Yet Another Hat Game

TL;DR

This paper reviews existing hat games, introduces a new hybrid game, and provides an optimal, simple strategy with a combinatorial proof, contributing to the understanding of strategic guessing in information-limited scenarios.

Contribution

It introduces a novel hybrid hat game and derives an optimal strategy with a combinatorial proof, expanding the strategic framework of hat games.

Findings

Introduced a new hybrid hat game

Derived an optimal strategy for the new game

Provided a combinatorial proof of strategy optimality

Abstract

Several different "hat games" have recently received a fair amount of attention. Typically, in a hat game, one or more players are required to correctly guess their hat colour when given some information about other players' hat colours. Some versions of these games have been motivated by research in complexity theory and have ties to well-known research problems in coding theory, and some variations have led to interesting new research. In this paper, we review Ebert's Hat Game, which garnered a considerable amount of publicity in the late 90's and early 00's, and the Hats-on-a-line Game. Then we introduce a new hat game which is a "hybrid" of these two games and provide an optimal strategy for playing the new game. The optimal strategy is quite simple, but the proof involves an interesting combinatorial argument.