A probabilistic look at the infinite hat-guessing game
Nathaniel Eldredge
TL;DR
The paper analyzes a two-color hat-guessing game with countably infinite players, contrasting nonconstructive AC-based strategies that can make all but finitely many guesses correct with measurable probabilistic approaches that bound performance. It proves an almost-sure bound for measurable strategies: $L \le \tfrac{1}{2} \le U$, demonstrating that unbalanced asymptotic densities are impossible without AC, while AC-based constructions (Gabay-O'Connor, Lenstra) exhibit far stronger, non-measurable success. The work explores explicit measurable strategies that realize the full range of feasible $(L,U)$ pairs, examines the growth of $S_n - \tfrac{n}{2}$, and extends the discussion to variants, including more colors and limited visibility. Together, it clarifies the divide between what can be achieved with and without the axiom of choice and highlights the probabilistic structure underpinning hat-guessing in infinite settings.
Abstract
In this article, we look at a hat-guessing game, in which each player must guess the color of their own hat while only seeing the hats of the other players. We focus on the case of two hat colors and a countably infinite number of players. By strategizing in advance, the players can, in some ways, do much better than random guessing; using the axiom of choice, they can in fact achieve highly counter-intuitive success. We review some of these results. Then, we use tools from probability to obtain bounds on how successful a strategy can be under a measurability hypothesis, in terms of the asymptotic density of the set of correctly guessing players. As we discuss, this illustrates that the full axiom of choice is truly necessary for the counter-intuitively successful strategies, and that there is a wide gap between what can be achieved with and without choice.