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Part of the book series: Mathematics and Its Applications ((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].

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References

  1. Bagchi, A.: Stackelberg Differential Games in Economic Models, Springer Verlag, Berlin, 1984.

    Book  MATH  Google Scholar 

  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. Basar, T. and Olsder, G.: Dynamic Noncooperative Game Theory, Academic Press, New York, 1982.

    MATH  Google Scholar 

  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.

    Chapter  Google Scholar 

  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. 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. Cavazzuti, E. and Pacchiarotti, N.: Convergence of Nash Equilibria, Boll. Unione Mat. Ital. 5-B(1986), 247–266.

    MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  9. Dasgupta, P. and Maskin, E.: The existence of equilibrium in discontinuous economic games, I: Theory, Review of Economic Studies 53(1986), 1–26.

    Article  MathSciNet  MATH  Google Scholar 

  10. Dontchev, A.L. and Zolezzi, T.: Well-Posed Optimization Problems, Lecture Notes in Mathematics. Vol. 1543, Springer Verlag, Berlin, 1993.

    MATH  Google Scholar 

  11. Fishburn, P.C.: Utility Theory for Decision Making, Wiley, New York, 1970.

    MATH  Google Scholar 

  12. Fudenberg, D. and Levine, D.: Subgame-Perfect Equilibria of Finite- and InfiniteHorizon Games, J. Econ. Theory 31(1983), 227–256.

    Article  MathSciNet  MATH  Google Scholar 

  13. Fudenberg, D. and Tirole, J.: Game Theory, MIT Press, Cambridge (Massachusetts), 1991.

    Google Scholar 

  14. Furi, M. and Vignoli, A.: About well-posed optimization problems for functionals in metric spaces, J. Optimization Theory Appl. 5(1970), 225–229.

    Article  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  16. Ichiishi, T.: Game Theory for Economic Analysis, Academic Press, New York, 1983.

    MATH  Google Scholar 

  17. Jurg, P. and Tijs, S.H.: On the Determinateness of Semi-Infinite Bimatrix Games, Intern. J. Game Theory 21(1993), 361–369.

    Article  MathSciNet  MATH  Google Scholar 

  18. Kalai, E. and Stanford, W.: Finite rationality and interpersonal complexity in repeated games, Econometrica 56(1988), 397–410.

    Article  MathSciNet  MATH  Google Scholar 

  19. Kats, A. and Thisse, J.-F.: Unilaterally Competitive Games, Intern. J. Game Theory 21(1992), 291–299.

    Article  MathSciNet  MATH  Google Scholar 

  20. Kelley, J.L.: General Topology, Van Nostrand, Princeton, 1955.

    MATH  Google Scholar 

  21. Kenderov, P.S. and Lucchetti, R.: Generic well-posedness of semicontinuous functions, preprint 1994.

    Google Scholar 

  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. Lassonde, M. and Schenkel, C.: KKM Principle, Fixed Points, and Nash Equilibria, J. Math. Anal. Appl. 164(1992), 542–548.

    Article  MathSciNet  MATH  Google Scholar 

  24. Loridan, P.: Well-Posedness in Vector Optimization, this volume.

    Google Scholar 

  25. Loridan, P. and Morgan, J.: Penalty functions in ε-programming and e-minimax problems. Math. Programnmning 26(1983), 213–231.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  29. Lucchetti, R., Patrone, F. and Tijs, S.H.: Determinateness of two-person games, Boll. Unione Mat. Ital. 5-B(1986), 907–924.

    MathSciNet  Google Scholar 

  30. Mallozzi, L. and Morgan, J.: ε-Mixed Strategies for Static Continuous Stackelberg Problem, J. Optimization Theory Appl. 78(1993), 303–316.

    Article  MathSciNet  MATH  Google Scholar 

  31. McLinden, L.: An application of Ekeland’s Theorem to Minimax Problems, Nonlin. Anal. T.M.A. 6(1982), 189–196.

    Article  MathSciNet  MATH  Google Scholar 

  32. McLinden, L.: A Minimax Theorem, Math. of Oper. Res. 9(1984), 576–591.

    Article  MathSciNet  MATH  Google Scholar 

  33. Monderer, D. and Samet, D.: Approximating common knowledge with common beliefs, Games and Econ. Behavior 1(1989), 170–190.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  36. Myerson, R.B.: Game Theory: Analysis of Conflict, Harvard University Press, Cambridge, MA, 1991.

