The Topological Setting

Abstract

In this chapter we start to move further away from the hat problem metaphor and think instead of trying to predict a function’s value at a point based on knowing (something about) its values on nearby points. The most natural setting for this is a topological space and if we wanted to only consider continuous colorings, then the limit operator would serve as a unique optimal predictor. But we want to consider arbitrary colorings. Thus we have each point in a topological space representing an agent, and if f and g are two colorings, then f and g are indistinguishable to agent a if f and g agree on some deleted neighborhood of the point a. It turns out that an optimal predictor in this case is wrong only on a set that is “scattered” (a concept with origins going back to Cantor).

To illustrate one corollary of this topological result, consider the hat problem in which the agents are indexed by real numbers, and each agent sees the hats worn by those to his left (that is, those indexed by smaller real numbers). The set of hat colors is some arbitrary set K. The question is whether or not there is a predictor ensuring that the set of agents guessing incorrectly is a small infinite set—e.g., indexed by a set of reals that is countable and nowhere dense. The answer here (again assuming the axiom of choice) is yes. In fact, there is a predictor guaranteeing the set of agent guessing incorrectly is a set of reals that is well ordered by the usual ordering of the reals. Moreover, this is an optimal predictor.