Biró, P. and Kern, W. and Paulusma, D. and Wojuteczky, P. (2016) 'The stable fixtures problem with payments.', in *Graph-theoretic concepts in computer science : 41st international workshop, WG 2015, Garching, Germany, June 17-19, 2015 ; revised papers.* Berlin: Springer, pp. 49-63. Lecture notes in computer science. (9224).

## Abstract

We generalize two well-known game-theoretic models by introducing multiple partners matching games, defined by a graph G=(N,E)G=(N,E), with an integer vertex capacity function b and an edge weighting w. The set N consists of a number of players that are to form a set M⊆EM⊆E of 2-player coalitions ij with value w(ij), such that each player i is in at most b(i) coalitions. A payoff is a mapping p:N×N→Rp:N×N→R with p(i,j)+p(j,i)=w(ij)p(i,j)+p(j,i)=w(ij) if ij∈Mij∈M and p(i,j)=p(j,i)=0p(i,j)=p(j,i)=0 if ij∉Mij∉M. The pair (M, p) is called a solution. A pair of players i, j with ij∈E∖Mij∈E∖M blocks a solution (M, p) if i, j can form, possibly only after withdrawing from one of their existing 2-player coalitions, a new 2-player coalition in which they are mutually better off. A solution is stable if it has no blocking pairs. We give a polynomial-time algorithm that either finds that no stable solution exists, or obtains a stable solution. Previously this result was only known for multiple partners assignment games, which correspond to the case where G is bipartite (Sotomayor 1992) and for the case where b≡1b≡1 (Biro et al. 2012). We also characterize the set of stable solutions of a multiple partners matching game in two different ways and initiate a study on the core of the corresponding cooperative game, where coalitions of any size may be formed.

Item Type: | Book chapter |
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Full text: | (AM) Accepted Manuscript Download PDF (317Kb) |

Status: | Peer-reviewed |

Publisher Web site: | http://dx.doi.org/10.1007/978-3-662-53174-7_4 |

Publisher statement: | The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-662-53174-7_4 |

Date accepted: | 12 August 2015 |

Date deposited: | 21 August 2015 |

Date of first online publication: | 05 August 2016 |

Date first made open access: | 05 August 2017 |

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