Simple Games

Desirability Relations, Trading, Pseudoweightings

Alan D. Taylor, William S. Zwicker

• Publisher: Princeton University Press
• Year: 2021
• Print ISBN: 9780691001203
• eBook ISBN: 9780691223896
• Topic: Mathematics
• Subtopic: Game Theory
• Index: Mathematics

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.

Table of contents

• Cover Page
• Half-title Page
• Title Page
• Copyright Page
• Dedication Page
• Contents
• Preface
• Acknowledgments
• Chapter 1 - Fundamentals
• Chapter 2 - General Trading: Weighted Games
• Chapter 3 - Pairwise Trading: Linear Games and Winder Games
• Chapter 4 - Cycle Trading: Weakly Acyclic Games and Strongly Acyclic Games
• Chapter 5 - Almost General Trading: Chow Games, Completely Acyclic Games, and Weighted Games
• Appendix I: Systems of Linear Inequalities
• Appendix II: Separating Hyperplanes
• Appendix III: Duality and Transitivity for Binary Relations
• References
• Index