Minimal predictors in hat problems

Abstract

We consider a combinatorial problem related to guessing the values of a function at various points based on its values at certain other points, often presented by way of a hat-problem metaphor: there are a number of players who will have colored hats placed on their heads, and they wish to guess the colors of their own hats. A visibility relation specifies who can see which hats. This paper focuses on the existence of minimal predictors: strategies guaranteeing at least one player guesses correctly, regardless of how the hats are colored. We first present some general results, in particular showing that transitive visibility relations admit a minimal predictor exactly when they contain an infinite chain, regardless of the number of colors. In the more interesting nontransitive case, we focus on a particular nontransitive relation on ω that is elementary, yet reveals unexpected phenomena not seen in the transitive case. For this relation, minimal predictors always exist for two colors but never for ℵ2 colors. For ℵ0 colors, the existence of minimal predictors is independent of ZFC plus a fixed value of the continuum, and turns out to be closely related to certain cardinal invariants involving meager sets of reals.

The mathematics of coordinated inference: a study of generalized hat problems

“…There is a version of Theorem 7.2.5 for the case in which the space is not T 0 . However, it turns out to be somewhat of a disjoint union of what we have done here and the finite case in [HT10].…”

“…Our starting point for discussing EO is the following theorem; the n = 2 case appeared in [HT10] and the general case was obtained by D. J. Velleman [Vel11]. The proof relies on Theorem 2.2.3.…”

“…We conclude this section with one other result about minimal predictors. It was first established in [BHKL08] and later rediscovered by Daniel J. Velleman [Vel11] in his solution to a question left open in [HT10]. Velleman's result is in Chapter 4.…”

The even-odd hat problem

“…Introduction. The purpose of this paper is to answer a question stated by C. Hardin and A. Taylor in their paper [3]. Hardin and Taylor study hat problems, in which a set of players have hats of a variety of different colors placed on their heads.…”

“…The question Hardin and Taylor ask in each case is whether or not there are strategies the players can use that will guarantee that at least one player guesses correctly. For more background on hat problems we refer the reader to [2] and [3].…”