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Well-Posedness for Nash Equilibria and Related Topics

  • F. Patrone
Part of the Mathematics and Its Applications book series (MAIA, volume 331)

Abstract

Most of this survey is in the context of non-cooperative games in strategic form, and is essentially devoted to concepts which gravitate around the idea of Nash equilibrium (briefly: NE): for standard terminology in game theory and for general reference, see [36] or [13].

Keywords

Nash Equilibrium Variational Inequality Saddle Point Problem Noncooperative Game Nash Equilibrium Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bagchi, A.: Stackelberg Differential Games in Economic Models, Springer Verlag, Berlin, 1984.zbMATHCrossRefGoogle Scholar
  2. 2.
    Baiocchi, C. and Capelo, A.C.: Disequazioni variazionali e quasivariazionali. Applicazioni a problemi di frontiera libera, Vol. 2, Pitagora, Bologna, 1978.Google Scholar
  3. 3.
    Basar, T. and Olsder, G.: Dynamic Noncooperative Game Theory, Academic Press, New York, 1982.zbMATHGoogle Scholar
  4. 4.
    Cavazzuti, E.: Convergence of Equilibria in the Theory of Games, in R. Conti, E. De Giorgi, and F. Giannessi (eds.), Optimization and Related Fields, Proc. Conf. Erice/Italy 1984, Lecture Notes in Mathematics, Vol. 1190, Springer Verlag, Berlin, 1986, pp.95–130.CrossRefGoogle Scholar
  5. 5.
    Cavazzuti, E.: Cobwebs and something else, in G. Ricci (ed.), Decision Processes in Economics, Proc. Conf. Modena/Italy 1989, Springer Verlag, Berlin, 1990, pp.34–43.Google Scholar
  6. 6.
    Cavazzuti, E. and Morgan, J.: Well-Posed Saddle Point Problems, in J.-B. Hiriart Urruty, W. Oettli and J. Stoer (eds.), Optimization, theory and algorithms, Proc. Conf. Confolant/France 1981, Lecture Notes in Pure and Applied Mathematics, Vol. 86. Marcel Dekker, New York and Basel, 1983, pp.61–76.Google Scholar
  7. 7.
    Cavazzuti, E. and Pacchiarotti, N.: Convergence of Nash Equilibria, Boll. Unione Mat. Ital. 5-B(1986), 247–266.MathSciNetGoogle Scholar
  8. 8.
    Čoban, M.M., Kenderov, P.S. and Revalski, J.P.: Generic well-posedness of optimization problems in topological spaces, Mathematika 36(1989), 301–324.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dasgupta, P. and Maskin, E.: The existence of equilibrium in discontinuous economic games, I: Theory, Review of Economic Studies 53(1986), 1–26.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dontchev, A.L. and Zolezzi, T.: Well-Posed Optimization Problems, Lecture Notes in Mathematics. Vol. 1543, Springer Verlag, Berlin, 1993.zbMATHGoogle Scholar
  11. 11.
    Fishburn, P.C.: Utility Theory for Decision Making, Wiley, New York, 1970.zbMATHGoogle Scholar
  12. 12.
    Fudenberg, D. and Levine, D.: Subgame-Perfect Equilibria of Finite- and InfiniteHorizon Games, J. Econ. Theory 31(1983), 227–256.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Fudenberg, D. and Tirole, J.: Game Theory, MIT Press, Cambridge (Massachusetts), 1991.Google Scholar
  14. 14.
    Furi, M. and Vignoli, A.: About well-posed optimization problems for functionals in metric spaces, J. Optimization Theory Appl. 5(1970), 225–229.zbMATHCrossRefGoogle Scholar
  15. 15.
    Glicksberg, I.L.: A Further Generalization of the Kakutani Fixed Point Theorem with Application to Nash Equilibrium Points, Proc. Amer. Math. Soc. 3(1952), 170–174.MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ichiishi, T.: Game Theory for Economic Analysis, Academic Press, New York, 1983.zbMATHGoogle Scholar
  17. 17.
