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Nondominated Nash points: application of biobjective mixed integer programming

Author

Listed:
  • Hadi Charkhgard

    (University of South Florida)

  • Martin Savelsbergh

    (Georgia Institute of Technology)

  • Masoud Talebian

    (Sharif University of Technology)

Abstract

We study the connection between biobjective mixed integer linear programming and normal form games with two players. We first investigate computing Nash equilibria of normal form games with two players using single-objective mixed integer linear programming. Then, we define the concept of efficient (Pareto optimal) Nash equilibria. This concept is precisely equivalent to the concept of efficient solutions in multi-objective optimization, where the solutions are Nash equilibria. We prove that the set of all points in the payoff (or objective) space of a normal form game with two players corresponding to the utilities of players in an efficient Nash equilibrium, the so-called nondominated Nash points, is finite. We demonstrate that biobjective mixed integer linear programming, where the utility of each player is an objective function, can be used to compute the set of nondominated Nash points. Finally, we illustrate how the nondominated Nash points can be used to determine the disagreement point of a bargaining problem.

Suggested Citation

  • Hadi Charkhgard & Martin Savelsbergh & Masoud Talebian, 2018. "Nondominated Nash points: application of biobjective mixed integer programming," 4OR, Springer, vol. 16(2), pages 151-171, June.
    • Handle: RePEc:spr:aqjoor:v:16:y:2018:i:2:d:10.1007_s10288-017-0354-2
    DOI: 10.1007/s10288-017-0354-2
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    1. Chalmet, L. G. & Lemonidis, L. & Elzinga, D. J., 1986. "An algorithm for the bi-criterion integer programming problem," European Journal of Operational Research, Elsevier, vol. 25(2), pages 292-300, May.
    2. John C. Harsanyi & Reinhard Selten, 1988. "A General Theory of Equilibrium Selection in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262582384, December.
    3. Nash, John, 1953. "Two-Person Cooperative Games," Econometrica, Econometric Society, vol. 21(1), pages 128-140, April.
    4. Chun, Youngsub & Thomson, William, 1990. "Nash solution and uncertain disagreement points," Games and Economic Behavior, Elsevier, vol. 2(3), pages 213-223, September.
    5. Porter, Ryan & Nudelman, Eugene & Shoham, Yoav, 2008. "Simple search methods for finding a Nash equilibrium," Games and Economic Behavior, Elsevier, vol. 63(2), pages 642-662, July.
    6. Matthias Ehrgott, 2006. "A discussion of scalarization techniques for multiple objective integer programming," Annals of Operations Research, Springer, vol. 147(1), pages 343-360, October.
    7. Serpil Say{i}n & Panos Kouvelis, 2005. "The Multiobjective Discrete Optimization Problem: A Weighted Min-Max Two-Stage Optimization Approach and a Bicriteria Algorithm," Management Science, INFORMS, vol. 51(10), pages 1572-1581, October.
    8. Nash, John, 1950. "The Bargaining Problem," Econometrica, Econometric Society, vol. 18(2), pages 155-162, April.
    9. John Conley & Bhaskar Chakravorti, 2004. "Bargaining efficiency and the repeated prisoners' dilemma," Economics Bulletin, AccessEcon, vol. 3(3), pages 1-8.
    10. Ted Ralphs & Matthew Saltzman & Margaret Wiecek, 2006. "An improved algorithm for solving biobjective integer programs," Annals of Operations Research, Springer, vol. 147(1), pages 43-70, October.
    11. Mark Voorneveld & Sofia Grahn & Martin Dufwenberg, 2000. "Ideal equilibria in noncooperative multicriteria games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 52(1), pages 65-77, September.
    12. David Avis & Gabriel Rosenberg & Rahul Savani & Bernhard Stengel, 2010. "Enumeration of Nash equilibria for two-player games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 42(1), pages 9-37, January.
    13. I. Nishizaki & T. Notsu, 2007. "Nondominated Equilibrium Solutions of a Multiobjective Two-Person Nonzero-Sum Game and Corresponding Mathematical Programming Problem," Journal of Optimization Theory and Applications, Springer, vol. 135(2), pages 217-239, November.
    14. Natashia Boland & Hadi Charkhgard & Martin Savelsbergh, 2015. "A Criterion Space Search Algorithm for Biobjective Integer Programming: The Balanced Box Method," INFORMS Journal on Computing, INFORMS, vol. 27(4), pages 735-754, November.
    15. Kirlik, Gokhan & Sayın, Serpil, 2014. "A new algorithm for generating all nondominated solutions of multiobjective discrete optimization problems," European Journal of Operational Research, Elsevier, vol. 232(3), pages 479-488.
    16. Banu Lokman & Murat Köksalan, 2013. "Finding all nondominated points of multi-objective integer programs," Journal of Global Optimization, Springer, vol. 57(2), pages 347-365, October.
    17. Ehrgott, Matthias & Wang, Judith Y.T. & Watling, David P., 2015. "On multi-objective stochastic user equilibrium," Transportation Research Part B: Methodological, Elsevier, vol. 81(P3), pages 704-717.
    18. Natashia Boland & Hadi Charkhgard & Martin Savelsbergh, 2015. "A Criterion Space Search Algorithm for Biobjective Mixed Integer Programming: The Triangle Splitting Method," INFORMS Journal on Computing, INFORMS, vol. 27(4), pages 597-618, November.
    19. Wang, Judith Y.T. & Ehrgott, Matthias, 2013. "Modelling route choice behaviour in a tolled road network with a time surplus maximisation bi-objective user equilibrium model," Transportation Research Part B: Methodological, Elsevier, vol. 57(C), pages 342-360.
    20. McKelvey, Richard D. & McLennan, Andrew, 1996. "Computation of equilibria in finite games," Handbook of Computational Economics, in: H. M. Amman & D. A. Kendrick & J. Rust (ed.), Handbook of Computational Economics, edition 1, volume 1, chapter 2, pages 87-142, Elsevier.
    21. repec:ebl:ecbull:v:3:y:2004:i:3:p:1-8 is not listed on IDEAS
    22. Kohlberg, Elon & Mertens, Jean-Francois, 1986. "On the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 54(5), pages 1003-1037, September.
    23. Todd R. Kaplan & John Dickhaut, "undated". "A Program for Finding Nash Equilibria," Working papers _004, University of Minnesota, Department of Economics.
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