Generalized three and four person hat game

Abstract

Ebert's hat problem with two colors and equal probabilities has, remarkable, the same optimal winning probability for three and four players. This paper studies Ebert's hat problem for three and four players, where the probabilities of the two colors may be different for each player. Our goal is to maximize the probability of winning the game and to describe winning strategies. We obtain different results for games with three and four players. We use the concept of an adequate set. The construction of adequate sets is independent of underlying probabilities and we can use this fact in the analysis of our general case. The computational complexity of the adequate set method is dramatically lower than by standard methods.