    Jurg, P. and Tijs, S.H.: On the Determinateness of Semi-Infinite Bimatrix Games, Intern. J. Game Theory 21(1993), 361–369.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Kalai, E. and Stanford, W.: Finite rationality and interpersonal complexity in repeated games, Econometrica 56(1988), 397–410.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Kats, A. and Thisse, J.-F.: Unilaterally Competitive Games, Intern. J. Game Theory 21(1992), 291–299.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Kelley, J.L.: General Topology, Van Nostrand, Princeton, 1955.zbMATHGoogle Scholar
  21. 21.
    Kenderov, P.S. and Lucchetti, R.: Generic well-posedness of semicontinuous functions, preprint 1994.Google Scholar
  22. 22.
    Kenderov, P.S. and Ribarska, N.K.: Most of the two-person zero-sum games have unique solution, in S. Fitzpatrick and G. Giles (eds.), Functional analysis and optimization, Proc. Conf. Canberra/Australia, Australian National Univ., Centre for Mathematical Analysis, Canberra, 1988, pp.73–83.Google Scholar
  23. 23.
    Lassonde, M. and Schenkel, C.: KKM Principle, Fixed Points, and Nash Equilibria, J. Math. Anal. Appl. 164(1992), 542–548.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Loridan, P.: Well-Posedness in Vector Optimization, this volume.Google Scholar
  25. 25.
    Loridan, P. and Morgan, J.: Penalty functions in ε-programming and e-minimax problems. Math. Programnmning 26(1983), 213–231.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Lucchetti, R.: Well-posedness, towards vector optimization, Lecture Notes in Economics and Mathematical Systems, Vol. 294, Springer Verlag, Berlin, 1983, pp. 194–207.Google Scholar
  27. 27.
    Lucchetti, R. and Patrone, F.: A characterization of Tyhonov well-posedness for minimum problems, with applications to variational inequalities, Numner. Funct. Analysis Optimiz. 3(1981), 461–476.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Lucchetti, R. and Patrone, F.: Some properties of “well-posed” variational inequalities governed by linear operators, Nurner. Funct. Analysis Optirniz. 5(1982–83), 349–361.MathSciNetCrossRefGoogle Scholar
  29. 29.
    Lucchetti, R., Patrone, F. and Tijs, S.H.: Determinateness of two-person games, Boll. Unione Mat. Ital. 5-B(1986), 907–924.MathSciNetGoogle Scholar
  30. 30.
    Mallozzi, L. and Morgan, J.: ε-Mixed Strategies for Static Continuous Stackelberg Problem, J. Optimization Theory Appl. 78(1993), 303–316.MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    McLinden, L.: An application of Ekeland’s Theorem to Minimax Problems, Nonlin. Anal. T.M.A. 6(1982), 189–196.MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    McLinden, L.: A Minimax Theorem, Math. of Oper. Res. 9(1984), 576–591.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Monderer, D. and Samet, D.: Approximating common knowledge with common beliefs, Games and Econ. Behavior 1(1989), 170–190.MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Morgan, J.: Constrained Well-Posed Two-Level Optimization Problems, in F.H. Clarke, V.F. Dem’yanov and F. Giannessi (eds.), Non-Smooth Optimization and Related Topics, Plenum Press, New York, 1989.Google Scholar
  35. 35.
    Moulin, H. and Vial, J.-P.: Strategically Zero-Sum Games: The Class of Games Whose Completely Mixed Equilibria Cannot be Improved Upon, Intern. J. Game Theory 7(1978), 201–221.MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Myerson, R.B.: Game Theory: Analysis of Conflict, Harvard University Press, Cambridge, MA, 1991.zbMATHGoogle Scholar
  37. 37.
    Nash, J.F.: Non-Cooperative Games, Ann. of Math. 54(1951), 286–295.MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Neyman, A.: Bounded complexity justifies cooperation in the finitely repeated prisoner’s dilemma, Economic Letters 19(1985), 227–229.MathSciNetCrossRefGoogle Scholar
  39. 39.
