Ebert's asymmetric Hat Game

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. All players guess simultaneously the color of their own head 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 or her own hat color correct and none of the players has an incorrect guess. This paper studies Ebert's hat problem, where the probabilities of the colors may be different (asymmetric case). Our goal is to maximize the probability of winning the game and to describe winning strategies. In this paper we introduce the notion of an adequate set. The construction of adequate sets is independent of underlying probabilities and we 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.