## Models in Cooperative Game Theory (Google eBook)This book investigates models in cooperative game theory in which the players have the possibility to cooperate partially. In a crisp game the agents are either fully involved or not involved at all in cooperation with some other agents, while in a fuzzy game players are allowed to cooperate with infinite many different participation levels, varying from non-cooperation to full cooperation. A multi-choice game describes the intermediate case in which each player may have a fixed number of activity levels. Different set and one-point solution concepts for these games are presented. The properties of these solution concepts and their interrelations on several classes of crisp, fuzzy, and multi-choice games are studied. Applications of the investigated models to many economic situations are indicated as well. The second edition is highly enlarged and contains new results and additional sections in the different chapters as well as one new chapter. |

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### Contents

5 | |

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 |

200 | |

### Common terms and phrases

a e I(v Aubin core bi-monotonic CFGN characteristic function clan members constrained egalitarian solution convex combination convex fuzzy games convex games convex set cooperative crisp games cooperative games core element Cores and Stable corresponding cr(v crisp coalition DC(v defined Definition denote Div(e EDC(v equal split-off set Example fuzzy clan game fuzzy coalitions Game Theory game v e game ve G grand coalition Hence hypercube i e car(s i e N implies inequality follows introduced Lemma Let a e Let v e G marginal contribution marginal vectors MCN,m Monotonic Allocation Schemes non-clan members non-empty Note obtain ordered partition p-core participation level payoff vector player set pmas Proof Proposition prove satisfies Shapley value solution concepts stable set subgame superadditive supermodularity Suppose Theorem total clan games unanimity game union property Weber set XDeN XDies zero-normalized

### 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.

### References to this book

The Social Science Jargon Buster: The Key Terms You Need to Know Zina O'Leary Limited preview - 2007 |