Elements of Fuzzy Game Theory

  • Antoine Billot
Part of the The Handbooks of Fuzzy Sets Series book series (FSHS, volume 1)

Abstract

This chapter presents in a first section the intuitional relationships between games and fuzziness; then, in a second section, we propose some arguments to justify and illustrate the two main concepts of fuzzy information and fuzzy coalition, on which cooperative and noncooperative analyses are founded; next, in the two following sections, we develop the noncooperative framework essentially from Butnariu’s contributions, gathering the literature and presenting the results, and finally the cooperative model essentially from Aubin’s works, Hüsseinov’s and Billot’s.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abreu D. & A. Rubinstein 1988. ‘The Structure of Nash Equilibrium in Repeated Games with Finite Automata’, Econometrica, 56, pp. 1259–1282.MathSciNetMATHCrossRefGoogle Scholar
  2. Aubin J. P. 1974. ‘Coeur et Valeur des Jeux Flous’, Compte-rendus de l’Académie des Sciences de Paris, 279, pp. 891–894.MathSciNetMATHGoogle Scholar
  3. Aubin J. P. 1976. ‘Fuzzy Core and Equilibrium in Games Defined in Strategic Form’, in Directions in Large Scale Systems, Y. C. Ho & S. K. Miller Eds, Plenum Press: New York.Google Scholar
  4. Aubin J. P. 1979. Mathematical Methods in Economics and Game Theory. North-Holland: Amsterdam.Google Scholar
  5. Aubin J. P. 1981. ‘Locally Lipschitz Cooperative Games’, Journal of Mathematical Economics, 8, pp. 241–262.MathSciNetMATHCrossRefGoogle Scholar
  6. Aubin J. P. 1986. L’Analyse non Linéaire et Ses Motivations Economiques. Masson: Paris.Google Scholar
  7. Aumann R. J. 1964. ‘Values of Market with a Continuum of Traders’, Econometrica, 83, pp. 611–646.Google Scholar
  8. Aumann R. J. 1976. ‘Agreeing to Disagree’, The Annals of Statistics, 4, pp. 1236–1239.MathSciNetMATHCrossRefGoogle Scholar
  9. Badard R. 1984. ‘Fixed Point Theorems for Fuzzy Numbers’, Fuzzy Sets and Systems, 13, pp. 291–302.MathSciNetMATHCrossRefGoogle Scholar
  10. Billot A. 1986. ‘A Contribution to A Mathematical Theory of Fuzzy Games’ in Fuzzy Economics and Spatial Analysis,C. Ponsard & B. Fustier Eds, Librairie de l’Université, Dijon, pp. 47–56.Google Scholar
  11. Billot A. 1987. Préférence et Utilité Floues. Presses Universitaires de France: Paris.Google Scholar
  12. Billot A. 1988. Préférence Imprécise et Equilibres Economiques: Une Analyse Axiomatique. Chs 3 & 4, Ph.D. in Economics, Université de Bourgogne, France.Google Scholar
  13. Billot A. 1990. ‘Peripherial Core of an Exchange Economy Represented as a Fuzzy Game’ in Multiperson Decision-Making Using Fuzzy Sets ans Possibility Theory, J. Kacprzyck & M. Fedrizzi Eds, Kluwer: Boston, pp. 311–335.CrossRefGoogle Scholar
  14. Billot A. 1991. ‘Aggregation of Preferences: The Fuzzy Case’, Theory and Decision, 30, pp. 51–93.MathSciNetMATHCrossRefGoogle Scholar
  15. Billot A. 1992. ‘From Fuzzy Set Theory to NonAdditive Probabilities: How Have Economists Reacted?’, Fuzzy Sets and Systems, 49, pp. 75–90.MathSciNetMATHCrossRefGoogle Scholar
  16. Billot A. 1995a. ‘Fuzzy Utility Function: A New Elementary Proof’, Fuzzy Sets and Systems, 74, pp. 271–276.MathSciNetMATHCrossRefGoogle Scholar
  17. Billot A. 1995b. Economic Theory of Fuzzy Equilibria. Springer-Verlag: Berlin, New York.MATHCrossRefGoogle Scholar
  18. Billot A. 1996. ‘Fuzzy Decision Theory’, to appear in Decision Under Uncertainty, International School of Economic Research, J. D. Hey, F.Hahn & L. Luini Eds, Oxford University Press: Oxford.Google Scholar
  19. Billot A.& B. Walliser 1995. ‘A Mixed Knowledge Hierarchy’, CERAS Working-Paper 95-17, Ecole des Ponts et Chaussées, Paris, France.Google Scholar
  20. Border K. C. 1985. Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press, Cambridge.MATHCrossRefGoogle Scholar
  21. Bose R. K. & D. Sahani 1987. ‘Fuzzy Maapings and Fixed Point Theorems’, Fuzzy Sets and Systems, 21, pp. 53–58.MathSciNetMATHCrossRefGoogle Scholar
  22. Brouwer L. E. J. 1912. ‘Uber Abbildung von Mannigfaltikeiten’, Mathematische Annalen, 71, pp. 97–115.MATHCrossRefGoogle Scholar
  23. Butnariu D.1978. ‘Fuzzy Games. A Description of the Concept’, Fuzzy Sets and Systems, 1, pp. 181–192.Google Scholar
