Acta Applicandae Mathematica

, Volume 54, Issue 2, pp 135–166

The Study of Minimax Inequalities, Abstract Economics and Applications to Variational Inequalities and Nash Equilibria

  • George Xian-Zhi Yuan
  • George Isac
  • Kok-Keong Tan
  • Jian Yu
Article

Abstract

In this survey, a new minimax inequality and one equivalent geometricform are proved. Next, a theorem concerning the existence of maximalelements for an LC-majorized correspondence is obtained.By the maximal element theorem, existence theorems of equilibrium point fora noncompact one-person game and for a noncompact qualitative game withLC-majorized correspondences are given. Using the lastresult and employing 'approximation approach', we prove theexistence of equilibria for abstract economies in which the constraintcorrespondence is lower (upper) semicontinuous instead of having lower(upper) open sections or open graphs in infinite-dimensional topologicalspaces. Then, as the applications, the existence theorems of solutions forthe quasi-variational inequalities and generalized quasi-variationalinequalities for noncompact cases are also proven. Finally, with theapplications of quasi-variational inequalities, the existence theorems ofNash equilibrium of constrained games with noncompact are given. Our resultsinclude many results in the literature as special cases.

KKM principle minimax inequality lower (upper) open sections lower (upper) semicontinuous class LC LC-majorant LC-majorized qualitative game abstract economics equilibrium Nash equilibrium the property (K) quasi-variational inequality 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aliprantis, C. and Brown, D.: Equilibria inmarkets with a Riesz space of commodities, J. Math. Econom. 11 (1983), 189–207.Google Scholar
  2. 2.
    Arrow, K. J. and Debreu, G.: Existence of an equilibrium for a competitive economy, Econometrica 22 (1954), 265–290.Google Scholar
  3. 3.
    Aubin, J. P.: Mathematical Methods of Game and Economics Theory, Rev. edn, North-Holland, Amsterdam, 1982.Google Scholar
  4. 4.
    Aubin, J. P. and Ekeland, I.: Applied Nonlinear Analysis, Wiley, New York, 1984.Google Scholar
  5. 5.
    Bergstrom, T. R., Parks, R. and Rader, T.: Preferences which have open graphs, J. Math. Econom. 3 (1976), 265–268.Google Scholar
  6. 6.
    Borglin, A. and Keiding, H.: Existence of equilibrium actions of equilibrium, 'A note the "new" existence theorems', J. Math. Econom. 3 (1976), 313–316.Google Scholar
  7. 7.
    Bewley, T. F.: Existence of equilibria in economics with infinite many commodities, J. Econom. Theory 4 (1972), 514–540.Google Scholar
  8. 8.
    Border, K. C.: Fixed Point Theorems with Applications to Economics and Game Theory, Cambridge University Press, 1985.Google Scholar
  9. 9.
    Chang, S. Y.: On the Nash equilibrium, Soochow J. Math. 16 (1990), 241–248.Google Scholar
  10. 10.
    Cullen, H. F.: Introduction to General Topology, D. C. Heath, Boston, 1968.Google Scholar
  11. 11.
    Debreu, G.: A social equilibrium existence theorem, Proc. Nat. Acad. Sci. USA 38 (1952), 386–393.Google Scholar
  12. 12.
    Ding, X. P. and Tan, K. K.: A minimax inequality with applications to existence of equilibrium point and fixed point theorems, Colloq. Math. 63 (1992), 233–247.Google Scholar
  13. 13.
    Ding, X. P. and Tan, K. K.: On equilibria of noncompact generalized games, J. Math. Anal. Appl. 167 (1993).Google Scholar
  14. 14.
