Asymmetric Hat Game with three players and three colors

The Hat Game (Ebert’s Hat Problem) got much attention in the beginning of this century; not in the last place by its connections to coding theory and computer science. This paper studies Ebert’s hat problem with three persons and three colors, where the probabilities of the colors may be different (asymmetric case) . All players guess simultaneously the color of their own hat observing only the hat colors of the other players. It is also allowed for each player to pass: no color is guessed. The team wins if at least one player guesses his hat color correct and none of the players has an incorrect guess. Our goal is to maximize the probability of winning the game and to describe winning strategies. Exact winning probability in case of three or more players and three or more colors is only known in the symmetric case with three players and three colors. In this paper we extend the notion of adequate set to more colors. The construction of adequate sets is independent of underlying probabilities and we can use this fact in the analysis of the asymmetric case. Another point of interest is the fact that computational complexity using adequate sets is much less than using standard methods.