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

Abstract: 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 $\lfloor n/2 \rfloor$ of the $n$ player guess their hat color correct, our strategy ensures that the players produce $\max{r,b} - 1.2 n^{2/3} -2$ 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 $f(n) = \Omega(\sqrt n)$.