On the hat problem, its variations, and their applications

The topic of our paper is the hat problem in which each of n players is randomly fitted with a blue or red hat. Then everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The aim is to maximize the probability of a win. There are known many variations of the hat problem. In this paper we give a comprehensive list of variations considered in the literature. We describe the applications of the hat problem and its variations, and their connections to different areas of science. We give the full bibliography of any papers, books, and electronic publications about the hat problem.