for instance, whether the center hex is a winning first move on an odd size board. The development of artificial intelligence for Hex is a notoriously rich and challenging problem, and has been an active area of research for over thirty years [Davis 1975/76; Nishizawa 1976; Anshelevich 2000; 2002a; 2002b; Cazenave 2001; Rasmussen and Maire 2004], yet the best programs play only at the level of an intermediate human [Melis and Hayward 2003]. Complete analysis of Hex is essentially intractable; the problem of determining which player has a winning strategy from a given board position is PSPACE-complete [Reisch 1981], and the problem of determining whether a given empty hex is dead, or irrelevant to the outcome of the game, is NP-complete [Björnsson et al. 2007]. Some recent research has focused on explicit solutions for small boards [Hayward et al. 2004; Hayward et al. 2005], but it is unclear whether such techniques will eventually extend to the standard 11x11 board.
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