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Extra-proximal methods for solving two-person nonzero-sum games

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Abstract

We consider two-person nonzero-sum game, both in the classical form and in the form of a game with coupled variables. An extra-proximal approach for finding the game’s solutions is suggested and justified. We provide our algorithm with an analysis of its convergence.

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References

  1. Antipin A.S. (1986). Extrapolation methods of computing the saddle point of Lagrange function and application to problems with block-separable structure. U.S.S.R. Comput. Maths. Math. Phys. 26(1): 96

    Article  Google Scholar 

  2. Antipin A.S. (1994). Feedback-Controlled Saddle Gradient Processes. Autom. Remote Control 55(3): 311–320

    MATH  MathSciNet  Google Scholar 

  3. Antipin A.S. (1995). The convergence of proximal methods to fixed points of extreme mappings and estimates of their rate of convergence. Comp. Maths. Math. Phys. 35(5): 539–551

    MATH  MathSciNet  Google Scholar 

  4. Antipin A.: Equilibrium programming problems: prox-regularization and prox-methods. In book: Recent Advances in Optimization, Lecture Notes in Economics and Mathematical Systems, vol 452, pp. 1–18. Springer, Heidelberg (1997)

  5. Antipin A.S. (2002). Gradient approach of computing fixed points of equilibrium problems. J. Global Optim. 24(3): 285–309

    Article  MATH  MathSciNet  Google Scholar 

  6. Antipin A. (2003). Solving two-person nonzero-sum game with the help of differential equations. Differ. Equat. 39(1): 11–22

    Article  MathSciNet  Google Scholar 

  7. Antipin A.: Extragradient approach to the solution of two-person nonzero-sum game. In the book: Optimization and Optimal Control, pp. 1–28. World Scientific Publishing Co., Singapore (2003)

  8. Aubin J.-P., Frankowska H. (1990). Set Valued Analysis. Boston etc., Birkhauser

    MATH  Google Scholar 

  9. Bakushinsky A., Goncharsky A. (1994). Ill-Posed Problems: Theory and Applications. Kluwer, Dordrecht

    Google Scholar 

  10. Brown G.W.: Iterative Solutions of Games by Fictitions Play. Activity Analysis of Production and Allocation. Cowles Commission for Research in Economics, vol. 13, pp. 74–376. Wiley, New York (1951)

  11. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer-Verlag, New York

  12. Flam S.D. (2002). Equilibrium, evolutionary stability and gradient dynamics. Int. Game Theory Rev. 4(4): 1–14

    Article  MathSciNet  Google Scholar 

  13. Garcia C.B., Zangwill W.I. (1981). Pathways to Solutions, Fixed Points and Equilibria. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  14. Golshtein E.G., Tretyakov N.V. (1996). Modified Lagrangians and Monotone Maps in Optimization. Wiley, New York

    MATH  Google Scholar 

  15. Hansen T. (1974). On the Approximation of Nash Equilibrium Points in an N-person Noncooperative Game. SIAM J. Appl. Math. 26(3): 622–637

    Article  MATH  MathSciNet  Google Scholar 

  16. Mangasarian O.L., Stone H. (1964). Two-Person Nonzero-Sum Games and Quadratic Programming. J. Math. Anal. Appl. 9: 348–355

    Article  MATH  MathSciNet  Google Scholar 

  17. Mills H. (1960). Equilibrium Points in Finite Games. J. Soc. Ind. Appl. Math. 8(2): 397–402

    Article  MATH  MathSciNet  Google Scholar 

  18. Nash J.F. Jr.(1950). Equilibrium points in n-person games. Proc. Nat. Acad. Sci. USA 36: 48–49

    Article  MATH  MathSciNet  Google Scholar 

  19. Nikaido H., Isoda K. (1955). Note on noncooperative convex game. Pacific J. Math. 5(Supp. 1): 807–815

    MathSciNet  Google Scholar 

  20. Polyak B.Th. (1973). Introduction to Optimization. Nauka, Moscow

    Google Scholar 

  21. Robinson J. (1951). Iterative method of solving a game. Ann. Math. 54: 296–301

    Article  Google Scholar 

  22. Rockafellar R.T. (1976). Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5): 877–898

    Article  MATH  MathSciNet  Google Scholar 

  23. Rosen J.B. (1965). Existance and Uniqueness of Equilibrium Points for Concave N-Person Games. Econometrica 33(3): 520–534

    Article  MATH  MathSciNet  Google Scholar 

  24. Stengel B.: Computing Equilibria for Two-Person Games. In book: Handbook of Game Theory, vol. 3, pp. 1–39. North-Holland, Amsterdam (2002)

  25. Vasil’ev F.P. (2002). Optimization Methods. Factorial-Press, Moscow

    Google Scholar 

  26. Zukhovitsky S.I., Polyak R.A., Primak M.E.: Concave N-person games (Numerical Methods) (in Russian). Economica i Matematich. Metody, VII(6), 888–900 (1971)

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Correspondence to Anatoly Antipin.

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Antipin, A. Extra-proximal methods for solving two-person nonzero-sum games. Math. Program. 120, 147–177 (2009). https://doi.org/10.1007/s10107-007-0152-2

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  • DOI: https://doi.org/10.1007/s10107-007-0152-2

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