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The Mathematics of Coordinated Inference

Christopher S. Hardin and Alan D. Taylor
Publication Date: 
Number of Pages: 
Developments in Mathematics 33
[Reviewed by
Miklós Bóna
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The title is certainly accurate. This 109-page book is entirely focused on the generalizations of the following problem. Two prisoners are put in a room where they cannot communicate with each other, but can see each other. Each of them wears a hat, either red or green. If at least one of them correctly guesses the color of his own hat, the prisoners will be released. The prisoners are allowed a strategy session beforehand. What should their strategy be?

In this easy case, the strategy should be that one prisoner should guess the color he sees, and the other one should guess the color he does not see (equivalently, one prisoner should assume that the two hats are the same color, while the other prisoner should assume that the hats are of different colors.

The authors walk us through the generalization of this problem. There could be more prisoners, more hats, the function that describes the colors can be more general as well. The prisoners may see all the hats, just some of them, or almost all of them. Or we can generalize the original problem by adding a graph, or adding some topology. On infinite sets, some generalizations of the hat problem may be equivalent to the Axiom of Choice. A surprising example is about a method to predict the value of \(f(a)\) for a function \(f:\mathbb{R}\to\mathbb{R}\) if we are only given the values of \(f\) on the interval \( (a-1,a)\). Most chapters contain a number of difficult-sounding open problems. 

Miklós Bóna is Professor of Mathematics at the University of Florida.

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 μ-Predictor.- 9. Generalizations and Galois-Tukey Connections.- Bibliography.- Index.