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Solving Chance-Constrained Games Using Complementarity Problems

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Book cover Operations Research and Enterprise Systems (ICORES 2016)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 695))

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Abstract

In this paper, we formulate the random bimatrix game as a chance-constrained game using chance constraint. We show that a Nash equilibrium problem, corresponding to independent normally distributed payoffs, is equivalent to a nonlinear complementarity problem. Further if the payoffs are also identically distributed, a strategy pair where each player’s strategy is the uniform distribution over his action set, is a Nash equilibrium. We show that a Nash equilibrium problem corresponding to independent Cauchy distributed payoffs, is equivalent to a linear complementarity problem.

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References

  1. Bazaraa, M., Sherali, H., Shetty, C.: Nonlinear Programming Theory and Algorithms, 3rd edn. Wiley, New York (2006)

    Book  MATH  Google Scholar 

  2. Blau, R.A.: Random-payoff two person zero-sum games. Oper. Res. 22(6), 1243–1251 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cassidy, R.G., Field, C.A., Kirby, M.J.L.: Solution of a satisficing model for random payoff games. Manage. Sci. 19(3), 266–271 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  4. Charnes, A., Cooper, W.W.: Deterministic equivalents for optimizing and satisficing under chance constraints. Oper. Res. 11(1), 18–39 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  5. Charnes, A., Kirby, M.J.L., Raike, W.M.: Zero-zero chance-constrained games. Theory Probab. Appl. 13(4), 628–646 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheng, J., Lisser, A.: A second-order cone programming approach for linear programs with joint probabilistic constraints. Oper. Res. Lett. 40(5), 325–328 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Collins, W.D., Hu, C.: Studying interval valued matrix games with fuzzy logic. Soft. Comput. 12, 147–155 (2008)

    Article  MATH  Google Scholar 

  8. Couchman, P., Kouvaritakis, B., Cannon, M., Prashad, F.: Gaming strategy for electric power with random demand. IEEE Trans. Power Syst. 20(3), 1283–1292 (2005)

    Article  Google Scholar 

  9. DeMiguel, V., Xu, H.: A stochastic multiple leader stackelberg model: analysis, computation, and application. Oper. Res. 57(5), 1220–1235 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, D.-F., Nan, J.-X., Zhang, M.J.: Interval programming models for matrix games with interval payoffs. Optim. Meth. Softw. 27(1), 1–16 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ferris, M.C., Munson, T.S.: Complementarity problems in GAMS and the PATH solver. J. Econ. Dyn. Control 24, 165–188 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jadamba, B., Raciti, F.: Variational inequality approach to stochastic Nash equilibrium problems with an application to cournot oligopoly. J. Optim. Theor. Appl. 165(3), 1050–1070 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions, vol. 1, 2nd edn. Wiley, New York (1994)

    MATH  Google Scholar 

  14. Katzwer, R.: Lemke-Howson Algorithm for 2-Player Games (2013). File ID: #44279 Version: 1.3

    Google Scholar 

  15. Lemke, C.E.: Bimatrix equilibrium points and mathematical programming. Manage. Sci. 11(7), 681–689 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lemke, C., Howson, J.: Equilibrium points of bimatrix games. SIAM J. 12, 413–423 (1964)

    MathSciNet  MATH  Google Scholar 

  17. Li, D.F.: Linear programming approach to solve interval-valued matrix games. J. Omega 39(6), 655–666 (2011)

    Article  Google Scholar 

  18. Mazadi, M., Rosehart, W.D., Zareipour, H., Malik, O.P., Oloomi, M.: Impact of wind integration on electricity markets: a chance-constrained Nash Cournot model. Int. Trans. Electr. Energy Syst. 23(1), 83–96 (2013)

    Article  Google Scholar 

  19. Mitchell, C., Hu, C., Chen, B., Nooner, M., Young, P.: A computational study of interval-valued matrix games. In: International Conference on Computational Science and Computational Intelligence (2014)

    Google Scholar 

  20. Munson, T.S.: Algorithms and environments for complementarity. Ph.D. thesis, University of Wisconsin, Madison (2000)

    Google Scholar 

  21. Nash, J.F.: Equilibrium points in n-person games. Proc. Natl. Acad. Sci. 36(1), 48–49 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  22. Neumann, J.V.: Zur theorie der gesellschaftsspiele. Math. Annalen 100(1), 295–320 (1928)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  24. Prékopa, A.: Stochastic Programming. Springer, Netherlands (1995)

    Book  MATH  Google Scholar 

  25. Ravat, U., Shanbhag, U.V.: On The characterization of solution sets of smooth and nonsmooth convex stochastic Nash games. SIAM J. Optim. 21(3), 1168–1199 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Schmelzer, S.: COMPASS: a free solver for mixed complementarity problems. Master’s thesis, Universität Wien (2012)

    Google Scholar 

  27. Singh, V.V., Jouini, O., Lisser, A.: Existence of Nash equilibrium for chance-constrained games. Oper. Res. Lett. 44(5), 640–644 (2016)

    Article  MathSciNet  Google Scholar 

  28. Singh, V.V., Jouini, O., Lisser, A.: Distributionally robust chance-constrained games: existence and characterization of Nash equilibrium. Optim. Lett. (2016). doi:10.1007/s11590-016-1077-6

  29. Song, T.: On random payoff matrix games. In: Phillips, F.Y., Rousseau, J.J. (eds.) Systems and Management Science by Extremal Methods, pp. 291–308. Springer, Heidelberg (1992)

    Chapter  Google Scholar 

  30. Valenzuela, J., Mazumdar, M.: Cournot prices considering generator availability and demand uncertainty. IEEE Trans. Power Syst. 22(1), 116–125 (2007)

    Article  Google Scholar 

  31. Wolf, D.D., Smeers, Y.: A stochastic version of a Stackelberg-Nash-Cournot equilibrium model. Manage. Sci. 43(2), 190–197 (1997)

    Article  MATH  Google Scholar 

  32. Xu, H., Zhang, D.: Stochastic Nash equilibrium problems: sample average approximation and applications. Comput. Optim. Appl. 55(3), 597–645 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research was supported by Fondation DIGITEO, SUN grant No. 2014-0822D.

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Authors and Affiliations

  1. Laboratoire de Recherche en Informatique, Université Paris Sud XI, Bât 650, 91405, Orsay, France

    Vikas Vikram Singh & Abdel Lisser

  2. Laboratoire Génie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, 92290, Châtenay-Malabry, France

    Oualid Jouini

Authors
  1. Vikas Vikram Singh
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  2. Oualid Jouini
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  3. Abdel Lisser
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Corresponding author

Correspondence to Vikas Vikram Singh .

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Editors and Affiliations

  1. Complutense University, Madrid, Spain

    Begoña Vitoriano

  2. INFORMS, Catonsville, Maryland, USA

    Greg H. Parlier

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Singh, V.V., Jouini, O., Lisser, A. (2017). Solving Chance-Constrained Games Using Complementarity Problems. In: Vitoriano, B., Parlier, G. (eds) Operations Research and Enterprise Systems. ICORES 2016. Communications in Computer and Information Science, vol 695. Springer, Cham. https://doi.org/10.1007/978-3-319-53982-9_4

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  • DOI: https://doi.org/10.1007/978-3-319-53982-9_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-53981-2

  • Online ISBN: 978-3-319-53982-9

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