Finite and Innite Hat Problems

This project will discuss hat problems and the Axiom of Choice. A basic example of a …nite hat problem is helpful in introducing the concept. Assume that there are two players and two possible hat colors, red and blue. The players will be given a certain amount of time before the game begins to communicate and create a strategy for determining their own hat colors. After this initial strategy session, the game begins and no communication of any kind is allowed between the players. A third party will place a red or a blue hat on each of the players. Each player will be able to see his teammate’s hat color, but will not be able to see his own hat color. The object of the game is for a player to correctly guess his own hat color. Each player’s guess must be independent of his teammate’s guess. Both players win the game if at least one of them guesses correctly. We want to know if there exists a strategy which ensures that at least one player correctly guesses the color of his own hat. Surprisingly, the answer to this question is yes. If one player guesses that his hat is the same color as his teammate’s hat and the other player guesses that his hat color is the opposite of his teammates hat color, then exactly one of the players will guess correctly. Although this is a very basic example, the solution is not immediately apparent. The question which this project deals with is whether a strategy exists which guarantees a win for the players under various scenarios. The articles by Christopher S. Hardin and Alan D. Taylor [3], [4] along with the text by Thomas Q. Sibley [5] will be used in the discussion of these problems. Before we can explore these more complex problems and solutions, we must de…ne a number of terms and concepts which will be used throughout this paper.