  24. Butnariu D.1979. ‘Solution Concepts for n-Person Fuzzy Games’ in Advances in Fuzzy Set Theory and Applications, M. M. Gupta, R. K. Ragade & R. R. Yager Eds, Klüwer: Boston.Google Scholar
  25. Butnariu D.1980. ‘Stability and Shapley-Value for n-Person Fuzzy Games’, Fuzzy Sets and; Systems, 7, pp. 63–72.Google Scholar
  26. Butnariu D.1982. ‘Fixed Points for Fuzzy Mappings’, Fuzzy Sets and Systems, 14, pp. 191–207.Google Scholar
  27. Butnariu D.1985. ‘NonAtomic Fuzzy Measures and Games’, Fuzzy Sets and Systems, 17, pp. 39–52.Google Scholar
  28. Butnariu D.1986. ‘Fuzzy Measurability and Integrability’, Journal of Mathematical Analysis and Applications, 117, pp. 385–410.Google Scholar
  29. Butnariu D. 1987. ‘Values and Cores of Fuzzy Games with Infinitely Many Players’, International Journal of Game Theory, 16, pp. 43–68.MathSciNetMATHCrossRefGoogle Scholar
  30. Chitra A. & P. V. Subrahmanyam 1987. ‘Fuzzy Sets and Fixed Points’, Journal of Mathematical Analysis and Applications, 124, 584–590.MathSciNetMATHCrossRefGoogle Scholar
  31. Chang C. L. 1968. ‘Fuzzy Topological Spaces’, Journal of Mathematical Analysis and Applications, 17, pp. 182–190.CrossRefGoogle Scholar
  32. Debreu G.1959. Theory of Value. Wiley: New York.Google Scholar
  33. Debreu G. & H. Scarf 1963. ‘A Limit Theorem on the Core of an Economy’, International Economic Review, 4, pp. 235–246.MATHCrossRefGoogle Scholar
  34. Ekeland I.1979. Economie Mathématique. Hermann, Paris.Google Scholar
  35. Friedman J. 1977. Oligopoly and the Theory of Games. North-Holland: AmsterdamMATHGoogle Scholar
  36. Heilpern S. 1981. ‘Fuzzy Mappings and Fixed Point Theorems’, Journal of Mathematical Analysis and Applications, 83, pp. 566–569.MathSciNetMATHCrossRefGoogle Scholar
  37. Husseinov F. 1994. ‘Interpretation of Aubin’s Fuzzy Coalitions and Their Extension: Relaxation of Finite Exchange Economy’, Journal of Mathematical Economics, 23, pp. 499–516.MathSciNetCrossRefGoogle Scholar
  38. Kakutani S. 1941. ‘A Generalization of Brouwer’s Fixed Point Theorem’, Duke Mathematical Journal, 8, pp. 416–427.MathSciNetCrossRefGoogle Scholar
  39. Kaufmann A. 1973. Introduction à la Théorie des Sous-Ensembles Flous, Vols 1 & 4. Masson: Paris.MATHGoogle Scholar
  40. Kaleva O. 1984. ‘A Note on Fixed Points for Fuzzy Mappings’, Fuzzy Sets and Systems, 15, pp. 99–100.MathSciNetCrossRefGoogle Scholar
  41. Liu Y. M. 1985. ‘A Note on Compactness in Fuzzy Unit Interval’, Kexue Tong-bao, 25, pp. 33–35.Google Scholar
  42. Mertens J. F. & S. Zamir 1985. ‘Formulation of Bayesian Analysis for Games with Incomplete Information’, International Journal of Game Theory, 14, pp. 1–29.MathSciNetMATHCrossRefGoogle Scholar
  43. Moulin H.1979. ‘Dominance-Solvable Voting Schemes’, Econometrica, 47, pp. 249–269.Google Scholar
  44. Moulin H. & F. Fogelman-Soulie 1979. La Convexité dans les Mathématiques de la Décision. Hermann: Paris.Google Scholar
  45. Ponsard C. 1980. ‘Fuzzy Economics Space’, First World Regional Science Congress, Harvard University, Cambridge, Massachusetts.Google Scholar
  46. Ponsard C. 1986. ‘Foundations of Soft Decision Theory’, in Management Decision Support Systems Using Fuzzy Sets and Possibility Theory, J. Kacprzyk & R. R. Yager Eds, Verlag T. U. V, Rheinland, pp. 27–37.Google Scholar
  47. Ponsard C. 1987. ‘Fuzzy Mathematical Models’, Fuzzy Sets and Systems, 10, pp. 302–313.MathSciNetGoogle Scholar
  48. Simon H. 1964. ‘Rationality’, in A Dictionary of the Social Sciences, J. Gould & W. L. Kolb Eds, Free Press, Glencoe, pp. 573–574.Google Scholar
  49. von Neumann J. & O. Morgenstern 1944. Theory of Games and Economic Behavior; Princeton University Press: Princeton.Google Scholar
  50. Walliser B. 1989. ‘Instrumental Rationality and Cognitive Rationality’, Theory and Decision, 27, pp. 7–36.CrossRefGoogle Scholar
  51. Weber S. 1979. ‘On ε-Cores of Balanced Games’, International Journal of Game Theory, 8, pp. 241–250.MathSciNetMATHCrossRefGoogle Scholar
  52. Wooders M.H. 1983. ‘The ε-Cores of a Large Replica Game’, Journal of Mathematical Economics, 11, pp. 277–300.MathSciNetMATHCrossRefGoogle Scholar
  53. Zimmermann H. J. 1985. Fuzzy Set Theory, and its Applications. Kluwer, Nijhoff Publishing, Boston.Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Antoine Billot

There are no affiliations available

Personalised recommendations