Existence of Nash Equilibria on Integer Programming Games

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 223)


We aim to investigate a new class of games, where each player’s set of strategies is a union of polyhedra. These are called integer programming games. To motivate our work, we describe some practical examples suitable to be modeled under this paradigm. We analyze the problem of determining whether or not a Nash equilibria exists for an integer programming game, and demonstrate that it is complete for the second level of the polynomial hierarchy.


Integer programming games Nash equilibria Computational complexity 



Part of this work was performed while the first author was in the Faculty of Sciences University of Porto and INESC TEC. The first author thanks the support of Institute for data valorisation (IVADO), the Portuguese Foundation for Science and Technology (FCT) through a PhD grant number SFRH/BD/79201/2011 and the ERDF European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme within project POCI-01-0145-FEDER-006961, and National Funds through the FCT (Portuguese Foundation for Science and Technology) as part of project UID/EEA/50014/2013. We thank the referees for comments and questions that helped clarifying the presentation.


  1. 1.
    A. Caprara, M. Carvalho, A. Lodi, G.J. Woeginger, A study on the computational complexity of the bilevel knapsack problem. SIAM J. Optim. 24(2), 823–838 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    M. Carvalho, Computation of equilibria on integer programming games, Ph.D. thesis, Faculdade de Ciências da Universidade do Porto (2016)Google Scholar
  3. 3.
    M. Carvalho, A. Lodi, J.P. Pedroso, A. Viana, Nash equilibria in the two-player kidney exchange game. Math. Programm. 161(1–2), 1–29 (2016). ISSN 1436-4646MathSciNetzbMATHGoogle Scholar
  4. 4.
    X. Chen, X. Deng, Settling the complexity of two-player nash equilibrium. in 47th Annual IEEE Symposium on Foundations of Computer Science, 2006. FOCS ’06 (2006), pp. 261–272.
  5. 5.
    C. Daskalakis, P. Goldberg, C. Papadimitriou, The complexity of computing a nash equilibrium. SIAM J. Comput. 39(1), 195–259 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    D. Fudenberg, J. Tirole, Game Theory (MIT Press, Cambridge, 1991)zbMATHGoogle Scholar
  7. 7.
    S.A. Gabriel, S. Ahmad Siddiqui, A.J. Conejo, C. Ruiz, Solving discretely-constrained nash-cournot games with an application to power markets. Netw. Spat. Econ. 13(3), 307–326 (2013). ISSN 1566-113XMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    I.L. Glicksberg, A further generalization of the kakutani fixed point theorem, with application to nash equilibrium points. Proc. Am. Math. Soc. 3(1), 170–174 (1952). ISSN 00029939MathSciNetzbMATHGoogle Scholar
  9. 9.
    M. Köppe, C. Thomas Ryan, M. Queyranne, Rational generating functions and integer programming games. Op. Res. 59(6), 1445–1460 (2011). ISSN 0030-364XMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    M.M. Kostreva, Combinatorial optimization in nash games. Comput. Math. Appl. 25(10–11), 27–34 (1993), URL ISSN 0898-1221
  11. 11.
    H. Li, J. Meissner, Competition under capacitated dynamic lot-sizing with capacity acquisition. Int. J. Prod. Econ. 131(2), 535–544 (2011), URL ISSN 0925-5273.
  12. 12.
    R.D. Mckelvey, A.M. Mclennan, T.L. Turocy, Gambit: software tools for game theory. Version 16.0.0 (2016),
  13. 13.
    J. Nash, Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    N. Nisan, T. Roughgarden, E. Tardos, V.V. Vazirani, Algorithmic Game Theory (Cambridge University Press, New York, 2007)CrossRefzbMATHGoogle Scholar
  15. 15.
    C.H. Papadimitriou, Computational Complexity (Addison-Wesley, Reading, 1994). ISBN 0201530821zbMATHGoogle Scholar
  16. 16.
    M.V. Pereira, S. Granville, M.H.C. Fampa, R. Dix, L.A. Barroso, Strategic bidding under uncertainty: a binary expansion approach. IEEE Trans. Power Syst. 20(1), 180–188 (2005). ISSN 0885-8950CrossRefGoogle Scholar
  17. 17.
    Y. Pochet, L.A. Wolsey, Production Planning by Mixed Integer Programming, Springer Series in Operations Research and Financial Engineering (Springer, New York, 2006). ISBN 0387299599zbMATHGoogle Scholar
  18. 18.
    N.D. Stein, A. Ozdaglar, P.A. Parrilo, Separable and low-rank continuous games. Int. J. Game Theory 37(4), 475–504 (2008). ISSN 0020-7276MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    B. von Stengel, Special issue of on computation of nash equilibria in finite games. Econ. Theory 42(1), 1–7 (2010). MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.IVADO FellowCanada Excellence Research Chair, École Polytechnique de MontréalMontrealCanada
  2. 2.Canada Excellence Research ChairÉcole Polytechnique de MontréalMontrealCanada
  3. 3.Faculdade de Ciências Universidade do Porto and INESC TECPortoPortugal

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