# Models in Cooperative Game Theory

Springer Science & Business Media, Mar 8, 2008 - Business & Economics - 204 pages
Cooperative game theory is a booming research area with many new developments in the last few years. So, our main purpose when prep- ing the second edition was to incorporate as much of these new dev- opments as possible without changing the structure of the book. First, this o?ered us the opportunity to enhance and expand the treatment of traditional cooperative games, called here crisp games, and, especially, that of multi-choice games, in the idea to make the three parts of the monograph more balanced. Second, we have used the opportunity of a secondeditiontoupdateandenlargethelistofreferencesregardingthe threemodels of cooperative games. Finally, we have bene?ted fromthis opportunity by removing typos and a few less important results from the ?rst edition of the book, and by slightly polishing the English style and the punctuation, for the sake of consistency along the monograph. The main changes are: (1) Chapter 3 contains an additional section, Section 3. 3, on the - erage lexicographic value, which is a recent one-point solution concept de?ned on the class of balanced crisp games. (2) Chapter 4 is new. It o?ers a brief overview on solution c- cepts for crisp games from the point of view of egalitarian criteria, and presents in Section 4. 2 a recent set-valued solution concept based on egalitarian considerations, namely the equal split-o? set. (3)Chapter5isbasicallyanenlargedversionofChapter4ofthe?rst edition because Section 5. 4 dealing with the relation between convex games and clan games with crisp coalitions is new.

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### Contents

 Preliminaries 5 Cores and Related Solution Concepts 13 22 The Core Cover the Reasonable Set and the Weber Set 20 The Shapley Value the τvalue and the Average Lexicographic Value 25 32 The τvalue 31 33 The Average Lexicographic Value 33 Egalitarianismbased Solution Concepts 36 42 The Equal SplitOff Set 38
 74 The Shapley Value and the Weber Set 94 75 Path Solutions and the Path Solution Cover 96 76 Compromise Values 100 Convex Fuzzy Games 103 82 Egalitarianism in Convex Fuzzy Games 110 83 Participation Monotonic Allocation Schemes 116 84 Properties of Solution Concepts 119 Fuzzy Clan Games 127

 421 The Equal SplitOff Set for General Games 39 422 The Equal SplitOff Set for Superadditive Games 41 Classes of Cooperative Crisp Games 43 512 Totally Balanced Games and Population Monotonic Allocation Schemes 45 52 Convex Games 46 522 Convex Games and Population Monotonic Allocation Schemes 49 523 The Constrained Egalitarian Solution for Convex Games 50 524 Properties of Solution Concepts 53 53 Clan Games 59 532 Total Clan Games and Monotonic Allocation Schemes 62 54 Convex Games versus Clan Games 65 541 Characterizations via Marginal Games 66 542 Dual Transformations 68 543 The Core versus the Weber Set 70 Preliminaries 77 Solution Concepts for Fuzzy Games 82 72 Cores and Stable Sets 85 73 Generalized Cores and Stable Sets 89
 92 Cores and Stable Sets for Fuzzy Clan Games 131 93 BiMonotonic Participation Allocation Rules 135 Preliminaries 142 Solution Concepts for MultiChoice Games 149 112 Marginal Vectors and the Weber Set 155 113 Shapleylike Values 159 114 The Equal SplitOff Set for MultiChoice Games 163 Classes of MultiChoice Games 165 122 Convex MultiChoice Games 170 1222 Monotonic Allocation Schemes 173 1223 The Constrained Egalitarian Solution 174 1224 Properties of Solution Concepts 180 123 MultiChoice Clan Games 182 1232 BiMonotonic Allocation Schemes 186 References 193 Index 200 Copyright

### Popular passages

Page vii - MP was supported by a Marie Curie Fellowship of the European Community Programme Improving the Human Research Potential and the Socio-economic Knowledge Base under contract number HPMT-CT-2001-00364 and by the grant GACR 201/03/0946.
Page 199 - R. (1988): Probabilistic values for games, in: Roth, AE (Ed.), The Shapley Value: Essays in Honour of Lloyd S. Shapley, Cambridge University Press, Cambridge, pp.
Page 4 - It is concerned primarily with "coalitions"— groups of players who coordinate their actions and perhaps even pool their winnings. A cooperative game can often be put into the form of a characteristic function, v(S), which expresses for each set of players 5 the amount they can get if they form a coalition excluding the other players. In an economic context, v(S) might represent the gross product...
Page 199 - S. and FAS Lipperts (1982): The hypercube and the core cover of n-person cooperative games, Cahiers du Centre d'Etudes de Researche Operationelle 24, 27-37.