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# The Stable Fixtures Problem with Payments

• Péter Biró
• Walter Kern
• Daniël Paulusma
• Péter Wojuteczky
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9224)

## Abstract

We generalize two well-known game-theoretic models by introducing multiple partners matching games, defined by a graph $$G=(N,E)$$, 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 $$M\subseteq E$$ 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 $$p: N \times N \rightarrow {\mathbb R}$$ with $$p(i,j)+p(j,i)=w(ij)$$ if $$ij\in M$$ and $$p(i,j)=p(j,i)=0$$ if $$ij\notin M$$. The pair (Mp) is called a solution. A pair of players ij with $$ij\in E\setminus M$$ blocks a solution (Mp) if ij 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 $$b\equiv 1$$ (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.

## Keywords

Stable Solution Cooperative Game Multiple Partner Grand Coalition Stable Match
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 2016

## Authors and Affiliations

• Péter Biró
• 1
• 2
• Walter Kern
• 3
• Daniël Paulusma
• 4
Email author
• Péter Wojuteczky
• 1
1. 1.Institute of EconomicsHungarian Academy of SciencesBudapestHungary
2. 2.Department of Operations Research and Actuarial SciencesCorvinus University of BudapestBudapestHungary
3. 3.Faculty of Electrical Engineering, Mathematics and Computer ScienceUniversity of TwenteEnschedeNetherlands
4. 4.School of Engineering and Computing SciencesDurham UniversityDurhamUK