, Volume 77, Issue 3, pp 836–866 | Cite as

The Price of Optimum: Complexity and Approximation for a Matching Game

  • Bruno Escoffier
  • Laurent GourvèsEmail author
  • Jérôme Monnot


This paper deals with a matching game in which the nodes of a simple graph are independent agents who try to form pairs. If we let the agents make their decision without any central control then a possible outcome is a Nash equilibrium, that is a situation in which no unmatched player can change his strategy and find a partner. However, there can be a big difference between two possible outcomes of the same instance, in terms of number of matched nodes. A possible solution is to force all the nodes to follow a centrally computed maximum matching but it can be difficult to implement this approach. This article proposes a tradeoff between the total absence and the full presence of a central control. Concretely, we study the optimization problem where the action of a minimum number of agents is centrally fixed and any possible equilibrium of the modified game must be a maximum matching. In algorithmic game theory, this approach is known as the price of optimum of a game. For the price of optimum of the matching game, deciding whether a solution is feasible is not straightforward, but we prove that it can be done in polynomial time. In addition, the problem is shown APX-hard, since its restriction to graphs admitting a perfect matching is equivalent, from the approximability point of view, to vertex cover. Finally we prove that this problem admits a polynomial 6-approximation algorithm in general graphs.


Approximation algorithm Srategic game Complexity Price of Anarchy Price of optimum Stackelberg strategy 



We would like to thank the anonymous reviewers for their valuable comments and suggestions.


  1. 1.
    Roughgarden, T., Tardos, É.: How bad is selfish routing? J. ACM 49, 236–259 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Koutsoupias, E., Papadimitriou, C.H.: Worst-case equilibria. Comput. Sci. Rev. 3, 65–69 (2009)CrossRefzbMATHGoogle Scholar
  3. 3.
    Papadimitriou, C.H.: Algorithms, games, and the internet. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, STOC ‘01, pp. 749–753. ACM, New York, NY, USA (2001). doi: 10.1145/380752.380883
  4. 4.
    Christodoulou, G., Koutsoupias, E., Nanavati, A.: Coordination mechanisms. Theory Comput. Sci. 410, 3327–3336 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kaporis, A.C., Spirakis, P.G.: Stackelberg games: the price of optimum. In: Kao, M.Y. (ed.) Encyclopedia of Algorithms. Springer, Berlin (2008)Google Scholar
  6. 6.
    Korilis, Y.A., Lazar, A.A., Orda, A.: Achieving network optima using stackelberg routing strategies. IEEE/ACM Trans. Netw. 5, 161–173 (1997)CrossRefGoogle Scholar
  7. 7.
    Roughgarden, T.: Stackelberg scheduling strategies. SIAM J. Comput. 33, 332–350 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kumar, V.S.A., Marathe, M.V.: Improved results for stackelberg scheduling strategies. In: Widmayer, P., Ruiz, F.T., Bueno, R.M., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) Lecture Notes in Computer Science. Springer, Berlin (2002)Google Scholar
  9. 9.
    Swamy, C.: The effectiveness of stackelberg strategies and tolls for network congestion games. In: Bansal, N., Pruhs, K., Stein, C. (eds.) SODA, pp. 1133–1142. SIAM, Philadelphia (2007)Google Scholar
  10. 10.
    Fotakis, D.: Stackelberg strategies for atomic congestion games. Theory Comput. Syst. 47, 218–249 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Sharma, Y., Williamson, D.P.: Stackelberg thresholds in network routing games or the value of altruism. Games Econ. Behav. 67, 174–190 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bonifaci, V., Harks, T., Schäfer, G.: Stackelberg routing in arbitrary networks. Math. Oper. Res. 35, 330–346 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nash, J.: Non-cooperative games. Ann. Math. 54, 286–295 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Aumann, R.J.: Acceptable points in general cooperative n-person games. Contrib. Theory Games 4, 287–324 (1959)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Andelman, N., Feldman, M., Mansour, Y.: Strong price of anarchy. Games Econ. Behav. 65, 289–317 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Manoussakis, Y.: Alternating paths in edge-colored complete graphs. Discret. Appl. Math. 56, 297–309 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hochbaum, D.S.: Approximation Algorithms for NP-Hard Problems. PWS Publishing Company, Boston, MA (1996)zbMATHGoogle Scholar
  18. 18.
    Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Ann. Math. 162, 439–485 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2-epsilon. J. Comput. Syst. Sci. 74, 335–349 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Alimonti, P., Kann, V.: Some apx-completeness results for cubic graphs. Theory Comput. Sci. 237, 123–134 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Garey, M., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, W. H (1979)zbMATHGoogle Scholar
  22. 22.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69, 9–15 (1962)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Bruno Escoffier
    • 1
  • Laurent Gourvès
    • 2
    • 3
    Email author
  • Jérôme Monnot
    • 2
    • 3
  1. 1.Sorbonne Universités, UPMC Université Paris 06, CNRS, LIP6 UMR 7606ParisFrance
  2. 2.CNRS, UMR 7243ParisFrance
  3. 3.LamsadeUniversité de Paris-DauphineParisFrance

Personalised recommendations