Mathematical Programming

, Volume 161, Issue 1–2, pp 389–417 | Cite as

Nash equilibria in the two-player kidney exchange game

  • Margarida Carvalho
  • Andrea Lodi
  • João Pedro Pedroso
  • Ana Viana
Full Length Paper Series A

Abstract

Kidney exchange programs have been set in several countries within national, regional or hospital frameworks, to increase the possibility of kidney patients being transplanted. For the case of hospital programs, it has been claimed that hospitals would benefit if they collaborated with each other, sharing their internal pools and allowing transplants involving patients of different hospitals. This claim led to the study of multi-hospital exchange markets. We propose a novel direction in this setting by modeling the exchange market as an integer programming game. The analysis of the strategic behavior of the entities participating in the kidney exchange game allowed us to prove that the most rational game outcome maximizes the social welfare and that it can be computed in polynomial time.

Keywords

Kidney exchange Nash equilibrium Social welfare  Matching 

Mathematics Subject Classification

91A80 05C85 90C27 

References

  1. 1.
    Abraham, D.J., Blum, A., Sandholm, T.: Clearing algorithms for barter exchange markets: enabling nationwide kidney exchanges. In: Proceedings of the 8th ACM Conference on Electronic Commerce EC ’07, pp. 295–304, ACM, New York, NY (2007)Google Scholar
  2. 2.
    Ashlagi, I., Fischer, F., Kash, I.A., Procaccia, A.D.: Mix and match: A strategyproof mechanism for multi-hospital kidney exchange. Games and Economic Behavior 91, 284–296 (2015)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Ashlagi, I., Roth, A.: Individual rationality and participation in large scale, multi-hospital kidney exchange. In: Proceedings of the 12th ACM Conference on Electronic Commerce, EC ’11, pp. 321–322, ACM, New York, NY (2011)Google Scholar
  4. 4.
    Ashlagi, I., Roth, A.: Individual rationality and participation in large scale, multi-hospital kidney exchange. Working Paper, http://web.mit.edu/iashlagi/www/papers/LargeScaleKidneyExchange_1_13 (2011)
  5. 5.
    Berge, C.: Two theorems in graph theory. Proc. Nat. Acad. Sci. 43(9), 842–844 (1957)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. Elsevier Science Publishing Co., Inc., Amsterdam (1976)CrossRefMATHGoogle Scholar
  7. 7.
    Caragiannis, I., Filos-Ratsikas, A., Procaccia, A.D.: An improved 2-agent kidney exchange mechanism. In: Chen, N., Elkind, E., Koutsoupias, E. (eds.) Internet and Network Economics, Vol. 7090 of Lecture Notes in Computer Science. Springer, Berlin, pp. 37–48 (2011)Google Scholar
  8. 8.
    Cechlárová, K., Fleiner, T., Manlove, D.: The kidney exchange game. In: Proceedings of the 8th International Symposium on Operational Research SOR, 5, pp. 77–83 (2005)Google Scholar
  9. 9.
    Constantino, M., Klimentova, X., Viana, A., Rais, A.: New insights on integer-programming models for the kidney exchange problem. Eur. J. Oper. Res. 231(1), 57–68 (2013)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    de Klerk, M., Keizer, K.M., Claas, F.H., Haase-Kromwijk, B.J.J.M., Weimar, W.: The Dutch national living donor kidney exchange program. Am. J. Transpl. 5, 2302–2305 (2005)CrossRefGoogle Scholar
  11. 11.
    Dickerson, J.P., Procaccia, A.D., Sandholm, T.: Price of Fairness in Kidney Exchange. In: Proceedings of the 2014 International Conference on Autonomous Agents and Multi-agent Systems AAMAS ’14, pp. 1013–1020 (2014)Google Scholar
  12. 12.
    Edmonds, J.: Paths, trees, and flowers. Canad. J. Math. 17, 449–467 (1965)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Fudenberg, D., Tirole, J.: Game Theory, 5th edn. MIT Press, Cambridge, MA (1996)MATHGoogle Scholar
  14. 14.
    Hajaj, C., Dickerson, J.P., Hassidim, A., Sandholm, T., Sarne, D.: Strategy-proof and efficient kidney exchange using a credit mechanism. In: Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, January 25–30, pp. 921–928, Austin, TX (2015)Google Scholar
  15. 15.
    Köppe, M., Ryan, C.T., Queyranne, M.: Rational generating functions and integer programming games. Oper. Res. 59(6), 1445–1460 (2011)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Manlove, D.F., O’Malley, G.: Paired and altruistic kidney donation in the UK: algorithms and experimentation. In: Klasing, R., (ed.), Experimental Algorithms, Vol. 7276 of Lecture Notes in Computer Science. Springer, Berlin pp. 271–282 (2012)Google Scholar
  17. 17.
    Monderer, D., Shapley, L.S.: Potential games. Games Econ. Behav. 14(1), 124–143 (1996)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Nash, J.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V.: Algorithmic Game Theory. Cambridge University Press, New York, NY (2007)CrossRefMATHGoogle Scholar
  20. 20.
    Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall Inc, Upper Saddle River, NJ (1982)MATHGoogle Scholar
  21. 21.
    Scalzo, V.: Pareto efficient Nash equilibria in discontinuous games. Econ. Lett. 107(3), 364–365 (2010)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  • Margarida Carvalho
    • 1
  • Andrea Lodi
    • 2
    • 3
  • João Pedro Pedroso
    • 1
  • Ana Viana
    • 4
  1. 1.INESC TEC and Faculdade de Ciências daUniversidade do PortoPortoPortugal
  2. 2.University of BolognaBolognaItaly
  3. 3.École Polytechnique de MontréalMontrealCanada
  4. 4.INESC TEC and Instituto Superior de Engenharia do PortoPortoPortugal

Personalised recommendations