Skip to main content

Part of the book series: The Handbooks of Fuzzy Sets 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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Abreu D. & A. Rubinstein 1988. ‘The Structure of Nash Equilibrium in Repeated Games with Finite Automata’, Econometrica, 56, pp. 1259–1282.

    Article  MathSciNet  MATH  Google Scholar 

  • Aubin J. P. 1974. ‘Coeur et Valeur des Jeux Flous’, Compte-rendus de l’Académie des Sciences de Paris, 279, pp. 891–894.

    MathSciNet  MATH  Google Scholar 

  • 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 

  • Aubin J. P. 1979. Mathematical Methods in Economics and Game Theory. North-Holland: Amsterdam.

    Google Scholar 

  • Aubin J. P. 1981. ‘Locally Lipschitz Cooperative Games’, Journal of Mathematical Economics, 8, pp. 241–262.

    Article  MathSciNet  MATH  Google Scholar 

  • Aubin J. P. 1986. L’Analyse non Linéaire et Ses Motivations Economiques. Masson: Paris.

    Google Scholar 

  • Aumann R. J. 1964. ‘Values of Market with a Continuum of Traders’, Econometrica, 83, pp. 611–646.

    Google Scholar 

  • Aumann R. J. 1976. ‘Agreeing to Disagree’, The Annals of Statistics, 4, pp. 1236–1239.

    Article  MathSciNet  MATH  Google Scholar 

  • Badard R. 1984. ‘Fixed Point Theorems for Fuzzy Numbers’, Fuzzy Sets and Systems, 13, pp. 291–302.

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Billot A. 1987. Préférence et Utilité Floues. Presses Universitaires de France: Paris.

    Google Scholar 

  • Billot A. 1988. Préférence Imprécise et Equilibres Economiques: Une Analyse Axiomatique. Chs 3 & 4, Ph.D. in Economics, Université de Bourgogne, France.

    Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • Billot A. 1991. ‘Aggregation of Preferences: The Fuzzy Case’, Theory and Decision, 30, pp. 51–93.

    Article  MathSciNet  MATH  Google Scholar 

  • Billot A. 1992. ‘From Fuzzy Set Theory to NonAdditive Probabilities: How Have Economists Reacted?’, Fuzzy Sets and Systems, 49, pp. 75–90.

    Article  MathSciNet  MATH  Google Scholar 

  • Billot A. 1995a. ‘Fuzzy Utility Function: A New Elementary Proof’, Fuzzy Sets and Systems, 74, pp. 271–276.

    Article  MathSciNet  MATH  Google Scholar 

  • Billot A. 1995b. Economic Theory of Fuzzy Equilibria. Springer-Verlag: Berlin, New York.

    Book  MATH  Google Scholar 

  • 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 

  • Billot A.& B. Walliser 1995. ‘A Mixed Knowledge Hierarchy’, CERAS Working-Paper 95-17, Ecole des Ponts et Chaussées, Paris, France.

    Google Scholar 

  • Border K. C. 1985. Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press, Cambridge.

    Book  MATH  Google Scholar 

  • Bose R. K. & D. Sahani 1987. ‘Fuzzy Maapings and Fixed Point Theorems’, Fuzzy Sets and Systems, 21, pp. 53–58.

    Article  MathSciNet  MATH  Google Scholar 

  • Brouwer L. E. J. 1912. ‘Uber Abbildung von Mannigfaltikeiten’, Mathematische Annalen, 71, pp. 97–115.

    Article  MATH  Google Scholar 

  • Butnariu D.1978. ‘Fuzzy Games. A Description of the Concept’, Fuzzy Sets and Systems, 1, pp. 181–192.

    Google Scholar 

  • 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 

  • Butnariu D.1980. ‘Stability and Shapley-Value for n-Person Fuzzy Games’, Fuzzy Sets and; Systems, 7, pp. 63–72.

    Google Scholar 

  • Butnariu D.1982. ‘Fixed Points for Fuzzy Mappings’, Fuzzy Sets and Systems, 14, pp. 191–207.

