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

TL;DR

This paper presents an improved polynomial-time strategy for Winkler's hat guessing game that outperforms previous methods in imbalanced distributions, achieving more correct guesses than the guaranteed half, with bounds on optimality.

Contribution

It introduces a new explicit strategy that yields better results for imbalanced hat distributions and analyzes the limitations of strategies guaranteeing a certain number of correct guesses.

Findings

Strategy guarantees more correct guesses in imbalanced cases

Explicit polynomial-time strategy provided

Lower bounds on the improvement possible with any strategy

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 $f(n) = \Omega(\sqrt n)$.