This book systematically presents the main solutions of cooperative games: the core, bargaining set, kernel, nucleolus, and the Shapley value of TU games, and the core, the main variants of the Shapley value, the ordinal bargaining set, and the Mas-Colell bargaining set of NTU games. To each solution the authors devote a separate chapter wherein they study its properties in full detail. Moreover, important variants are defined or even intensively analyzed. The authors also investigate in separate chapters continuity, dynamics, and geometric properties of solutions of TU games. The study culminates in uniform and coherent axiomatizations of all the foregoing solutions (excluding the bargaining set). Such axiomatizations have not appeared in any book. Moreover, the book contains a detailed analysis of the main results on cooperative games without side payments. Such analysis is very limited or non-existent in other books on game theory.
Introduction 1
Cooperative Games 1
Outline of the Book 2
Special Remarks 5
2 Coalitional TU Games and Solutions 9
Coalitional Games 9
Some Families of Games 13
Properties of Solutions 19
3 The Core 27
The Bondareva-Shapley Theorem 27
An Application to Market Games 32
Totally Balanced Games 34
Some Families of Totally Balanced Games 35
A Characterization of Convex Games 39
An Axiomatization of the Core 40
An Axiomatization of the Core on Market Games 42
The Core for Games with Various Coalition Structures 44
4 Bargaining Sets 51
The Bargaining Set M 52
Existence of the Bargaining Set 57
Balanced Superadditive Games and the Bargaining Set 62
Further Bargaining Sets 64
Non-monotonicity of Bargaining Sets 72
The Bargaining Set and Syndication: An Example 76
5 The Prekernel, Kernel, and Nucleolus 81
The Nucleolus and the Prenucleolus 82
The Reduced Game Property 86
Desirability, Equal Treatment, and the Prekernel 89
An Axiomatization of the Prekernel 91