    Pascoa, M.R.: Approximate equilibrium in pure strategies for non-atomic games, J. of Math. Econ. 22(1993), 223–241.MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Patrone, F.: Well-Posedness as an Ordinal Property, Rivista di Matematica pura ed applicata 1(1987). 95–104.MathSciNetzbMATHGoogle Scholar
  41. 41.
    Patrone, F.: Well-posed minimum problems for preorders, Rend. Sem. Mat. Univ. Padova 84(1990). 109–121.MathSciNetzbMATHGoogle Scholar
  42. 42.
    Patrone, F. and Pusillo Chicco, L.: Antagonism for two-person games: taxonomy and applications to Tikhonov well-posedness, preprint.Google Scholar
  43. 43.
    Patrone, F. and Revalski, J.P.: Constrained minimum problems for preorders: Tikhonov and Hadamard well-posedness, Boll. Unione Mat. Ital. 5-B(1991), 639–652.MathSciNetGoogle Scholar
  44. 44.
    Patrone F. and Revalski, J.P.: Characterization of Tikhonov Well-Posedness for Preorders, Math. Balkanica 5(1991), 146–155.MathSciNetzbMATHGoogle Scholar
  45. 45.
    Patrone, F. and Tijs, S.H.: Unified Approach to Approximate Solutions in Games and Multiobiective Programming, J. Optimization Theory Appl. 52(1987), 273–278.MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Patrone, F. and Torre, A.: Characterizations of Existence and Uniqueness for Saddle Point Problems and Related Topics, Boll. Unione Mat. Ital. 5-C(1986), 175–184.MathSciNetGoogle Scholar
  47. 47.
    Patrone, F. and Torre, A.: Topologies on von Neumann-Morgenstern preferences and apmlications to Game Theory. preprint.Google Scholar
  48. 48.
    Radner, R.: Collusive behavior in non-cooperative epsilon equilibria of oligopolies with long but finite lives, Journal of Economic Theory 22(1980), 121–157.CrossRefGoogle Scholar
  49. 49.
    Radzik, T.: Pure-strategy ε-Nash equilibrium in two-person non-zero-sum games, Games and Econ. Behavior 3(1991), 356–367.MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Radzik, T.: Nash Equilibria of Discontinuous Non-Zero-Sum Two-Person Games, Inter. J. of Game Theory 21(1993), 429–437.MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Revalski, J.P.: Variational inequalities with unique solution, in Mathematics and Education in Mathematics, Proc. 14th Spring Confer. of the Union of Bulgarian Mathematicians, Sofia, 1985, pp. 534–541.Google Scholar
  52. 52.
    Rubinstein, A.: Finite automata play the repeated prisoners’ dilemma, J. Econ. Theory 39(1986), 76–83.CrossRefGoogle Scholar
  53. 53.
    Simon, L.K. and Zame, W.R.: Discontinuous Games and Endogenous Sharing Rules, Econometrica 58(1990), 861–872.MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Sion, M. and Wolfe, P.: On a game without a value, Annals of Math. Studies 39(1957), 299–306.MathSciNetzbMATHGoogle Scholar
  55. 55.
    Tijs, S.H.: ε-Equilibrium point theorems for two-person games, Methods of Oper. Res. 26(1977), 755–766.MathSciNetGoogle Scholar
  56. 56.
    Tijs, S.H.: Nash equilibria for noncooperative n-person games in normal form, SIAM Review 23(1981), 225–237.MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Topkis, D.: Equilibrium points in nonzero-sum n-person submodular games, SIAM J. Control Optirn. 17(1979). 773–787.MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Van Damme, E.E.C.: Stability and Perfection of Nash Equilibria, Springer Verlag, Berlin, 1987; 2nd edition 1991.zbMATHGoogle Scholar
  59. 59.
    Vives, X.: Nash equilibrium with strategic complementarities, J. Math. Econ. 19(1990), 305–321.MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Wilson, R.: Computing equilibria of n-person games, SIAM J. Appl. Math. 21(1971), 80–87.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • F. Patrone
    • 1
  1. 1.Dipartimento di MatematicaUniversità di GenovaGenovaItaly

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