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.
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