A General Theorem for Non-Simultaneous Hat Guessing Puzzles
Souji Shizuma
TL;DR
This work develops a general framework for non-simultaneous hat-guessing puzzles, formalizing the games with set-theoretic constructs and predictors. It establishes finite- and infinite-prisoner results, showing that non-simultaneity enables strong strategies such as at most one error or finite-error predictors, often leveraging the Axiom of Choice to construct parity functions. The paper connects non-simultaneous results to the classical simultaneous case, clarifying where each paradigm yields different guarantees, and it highlights open questions about the necessity of certain structural conditions. The findings have implications for strategy design in coordination games and illuminate foundational aspects of choice and parity in infinite combinatorial settings.
Abstract
The prisoners and hats puzzle, or simply the hat puzzle, is a family of games in which a group of prisoners are each assigned a colored hat and are asked to guess the color of their own hat. Various versions of the puzzle arise depending on the number of prisoners, the number of possible hat colors, and the information available to them before and after the game begins. These puzzles are broadly classified according to whether the prisoners' declarations are made simultaneously or non-simultaneously. In this paper we present a general theorem concerning the existence of a winning strategy when the declarations are non-simultaneous. We also discuss the relationship between the construction of such strategies and the Axiom of Choice, as well as their connection to the simultaneous-declaration case.