Mathematics of coordinated inference : a study of generalized hat problems / Christopher S. Hardin and Alan D. Taylor.

Includes bibliographical references and index.

1. Introduction --
2. The Finite Setting --
3. The Denumerable Setting: Full Visibility --
4. The Denumerable Setting: One-Way Visibility --
5. Dual Hat Problems and the Uncountable --
6. Galvin's Setting: Neutral and Anonymous Predictors --
7. The Topological Setting --
8. Universality of the -Predictor --
9. Generalizations and Galois-Tukey Connections --
Bibliography --
Index--

Two prisoners are told that they will be brought to a room and seated so that each can see the other. Hats will be placed on their heads;each hat is either red or green. The two prisoners must simultaneouslysubmit a guess of their own hat color, and they both go free if atleast one of them guesses correctly. While no communication is allowedonce the hats have been placed, they will, however, be allowed to havea strategy session before being brought to the room. Is there astrategy ensuring their release? The answer turns out to be yes, and this is the simplest non-trivial example of a hat problem. This book deals with the question of how successfully one can predict the value of an arbitrary function at one or more points of its domainbased on some knowledge of its values at other points. Topics rangefrom hat problems that are accessible to everyone willing to thinkhard, to some advanced topics in set theory and infinitarycombinatorics. For example, there is a method of predicting the valuef(a) of a function f mapping the reals to the reals, based only onknowledge of f's values on the open interval (a 1, a), and for everysuch function the prediction is incorrect only on a countable set that is nowhere dense. The monograph progresses from topics requiring fewer prerequisites to those requiring more, with most of the text being accessible to any graduate student in mathematics. The broad range of readership includes researchers, postdocs, and graduate students in the fields of set theory, mathematical logic, and combinatorics, The hope is that this book will bring together mathematicians from different areas to think about set theory via a very broad array of coordinated inference problems.