Skip to main content
SpringerLink
Log in

Computing the nucleolus of weighted cooperative matching games in polynomial time

  • Full Length Paper
  • Series B
  • Published:
Mathematical Programming Submit manuscript

Abstract

We provide an efficient algorithm for computing the nucleolus for an instance of a weighted cooperative matching game. This resolves a long-standing open question posed in Faigle (Math Programm, 83: 555–569, 1998).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. It is common within the literature, for instance in [30], to exclude the coalitions for \(S = \varnothing \) and \(S = V\) in the definition of the nucleolus. On the other hand, one could also consider the definition of the nucleolus with all possible coalitions, including \(S = \varnothing \) and \(S = V\). We note that the two definitions of the nucleolus are equivalent in all instances of matching games except for the trivial instance of a graph consisting of two nodes joined by a single edge.

References

  1. Aumann, R.J., Maschler, M.: Game theoretic analysis of a bankruptcy problem from the talmud. J. Econ. Theory 36(2), 195–213 (1985)

    MathSciNet  MATH  Google Scholar 

  2. Bateni, M., Hajiaghayi, M., Immorlica, N., Mahini, H.: The cooperative game theory foundations of network bargaining games. In: International Colloquium on Automata, Languages, and Programming, pp. 67–78. Springer (2010)

  3. Biró, P., Kern, W., Paulusma, D.: Computing solutions for matching games. Int. J. Game Theory 41, 75–90 (2012)

    MathSciNet  MATH  Google Scholar 

  4. Biró, P., Kern, W., Paulusma, D., Wojuteczky, P.: The stable fixtures problem with payments. Games Econ. Behav. 11(9), 24–241 (2017)

    MATH  Google Scholar 

  5. Brânzei, R., Solymosi, T., Tijs, S.: Strongly essential coalitions and the nucleolus of peer group games. Int. J. Game Theory 33(3), 447–460 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Chen, N., Lu, P., Zhang, H.: Computing the nucleolus of matching, cover and clique games. In: AAAI (2012)

  7. Cook, K.S., Yamagishi, T.: Power in exchange networks: a power-dependence formulation. Soc. Netw. 14(3–4), 245–265 (1992)

    Google Scholar 

  8. Davis, M., Maschler, M.: The kernel of a cooperative game. Naval Res. Logist. Q. 12(3), 223–259 (1965)

    MathSciNet  MATH  Google Scholar 

  9. Deng, X., Fang, Q.: Algorithmic cooperative game theory. In: Pareto Optimality, Game Theory And Equilibria, pp. 159–185. Springer (2008)

  10. Deng, X., Fang, Q., Sun, X.: Finding nucleolus of flow game. J. Comb. Optim. 18(1), 64–86 (2009)

    MathSciNet  MATH  Google Scholar 

  11. Deng, X., Ibaraki, T., Nagamochi, H.: Algorithmic aspects of the core of combinatorial optimization games. Math. Oper. Res. 24(3), 751–766 (1999)

    MathSciNet  MATH  Google Scholar 

  12. Easley, D., Kleinberg, J.: Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  13. Edmonds, J.: Maximum matching and a polyhedron with 0, 1-vertices. J. Res. Natl. Bureau Stand. B 69(125–130), 55–56 (1965)

    MathSciNet  Google Scholar 

  14. Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17(3), 449–467 (1965)

    MathSciNet  MATH  Google Scholar 

  15. Elkind, E., Goldberg, L.A., Goldberg, P., Wooldridge, M.: Computational complexity of weighted threshold games. In: Proceedings of the National Conference on Artificial Intelligence, p. 718 (2007)

  16. Eriksson, K., Karlander, J.: Stable outcomes of the roommate game with transferable utility. Int. J. Game Theory 29(4), 555–569 (2001)

    MathSciNet  MATH  Google Scholar 

  17. Faigle, U., Kern, W., Fekete, S.P., Hochstättler, W.: The nucleon of cooperative games and an algorithm for matching games. Math. Program. 83(1–3), 195–211 (1998)

    MathSciNet  MATH  Google Scholar 

  18. Faigle, U., Kern, W., Kuipers, J.: Note computing the nucleolus of min-cost spanning tree games is np-hard. Int. J. Game Theory 27(3), 443–450 (1998)

    MATH  Google Scholar 

  19. Faigle, U., Kern, W., Kuipers, J.: On the computation of the nucleolus of a cooperative game. Int. J. Game Theory 30(1), 79–98 (2001)

    MathSciNet  MATH  Google Scholar 

  20. Faigle, U., Kern, W., Kuipers, J.: Computing an element in the lexicographic kernel of a game. Math. Methods Oper. Res. 63(3), 427–433 (2006)

    MathSciNet  MATH  Google Scholar 

  21. Faigle, U., Kern, W., Paulusma, D.: Note on the computational complexity of least core concepts for min-cost spanning tree games. Math. Methods Oper. Res. 52(1), 23–38 (2000)

    MathSciNet  MATH  Google Scholar 

  22. Farczadi, L.: Matchings and Games on Networks. University of Waterloo, Waterloo (2015)

    Google Scholar 

  23. Farczadi, L., Georgiou, K., Könemann, J.: Network bargaining with general capacities. In: European Symposium on Algorithms, pp. 433–444. Springer (2013)

  24. Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Monthly 69(1), 9–15 (1962)

