The Stable Fixtures Problem with Payments

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


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.


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.


  1. 1.
    Baïou, M., Balinski, M.: Many-to-many matching: stable polyandrous polygamy (or polygamous polyandry). Discrete Appl. Math. 101, 1–12 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bellman, R.: On a routing problem. Q. Appl. Math. 16, 87–90 (1958)zbMATHGoogle Scholar
  3. 3.
    Biró, P., Bomhoff, M., Golovach, P.A., Kern, W., Paulusma, D.: Solutions for the stable roommates problem with payments. Theoret. Comput. Sci. 540–541, 53–61 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Biró, P., Kern, W., Paulusma, D.: Computing solutions for matching games. Int. J. Game Theory 41, 75–90 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cechlárová, K., Fleiner, T.: On a generalization of the stable roommates problem. ACM Trans. Algorithms 1, 143–156 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chalkiadakis, G., Elkind, E., Markakis, E., Polurkov, M., Jennings, N.R.: Cooperative games with overlapping coalitions. J. Artif. Intell. Res. 39, 179–216 (2010)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Deng, X., Ibaraki, T., Nagamochi, H.: Algorithmic aspects of the core of combinatorial optimization games. Math. Oper. Res. 24, 751–766 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Eriksson, K., Karlander, J.: Stable outcomes of the roommate game with transferable utility. Int. J. Game Theory 29, 555–569 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69, 9–15 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Garey, M., Johnson, D.: Computers and Intractability. A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  11. 11.
    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method, its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981). [corrigendum: Combinatorica 4, 291–295 (1984)]MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, 2nd edn. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  13. 13.
    Irving, R.W.: An efficient algorithm for the stable roommates problem. J. Algorithms 6, 577–595 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Irving, R.W., Scott, S.: The stable fixtures problem - a many-to-many extension of stable roommates. Discrete Appl. Math. 155, 2118–2129 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kern, W., Paulusma, D.: The new FIFA rules are hard: complexity aspects of sport competitions. Discrete Appl. Math. 108, 317–323 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Khachiyan, L.G.: A polynomial algorithm in linear programming. Soviet Math. Dokl. 20, 191–194 (1979)zbMATHGoogle Scholar
  17. 17.
    Lawler, E.: Combinatorial Optimization. Courier Dover Publ., New York (1976)zbMATHGoogle Scholar
  18. 18.
    Koopmans, T.C., Beckmann, M.: Assignment problems and the location of economic activities. Econometrica 25, 53–76 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Letchford, A.N., Reinelt, G., Theis, D.O.: Odd minimum cut sets and \(b\)-matchings revisited. SIAM J. Discrete Math. 22, 1480–1487 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Manlove, D.: Algorithmics of Matching Under Preferences, Series on Theoretical Computer Science, vol. 2. World Scientific (2013)Google Scholar
  21. 21.
    Megiddo, N.: Combinatorial optimization with rational objective functions. Math. OR 4, 414–424 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Roth, A.E., Sotomayor, M.: Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle Scholar
  23. 23.
    Shapley, L.S., Shubik, M.: The assignment game I: the core. Int. J. Game Theory 1, 111–130 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sotomayor, M.: The multiple partners game. In: Equilibrium and Dynamics: Essays in Honor of David Gale. Macmillan Press Ltd, New York (1992)Google Scholar
  25. 25.
    Tutte, W.T.: A short proof of the factor factor theorem for finite graphs. Can. J. Math. 6, 347–352 (1954)MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations