Introduction

Abstract

In addition to establishing notation and providing an overview of the monograph, this introductory chapter sets the stage for the kind of generalized hat problems in which we are interested. A reasonably general framework for these problems has a set A (of agents), a set K of (colors), and a set C of functions (colorings) mapping A to K. The goal is for the agents to construct coordinated strategies so that if each agent is given a certain piece of information about one of the colorings, then he can provide a guess as to some other aspect of the coloring. The collection of guesses, taken together over the set of agents, picks out a (possibly empty) set of colorings, those consistent with every agent’s guess. We think of this process of collecting together the guesses of the agents as a “predictor.” In most cases of interest, this prediction is a single coloring. The chapter also introduces one positive result—using the so-called “μ-predictor”—and two important negative results that will be used frequently in later chapters.