    MathSciNet  MATH  Google Scholar 

  25. Gillies, D.B.: Solutions to general non-zero-sum games. Contrib. Theory Games 4(40), 47–85 (1959)

    MathSciNet  MATH  Google Scholar 

  26. Granot, D., Granot, F., Zhu, W.R.: Characterization sets for the nucleolus. Int. J. Game Theory 27(3), 359–374 (1998)

    MathSciNet  MATH  Google Scholar 

  27. Granot, D., Maschler, M., Owen, G., Zhu, W.R.: The kernel/nucleolus of a standard tree game. Int. J. Game Theory 25(2), 219–244 (1996)

    MathSciNet  MATH  Google Scholar 

  28. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, Algorithms and Combinatorics: Study and Research Texts, vol. 2. Springer, Berlin (1988)

    MATH  Google Scholar 

  29. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, vol. 2. Springer, Berlin (2012)

    MATH  Google Scholar 

  30. Kern, W., Paulusma, D.: Matching games: the least core and the nucleolus. Math. Oper. Res. 28(2), 294–308 (2003)

    MathSciNet  MATH  Google Scholar 

  31. Kleinberg, J., Tardos, E.: Balanced outcomes in social exchange networks. In: Proceedings of the fourtieth annual ACM symposium on Theory of computing-STOC 08, p. 295. New York, New York, USA (2008)

  32. Koopmans, T.C., Beckmann, M.: Assignment problems and the location of economic activities. In: Econometrica: journal of the Econometric Society, pp. 53–76 (1957)

  33. Kopelowitz, A.: Computation of the kernels of simple games and the nucleolus of n-person games. Tech. rep., Hebrew University Jerusalem (Israel) Department of Mathematics (1967)

  34. Kuhn, H.W.: The hungarian method for the assignment problem. Naval Res. Logist. Q. 2(1–2), 83–97 (1955)

    MathSciNet  MATH  Google Scholar 

  35. Kuipers, J., Solymosi, T., Aarts, H.: Computing the nucleolus of some combinatorially-structured games. Math. Program. 88(3), 541–563 (2000)

    MathSciNet  MATH  Google Scholar 

  36. Lau, L.C., Ravi, R., Singh, M.: Iterative Methods in Combinatorial Optimization, vol. 46. Cambridge University Press, Cambridge (2011)

    MATH  Google Scholar 

  37. Lemaire, J.: An application of game theory: cost allocation. ASTIN Bull. J. IAA 14(1), 61–81 (1984)

    Google Scholar 

  38. Maschler, M., Peleg, B., Shapley, L.S.: Geometric properties of the kernel, nucleolus, and related solution concepts. Math. Oper. Res. 4(4), 303–338 (1979)

    MathSciNet  MATH  Google Scholar 

  39. Megiddo, N.: Computational complexity of the game theory approach to cost allocation for a tree. Math. Oper. Res. 3(3), 189–196 (1978)

    MathSciNet  MATH  Google Scholar 

  40. Nash Jr., J.F.: The bargaining problem. Econ. J. Econ. Soc. 10, 155–162 (1950)

    MathSciNet  MATH  Google Scholar 

  41. Paulusma, D.: Complexity Aspects of Cooperative Games. Twente University Press, New York (2001)

    Google Scholar 

  42. Potters, J., Reijnierse, H., Biswas, A.: The nucleolus of balanced simple flow networks. Games Econ. Behav. 54(1), 205–225 (2006)

    MathSciNet  MATH  Google Scholar 

  43. Rothvoß, T.: The matching polytope has exponential extension complexity. J. ACM 64(6), 41 (2017)

    MathSciNet  MATH  Google Scholar 

  44. Schmeidler, D.: The nucleolus of a characteristic function game. SIAM J. Appl. Math. 17(6), 1163–1170 (1969)

    MathSciNet  MATH  Google Scholar 

  45. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, vol. 24. Springer, Berlin (2002)

    MATH  Google Scholar 

  46. Shapley, L.S., Shubik, M.: The assignment game I: the core. Int. J. Game Theory 1(1), 111–130 (1971)

    MathSciNet  MATH  Google Scholar 

  47. Solymosi, T., Raghavan, T.E.: An algorithm for finding the nucleolus of assignment games. Int. J. Game Theory 23(2), 119–143 (1994)

    MathSciNet  MATH  Google Scholar 

  48. Stearns, R.E.: Convergent transfer schemes for n-person games. Trans. Am. Math. Soc. 134(3), 449–459 (1968)

    MathSciNet  MATH  Google Scholar 

  49. Willer, D.: Network Exchange Theory. Greenwood Publishing Group, Greenwood (1999)

    Google Scholar 

  50. Ziegler, G.M.: Lectures on Polytopes, vol. 152. Springer, Berlin (2012)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors thank Umang Bhaskar, Daniel Dadush, and Linda Farczadi for stimulating and insightful discussions related to this paper.

Author information

Authors and Affiliations

  1. University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Jochen Könemann & Justin Toth

  2. University of Ottawa, Ottawa, ON, K1N 6N5, Canada

    Kanstantsin Pashkovich

Authors
  1. Jochen Könemann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kanstantsin Pashkovich
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Justin Toth
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Justin Toth.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was done in part while the second author was visiting the Simons Institute for the Theory of Computing. Supported by DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF grant #CCF-1740425.

We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Könemann, J., Pashkovich, K. & Toth, J. Computing the nucleolus of weighted cooperative matching games in polynomial time. Math. Program. 183, 555–581 (2020). https://doi.org/10.1007/s10107-020-01483-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-020-01483-4

Keywords

Mathematics Subject Classification

Search

Navigation