Insights into Game Theory: An Alternative Mathematical Experience

by Ein-Ya Gura , Michael Maschler

• ISBN13: 9780521874229
• ISBN10: 052187422X
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-12-15
• Publisher: Cambridge University Press

Summary

Few branches of mathematics have been more influential in the social sciences than game theory. In recent years, it has become an essential tool for all social scientists studying the strategic behaviour of competing individuals, firms and countries. However, the mathematical complexity of game theory is often very intimidating for students who have only a basic understanding of mathematics. Insights into Game Theory addresses this problem by providing students with an understanding of the key concepts and ideas of game theory without using formal mathematical notation. The authors use four very different topics (college admission, social justice and majority voting, coalitions and co-operative games, and a bankruptcy problem from the Talmud) to investigate four areas of game theory. The result is a fascinating introduction to the world of game theory and its increasingly important role in the social sciences.

Table of Contents

Preface	p. xi
Introduction	p. xii
Mathematical Matching	p. 1
Introduction	p. 1
The Matching Problem	p. 2
Exercises	p. 7
Further Examples	p. 10
Exercises	p. 13
A Procedure for Finding Stable Matching Systems (The Gale-Shapley Algorithm)	p. 15
Exercises	p. 18
A Stable Matching System Always Exists	p. 19
The Maximum Number of Courtship Stages in the Gale-Shapley Algorithm	p. 21
Generalization	p. 26
Exercises	p. 33
The Gale-Shapley Algorithm and the Assignment Problem	p. 37
Exercises	p. 40
Optimality	p. 43
Exercises	p. 49
Condition for the Existence of a Unique Stable Matching System	p. 52
Exercises	p. 54
Discussion	p. 55
Review Exercises	p. 56
Social Justice	p. 59
Presentation of the Problem	p. 59
Mathematical Description of the Problem	p. 62
Exercises	p. 64
Social Choice Function	p. 67
Axioms for the Social Choice Function	p. 77
Exercises	p. 80
What Follows from Axioms 1-4?	p. 81
Exercises	p. 85
Arrow's Theorem	p. 87
What Next?	p. 92
Review Exercises	p. 93
The Shapley Value in Cooperative Games	p. 97
Introduction	p. 97
Cooperative Games	p. 98
Important Examples of Coalition Function Games	p. 101
Exercises	p. 105
Additive Games	p. 106
Superadditive Games	p. 107
Majority Games	p. 108
Exercises	p. 112
Symmetric Players	p. 113
Exercises	p. 115
Null Players	p. 116
Exercises	p. 117
The Sum of Games	p. 118
Exercises	p. 121
The Shapley Value	p. 124
Exercises	p. 133
Dissolving a Partnership	p. 133
Exercises	p. 141
The Shapley Value as the Average of Players' Marginal Contributions	p. 142
Exercises	p. 146
The Shapley Value as a Player's Index of Power in Weighted Majority Games	p. 148
Exercises	p. 153
The Shapley-Shubik Index as an Index for the Analysis of Parliamentary Phenomena	p. 153
Exercises	p. 156
The Security Council	p. 156
Exercises	p. 158
Cost Games	p. 159
Exercises	p. 162
Review Exercises	p. 164
Analysis of a Bankruptcy Problem from the Talmud	p. 166
Introduction	p. 166
The Contested Garment	p. 168
Exercises	p. 171
A Physical Interpretation of the Contested-Garment Principle	p. 172
Exercises	p. 176
A Bankruptcy Problem from the Talmud	p. 177
Exercises	p. 180
Existence and Uniqueness	p. 182
Divisions Consistent with the Contested-Garment Principle	p. 186
Exercises	p. 191
Consistency	p. 192
Exercises	p. 194
Rif's Law of Division	p. 194
Exercises	p. 196
Proportional Division	p. 196
O'Neill's Law of Division	p. 197
Exercises	p. 200
Discussion	p. 201
Review Exercises	p. 203
Answers to the Exercises	p. 205
Chapter 1	p. 205
Chapter 2	p. 213
Chapter 3	p. 220
Chapter 4	p. 229
Bibliography	p. 233
Index	p. 235
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