Alan D. Taylor – författare

Simple games are mathematical structures inspired by voting systems in which a single alternative, such as a bill, is pitted against the status quo. The first in-depth mathematical study of the subject as a coherent subfield of finite combinatorics--one with its own organized body of techniques and results--this book blends new theorems with some of the striking results from threshold logic, making all of it accessible to game theorists. Introductory material receives a fresh treatment, with an emphasis on Boolean subgames and the Rudin-Keisler order as unifying concepts. Advanced material focuses on the surprisingly wide variety of properties related to the weightedness of a game. A desirability relation orders the individuals or coalitions of a game according to their influence in the corresponding voting system. As Taylor and Zwicker show, acyclicity of such a relation approximates weightedness--the more sensitive the relation, the closer the approximation. A trade is an exchange of players among coalitions, and robustness under such trades is equivalent to weightedness of the game. Robustness under trades that fit some restrictive exchange pattern typically characterizes a wider class of simple games--for example, games for which some particular desirability order is acyclic. Finally, one can often describe these wider classes of simple games by weakening the total additivity of a weighting to obtain what is called a pseudoweighting. In providing such uniform explanations for many of the structural properties of simple games, this book showcases numerous new techniques and results.

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 simultaneously submit a guess of their own hat color, and they both go free if at least one of them guesses correctly. While no communication is allowed once the hats have been placed, they will, however, be allowed to have a strategy session before being brought to the room. Is there a strategy 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 domain based on some knowledge of its values at other points. Topics range from hat problems that are accessible to everyone willing to think hard, to some advanced topics in set theory and infinitary combinatorics. For example, there is a method of predicting the value f(a) of a function f mapping the reals to the reals, based only on knowledge of f''s values on the open interval (a – 1, a), and for every such 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.