## Abstract

One key question in cooperative game theory is that of coalitional stability. A coalition in such games is stable when no subset of the agents in it has a rational incentive to leave the coalition. Finding a division of the gains of the coalition (an imputation) lies at the heart of many cooperative game theory solution concepts, the most prominent of which is the core. However, some coalitional games have empty cores, and any imputation in such a game is unstable. We investigate the possibility of stabilizing the coalitional structure using external payments. In this case, a supplemental payment is offered to the grand coalition by an external party which is interested in having the members of the coalition work together. The sum of this payment plus the gains of the coalition, called the coalition's "adjusted gains", may be divided among the members of the coalition in a stable manner. We call a division of the adjusted gains a super- imputation, and define the cost of stability (CoS) as the minimal sum of payments that stabilizes the coalition. We examine the cost of stability in weighted voting games, where each agent has a weight, and a coalition is successful if the sum of its weights exceeds a given threshold. Such games offer a simple model of decision making in political bodies, and of cooperation in multiagent settings. We show that it is coNP-complcte to test whether a super-imputation is stable, but show that if either the weights or payments of agents are bounded then there exists a polynomial algorithm for this problem. We provide a polynomial approximation algorithm for computing the cost of stability.

Original language | American English |
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Title of host publication | 8th International Joint Conference on Autonomous Agents and Multiagent Systems 2009, AAMAS 2009 |

Publisher | International Foundation for Autonomous Agents and Multiagent Systems (IFAAMAS) |

Pages | 1092-1093 |

Number of pages | 2 |

ISBN (Print) | 9781615673346 |

State | Published - 2009 |

Event | 8th International Joint Conference on Autonomous Agents and Multiagent Systems 2009, AAMAS 2009 - Budapest, Hungary Duration: 10 May 2009 → 15 May 2009 |

### Publication series

Name | Proceedings of the International Joint Conference on Autonomous Agents and Multiagent Systems, AAMAS |
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Volume | 2 |

ISSN (Print) | 1548-8403 |

ISSN (Electronic) | 1558-2914 |

### Conference

Conference | 8th International Joint Conference on Autonomous Agents and Multiagent Systems 2009, AAMAS 2009 |
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Country/Territory | Hungary |

City | Budapest |

Period | 10/05/09 → 15/05/09 |

## Keywords

- Coalition formation
- Core
- Weighted voting games