On Hats and other Covers

TL;DR

This paper explores optimal strategies for a hat-guessing game, linking strategies to covering codes, extending to multiple colors, and achieving high success probabilities through novel constructions.

Contribution

It introduces the concept of strong covering for multi-color hats and provides efficient constructions that approach perfect success probability.

Findings

Strategies for two-color hats relate to binary covering codes.

Extended analysis for q-color hats introduces strong covering concepts.

Constructed strategies achieve success probabilities close to 1.

Abstract

We study a game puzzle that has enjoyed recent popularity among mathematicians, computer scientist, coding theorists and even the mass press. In the game, $n$ players are fitted with randomly assigned colored hats. Individual players can see their teammates' hat colors, but not their own. Based on this information, and without any further communication, each player must attempt to guess his hat color, or pass. The team wins if there is at least one correct guess, and no incorrect ones. The goal is to devise guessing strategies that maximize the team winning probability. We show that for the case of two hat colors, and for any value of $n$, playing strategies are equivalent to binary covering codes of radius one. This link, in particular with Hamming codes, had been observed for values of $n$ of the form $2^m-1$. We extend the analysis to games with hats of $q$ colors, $q\geq 2$, where 1-coverings are not sufficient to characterize the best strategies. Instead, we introduce the more appropriate notion of a {\em strong covering}, and show efficient constructions of these coverings, which achieve winning probabilities approaching unity. Finally, we briefly discuss results on variants of the problem, including arbitrary input distributions, randomized playing strategies, and symmetric strategies.