    Ding, X. P., Kim, W. K. and Tan, K. K.: Equilibria of noncompact generalized games with ℒ*-majorized preference correspondences, J. Math. Anal. Appl. 164 (1992), 508–517.Google Scholar
  15. 15.
    Fan, K.: Fixed points and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. USA 38 (1952), 131–136.Google Scholar
  16. 16.
    Fan, K.: Some properties of convex sets related to fixed point theorems, Math. Ann. 226 (1984), 519–537.Google Scholar
  17. 17.
    Flam, S. D.: Abstract economy and games, Soochow J. Math. 5 (1979), 155–162.Google Scholar
  18. 18.
    Florenzano, M.: On the existence of equilibrium in economics with an infinite dimensional commodity spaces, J. Math. Econom. 12 (1983), 207–219.Google Scholar
  19. 19.
    Gale, D. and Mas-Colell, A.: An equilibrium existence for a general model without ordered preferences, J. Math. Econom. 2 (1975), 9–15.Google Scholar
  20. 20.
    Gale, D. and Mas-Colell, A.: On the role of complete, transitive preferences in equilibrium theory, in: G. Schwödiauer (ed.), Equilibrium and Disequilibrium in Economics Theory, Reidel, Dordrecht, 1978, pp. 7–14.Google Scholar
  21. 21.
    Glicksberg, I. L.: A further generalization of the Kakutani fixed point theorem with applications to Nash equilibrium points, Proc. Amer. Math. Soc. 3 (1952), 170–174.Google Scholar
  22. 22.
    Kajii, A.: Note on equilibrium without ordered preferences in topological vector space, Econom. Lett. 27 (1988), 1–4.Google Scholar
  23. 23.
    Keiding, H.: Existence of economic equilibriums, Lecture Notes in Econom. and Math. Systems 226, Springer-Verlag, New York, 1984, pp. 223–243.Google Scholar
  24. 24.
    Khan, M. A. and Vohra, R.: Equilibrium in abstract economics without preferences and with a measure of agents, J. Math. Econom. 13 (1984), 133–142.Google Scholar
  25. 25.
    Khan, M. A. and Papageorgiou, R.: On cournot Nash equilibria in generalized quantitative games with an atomless measure spaces of agents, Proc. Amer. Math. Soc. 100 (1987), 505–510.Google Scholar
  26. 26.
    Kim, W. K.: Remark on a generalized quasi-variational inequality, Proc. Amer. Math. Soc. 103 (1988), 667–668.Google Scholar
  27. 27.
    Kim, T. and Richter, M. K.: Nontransitive-nontotal consumer theory, J. Econom. Theory 28 (1986), 324–363.Google Scholar
  28. 28.
    Kim, T., Prikry, K. and Yannelis, N. C.: Equilibrium in abstract economics with a measure space and with an infinite dimensional strategy space, J. Approx. Theory 56 (1989), 256–266.Google Scholar
  29. 29.
    Kim, W. K. and Tan, K. K.: A variational inequality in noncompact sets and its applications, Bull. Austral. Math. Soc. 46 (1992), 139–148.Google Scholar
  30. 30.
    Klein, E. and Thompson, A. C.: Theory of Correspondences: Including Applications to Mathematical Economics, Wiley, 1984.Google Scholar
  31. 31.
    Kneser, H.: Sur un théorème fondamental de la théorie des jeux, C.R. Acad. Sci. Paris 234 (1952), 2418–2420.Google Scholar
  32. 32.
    Isac, G.: Complementarity Problems, Lecture Notes in Math. 1528, Springer-Verlag, New York, 1992.Google Scholar
  33. 33.
    Michael, E.: Continuous selections I, Ann. Math. 63 (1956), 361–382.Google Scholar
  34. 34.
    Mosco, U.: Implicit problems and quasi-variational inequalities, in: Lecture Notes in Math.543, Springer-Verlag, New York, 1976, pp. 83–156.Google Scholar
  35. 35.
    Ricceri, B.: Un théorme d'existence pour les inéquations, C.R. Acad. Sci. Paris Ser. I Math. 301 (1985), 885–888.Google Scholar
  36. 36.