    MATH  Google Scholar 

  37. Nash, J.F.: Non-Cooperative Games, Ann. of Math. 54(1951), 286–295.

    Article  MathSciNet  MATH  Google Scholar 

  38. Neyman, A.: Bounded complexity justifies cooperation in the finitely repeated prisoner’s dilemma, Economic Letters 19(1985), 227–229.

    Article  MathSciNet  Google Scholar 

  39. Pascoa, M.R.: Approximate equilibrium in pure strategies for non-atomic games, J. of Math. Econ. 22(1993), 223–241.

    Article  MathSciNet  MATH  Google Scholar 

  40. Patrone, F.: Well-Posedness as an Ordinal Property, Rivista di Matematica pura ed applicata 1(1987). 95–104.

    MathSciNet  MATH  Google Scholar 

  41. Patrone, F.: Well-posed minimum problems for preorders, Rend. Sem. Mat. Univ. Padova 84(1990). 109–121.

    MathSciNet  MATH  Google Scholar 

  42. Patrone, F. and Pusillo Chicco, L.: Antagonism for two-person games: taxonomy and applications to Tikhonov well-posedness, preprint.

    Google Scholar 

  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.

    MathSciNet  Google Scholar 

  44. Patrone F. and Revalski, J.P.: Characterization of Tikhonov Well-Posedness for Preorders, Math. Balkanica 5(1991), 146–155.

    MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  47. Patrone, F. and Torre, A.: Topologies on von Neumann-Morgenstern preferences and apmlications to Game Theory. preprint.

    Google Scholar 

  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.

    Article  Google Scholar 

  49. Radzik, T.: Pure-strategy ε-Nash equilibrium in two-person non-zero-sum games, Games and Econ. Behavior 3(1991), 356–367.

    Article  MathSciNet  MATH  Google Scholar 

  50. Radzik, T.: Nash Equilibria of Discontinuous Non-Zero-Sum Two-Person Games, Inter. J. of Game Theory 21(1993), 429–437.

    Article  MathSciNet  MATH  Google Scholar 

  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. Rubinstein, A.: Finite automata play the repeated prisoners’ dilemma, J. Econ. Theory 39(1986), 76–83.

    Article  Google Scholar 

  53. Simon, L.K. and Zame, W.R.: Discontinuous Games and Endogenous Sharing Rules, Econometrica 58(1990), 861–872.

    Article  MathSciNet  MATH  Google Scholar 

  54. Sion, M. and Wolfe, P.: On a game without a value, Annals of Math. Studies 39(1957), 299–306.

    MathSciNet  MATH  Google Scholar 

  55. Tijs, S.H.: ε-Equilibrium point theorems for two-person games, Methods of Oper. Res. 26(1977), 755–766.

    MathSciNet  Google Scholar 

  56. Tijs, S.H.: Nash equilibria for noncooperative n-person games in normal form, SIAM Review 23(1981), 225–237.

    Article  MathSciNet  MATH  Google Scholar 

  57. Topkis, D.: Equilibrium points in nonzero-sum n-person submodular games, SIAM J. Control Optirn. 17(1979). 773–787.

    Article  MathSciNet  MATH  Google Scholar 

  58. Van Damme, E.E.C.: Stability and Perfection of Nash Equilibria, Springer Verlag, Berlin, 1987; 2nd edition 1991.

    MATH  Google Scholar 

  59. Vives, X.: Nash equilibrium with strategic complementarities, J. Math. Econ. 19(1990), 305–321.

    Article  MathSciNet  MATH  Google Scholar 

  60. Wilson, R.: Computing equilibria of n-person games, SIAM J. Appl. Math. 21(1971), 80–87.

    Article  MathSciNet  Google Scholar 

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Patrone, F. (1995). Well-Posedness for Nash Equilibria and Related Topics. In: Lucchetti, R., Revalski, J. (eds) Recent Developments in Well-Posed Variational Problems. Mathematics and Its Applications, vol 331. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8472-2_9

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  • DOI: https://doi.org/10.1007/978-94-015-8472-2_9

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