Insights into game theory : an alternative mathematical experience /
Ein-Ya Gura and Michael Maschler.
- Cambridge, UK ; New York : Cambridge University Press, 2008.
- xiv, 236 p. : ill. ; 24 cm.
Includes bibliographical references (p. [233]-234) and index.
Prefacep. xi Introductionp. xii Mathematical Matchingp. 1 Introductionp. 1 The Matching Problemp. 2 Exercisesp. 7 Further Examplesp. 10 Exercisesp. 13 A Procedure for Finding Stable Matching Systems (The Gale-Shapley Algorithm)p. 15 Exercisesp. 18 A Stable Matching System Always Existsp. 19 The Maximum Number of Courtship Stages in the Gale-Shapley Algorithmp. 21 Generalizationp. 26 Exercisesp. 33 The Gale-Shapley Algorithm and the Assignment Problemp. 37 Exercisesp. 40 Optimalityp. 43 Exercisesp. 49 Condition for the Existence of a Unique Stable Matching Systemp. 52 Exercisesp. 54 Discussionp. 55 Review Exercisesp. 56 Social Justicep. 59 Presentation of the Problemp. 59 Mathematical Description of the Problemp. 62 Exercisesp. 64 Social Choice Functionp. 67 Axioms for the Social Choice Functionp. 77 Exercisesp. 80 What Follows from Axioms 1-4?p. 81 Exercisesp. 85 Arrow's Theoremp. 87 What Next?p. 92 Review Exercisesp. 93 The Shapley Value in Cooperative Gamesp. 97 Introductionp. 97 Cooperative Gamesp. 98 Important Examples of Coalition Function Gamesp. 101 Exercisesp. 105 Additive Gamesp. 106 Superadditive Gamesp. 107 Majority Gamesp. 108 Exercisesp. 112 Symmetric Playersp. 113 Exercisesp. 115 Null Playersp. 116 Exercisesp. 117 The Sum of Gamesp. 118 Exercisesp. 121 The Shapley Valuep. 124 Exercisesp. 133 Dissolving a Partnershipp. 133 Exercisesp. 141 The Shapley Value as the Average of Players' Marginal Contributionsp. 142 Exercisesp. 146 The Shapley Value as a Player's Index of Power in Weighted Majority Gamesp. 148 Exercisesp. 153 The Shapley-Shubik Index as an Index for the Analysis of Parliamentary Phenomenap. 153 Exercisesp. 156 The Security Councilp. 156 Exercisesp. 158 Cost Gamesp. 159 Exercisesp. 162 Review Exercisesp. 164 Analysis of a Bankruptcy Problem from the Talmudp. 166 Introductionp. 166 The Contested Garmentp. 168 Exercisesp. 171 A Physical Interpretation of the Contested-Garment Principlep. 172 Exercisesp. 176 A Bankruptcy Problem from the Talmudp. 177 Exercisesp. 180 Existence and Uniquenessp. 182 Divisions Consistent with the Contested-Garment Principlep. 186 Exercisesp. 191 Consistencyp. 192 Exercisesp. 194 Rif's Law of Divisionp. 194 Exercisesp. 196 Proportional Divisionp. 196 O'Neill's Law of Divisionp. 197 Exercisesp. 200 Discussionp. 201 Review Exercisesp. 203 Answers to the Exercisesp. 205 Chapter 1p. 205 Chapter 2p. 213 Chapter 3p. 220 Chapter 4p. 229 Bibliographyp. 233 Indexp. 235 Table of Contents provided by Ingram. All Rights Reserved.
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.