On Hat Puzzle 2.1: From Dr. Todd Ebert's thesis

In this new section, we introduce another version of Hat Puzzle, which mathematicians credit to Dr. Todd Ebert, a computer science instructor at the UC, Irvine, who introduced it in his Ph.D. thesis [Applications of recursive operators to randomness and complexity].

The simple version of hat problem is stated as below:

Three players enter a room and a red or blue hat is placed on each person's head. The color of each hat is determined by a coin toss, with the outcome of one coin toss having no effect on the others. Each person can see the other players' hats but not his own.

No communication of any sort is allowed, except for an initial strategy session before the game begins. Once they have had a chance to look at the other hats, the players must simultaneously guess the color of their own hats or pass. The group shares a hypothetical $3 million prize if at least one player guesses correctly and no players guess incorrectly.

The same game can be played with any number of players. The general problem is to find a strategy for the group that maximizes its chances of winning the prize.

One obvious strategy for the players, for instance, would be for one player to always guess "red" while the other players pass. This would give the group a 50 percent chance of winning the prize. Can the group do better?

Most people think not. Since each person's hat color is independent of the other players' colors and no communication is allowed, it seems impossible for the players to learn anything just by looking at one another. All the players can do, it seems, is guess.