Winkler's Hat Guessing Game: Better Results for Imbalanced Hat Distributions

In this note, we give an explicit polynomial-time executable strategy for Peter Winkler’s hat guessing game that gives superior results if the distribution of hats is imbalanced. While Winkler’s strategy guarantees in any case that ⌊n/2⌋ of the n player guess their hat color correct, our strategy ensures that the players produce max{r, b} − 1.2n2/3 − 2 correct guesses for any distribution of r red and b = n − r blue hats. We also show that any strategy ensuring max{r, b} − f(n) correct guesses necessarily has f(n) = Ω( √ n). 1 The Hat Color Guessing Game In this note, we deal with the following game suggested by Peter Winkler [Win02]. In the simultaneous hat guessing game, there are n players each wearing a red or blue hat. Each player can see all hats except his own. Simultaneously, the players have to guess the color of their own hat. No communication is allowed during the game. The players may, however, discuss their strategy before they get to see the hats. It is easy to see from the rules that no player can make sure that he guesses his hat color, no matter what strategy the players agree on. Thus the following result in [Win02] is quite surprising. There is a strategy that guarantees that ⌊n/2⌋ players guess their hat color ∗This is an old result of mine never published except in my Habilitation thesis.