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.
KeywordsVisibility Graph Successful Predictor Strict Partial Order Optimal Predictor Infinite Case
- [Ebe98]Ebert, T.: Applications of recursive operators to randomness and complexity. PhD thesis, University of California at Santa Barbara (1998)Google Scholar
- [Gar61]Gardner, M.: The 2nd Scientific American Book of Mathematical Puzzles & Diversions. Simon and Schuster, New York (1961)Google Scholar
- [LS02]Lenstra, H., Seroussi, G.: On hats and other covers. In: IEEE International Symposium on Informations Theory, Lausanne (2002)Google Scholar