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# Simple Games

### Hardcover

Published: 22nd September 1999
Ships: 7 to 10 business days
RRP \$222.99
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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.

 Preface p. ix Acknowledgments p. xv Fundamentals p. 3 Introduction p. 3 Examples p. 8 The Dual Game p. 14 The Algebra of Simple Games p. 19 The Two-Point Constant-Sum Extension of a Game p. 26 Pregames and Weighted Graphs p. 29 Vector-Weighted Simple Games and Dimension Theory p. 34 The Voting Bloc and Bicameral Meet Characterization p. 39 The Game behind a Simple Game p. 40 General Trading: Weighted Games p. 43 Introduction p. 43 Trading Transforms and Trading Matrices p. 45 Sequential Transfers p. 54 The Trading Characterization of Weighted Games p. 56 Pregraphs and Graphs p. 63 The Traditional Approaches: Systems of Linear Inequalities and Separating Hyperplanes p. 68 The Gabelman Examples p. 74 A General Framework p. 79 Pairwise Trading: Linear Games and Winder Games p. 86 Introduction p. 86 The Desirability Relation on Individuals and Swap Robustness p. 87 Shift Minimal Winning Coalitions and the Ordinal Power Structure of a Simple Game p. 92 A Classification Theorem for Linear Games p. 97 Chvatal's Conjecture p. 103 The PSA Pseudoweighting Characterization of Linear Games p. 110 The Local Weighting Characterization of Linear Games p. 115 Two-Trade Robustness and Winder Games p. 120 A Weighting Characterization of Winder Games p. 122 The Hereditarily Dual-Comparable Characterization of Winder Games p. 123 Cycle Trading: Weakly Acyclic Games and Strongly Acyclic Games p. 125 Introduction p. 125 An Impossibility Result for Coalitional Desirability Relations p. 127 Possibilities, and More Impossibilities, from the Weight-Induced Order p. 134 Lapidot's Desirability Relation on Coalitions and Weakly Acyclic Games p. 139 The SSA Pseudoweighting Characterization of Weakly Acyclic Games, and a Generalization p. 142 An Inductive Construction of SSA Pseudoweightings for Weakly Acyclic Games p. 145 Winder's Desirability Relation on Coalitions and Strongly Acyclic Games p. 150 A Pseudoweighting Characterization of Strongly Acyclic Games p. 156 Sequential Transfer Trading for [precedes][subscript L] and [precedes][subscript W] p. 157 Peleg's Question on the Weightedness of Constant-Sum Acyclic Games p. 165 Almost General Trading: Chow Games, Completely Acyclic Games, and Weighted Games p. 178 Introduction p. 178 Chow Games and Chow-Lapidot Parameters p. 179 A Gabelman-Style, Nonweighted Chow Game p. 183 The Trading Version of Lapidot's Desirability Relation p. 190 The Trading Version of Winder's Desirability Relation p. 196 Multiweightings p. 201 Weighted Games and the Weight-Induced Order p. 205 Systems of Linear Inequalities p. 215 Separating Hyperplanes p. 220 Duality and Transitivity for Binary Relations p. 223 References p. 229 Index p. 235 Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780691001203
ISBN-10: 0691001200
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 264
Published: 22nd September 1999
Dimensions (cm): 23.7 x 16.3  x 2.172
Weight (kg): 0.504