    Shafer, W. and Sonnenschein, H.: Equilibria in abstract economics without ordered preferences, J. Math. Econom. 2 (1975), 345–348.Google Scholar
  37. 37.
    Shafer, W.: Equilibrium in economics without ordered preferences or free disposal, J. Math. Econom. 3 (1976), 135–137.Google Scholar
  38. 38.
    Shih, M. H. and Tan, K. K.: Generalized quasi-variational inequalities in locally convex topological vector spaces, J. Math. Anal. Appl. 108 (1985), 333–343.Google Scholar
  39. 39.
    Tan, K. K. and Yu, J.: New minimax inequality with applications to existence theorems of equilibrium points, J. Optim. Theory Appl. 82 (1994), 105–120.Google Scholar
  40. 40.
    Tan, K. K. and Yuan, X. Z.: Approximation method and equilibria of abstract economies, Proc. Amer. Math. Soc. 122 (1994), 503–510.Google Scholar
  41. 41.
    Tan, K. K., Yu, J. and Yuan, X. Z.: Existence theorems for saddle points of vector-valued maps, J. Optim. Theory Appl. 89 (1996), 731–747.Google Scholar
  42. 42.
    Tarafdar, E. and Mehta, G.: A generalized version of the Gale–Nikaido–Debreu theorem, Comm. Math. Univ. Carolin. 28 (1987), 655–659.Google Scholar
  43. 43.
    Tian, G.: On the existence of equilibria in generalized games, Internat. J. Game Theory 22 (1992), 247–254.Google Scholar
  44. 44.
    Tian, G. and Zhou, J.: Quasi-variational inequalities with noncompact sets, J. Math. Anal. Appl. {vn160} (1991), 583–595.Google Scholar
  45. 45.
    Toussaint, S.: On the existence of equilibria in economics with infinitely many commodities and without ordered preferences, J. Econom. Theory 33 (1984), 98–115.Google Scholar
  46. 46.
    Tulcea, C. I.: On the equilibriums of generalized games, The Center for Mathematical Studies in Economics and Management Science, Paper No. 696, 1986.Google Scholar
  47. 47.
    Tulcea, C. I.: On the approximation of upper semiocntinuous correspondences and the equilibrium of the generalized games, J. Math. Anal. Appl. 136 (1988), 267–289.Google Scholar
  48. 48.
    Yannelis, N. C.: Equilibria in noncooperative models of competition, J. Econom. Theory 41 (1987), 96–111.Google Scholar
  49. 49.
    Yannelis, N. C. and Prabhakar, N. D.: Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econom. 12 (1983), 233–245.Google Scholar
  50. 50.
    Yu, J.: On Nash equilibrium in N-person games over relefexive Banach spaces, J. Optim. Theory Appl. 73 (1992), 211–214.Google Scholar
  51. 51.
    Yuan, George X. Z.: The study of minimax inequalities and applications to economies and variational inequalities, Mem. Amer. Math. Soc. 132, No.625(1998), 1–140.Google Scholar
  52. 52.
    Yuan, X. Z. and Tarafdar, E.: Enayet non-compact Pareto equilibria for multiobjective games, J. Math. Anal. Appl. 204 (1996), 156–163.Google Scholar
  53. 53.
    Yuan, X. Z. and Tarafdar, E.: Existence of equilibria of generalized games without compactness and paracompactness, Nonlinear Anal. 26 (1996), 893–902.Google Scholar
  54. 54.
    Zhou, J. X. and Chen, G.: Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities, J. Math. Anal. Appl. 132 (1988), 213–225.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • George Xian-Zhi Yuan
    • 1
  • George Isac
    • 2
  • Kok-Keong Tan
    • 3
  • Jian Yu
    • 3
  1. 1.Department of MathematicsThe University of QueenslandBrisbaneAustralia
  2. 2.Department of Mathematics and Computer ScienceRoyal Military College of CanadaKingstonCanada
  3. 3.Department of Mathematics and StatisticsDalhousie UniversityHalifaxCanada

Personalised recommendations