    Google Scholar 

  • Butnariu D.1985. ‘NonAtomic Fuzzy Measures and Games’, Fuzzy Sets and Systems, 17, pp. 39–52.

    Google Scholar 

  • Butnariu D.1986. ‘Fuzzy Measurability and Integrability’, Journal of Mathematical Analysis and Applications, 117, pp. 385–410.

    Google Scholar 

  • Butnariu D. 1987. ‘Values and Cores of Fuzzy Games with Infinitely Many Players’, International Journal of Game Theory, 16, pp. 43–68.

    Article  MathSciNet  MATH  Google Scholar 

  • Chitra A. & P. V. Subrahmanyam 1987. ‘Fuzzy Sets and Fixed Points’, Journal of Mathematical Analysis and Applications, 124, 584–590.

    Article  MathSciNet  MATH  Google Scholar 

  • Chang C. L. 1968. ‘Fuzzy Topological Spaces’, Journal of Mathematical Analysis and Applications, 17, pp. 182–190.

    Article  Google Scholar 

  • Debreu G.1959. Theory of Value. Wiley: New York.

    Google Scholar 

  • Debreu G. & H. Scarf 1963. ‘A Limit Theorem on the Core of an Economy’, International Economic Review, 4, pp. 235–246.

    Article  MATH  Google Scholar 

  • Ekeland I.1979. Economie Mathématique. Hermann, Paris.

    Google Scholar 

  • Friedman J. 1977. Oligopoly and the Theory of Games. North-Holland: Amsterdam

    MATH  Google Scholar 

  • Heilpern S. 1981. ‘Fuzzy Mappings and Fixed Point Theorems’, Journal of Mathematical Analysis and Applications, 83, pp. 566–569.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Kakutani S. 1941. ‘A Generalization of Brouwer’s Fixed Point Theorem’, Duke Mathematical Journal, 8, pp. 416–427.

    Article  MathSciNet  Google Scholar 

  • Kaufmann A. 1973. Introduction à la Théorie des Sous-Ensembles Flous, Vols 1 & 4. Masson: Paris.

    MATH  Google Scholar 

  • Kaleva O. 1984. ‘A Note on Fixed Points for Fuzzy Mappings’, Fuzzy Sets and Systems, 15, pp. 99–100.

    Article  MathSciNet  Google Scholar 

  • Liu Y. M. 1985. ‘A Note on Compactness in Fuzzy Unit Interval’, Kexue Tong-bao, 25, pp. 33–35.

    Google Scholar 

  • Mertens J. F. & S. Zamir 1985. ‘Formulation of Bayesian Analysis for Games with Incomplete Information’, International Journal of Game Theory, 14, pp. 1–29.

    Article  MathSciNet  MATH  Google Scholar 

  • Moulin H.1979. ‘Dominance-Solvable Voting Schemes’, Econometrica, 47, pp. 249–269.

    Google Scholar 

  • Moulin H. & F. Fogelman-Soulie 1979. La Convexité dans les Mathématiques de la Décision. Hermann: Paris.

    Google Scholar 

  • Ponsard C. 1980. ‘Fuzzy Economics Space’, First World Regional Science Congress, Harvard University, Cambridge, Massachusetts.

    Google Scholar 

  • 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 

  • Ponsard C. 1987. ‘Fuzzy Mathematical Models’, Fuzzy Sets and Systems, 10, pp. 302–313.

    MathSciNet  Google Scholar 

  • 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 

  • von Neumann J. & O. Morgenstern 1944. Theory of Games and Economic Behavior; Princeton University Press: Princeton.

    Google Scholar 

  • Walliser B. 1989. ‘Instrumental Rationality and Cognitive Rationality’, Theory and Decision, 27, pp. 7–36.

    Article  Google Scholar 

  • Weber S. 1979. ‘On ε-Cores of Balanced Games’, International Journal of Game Theory, 8, pp. 241–250.

    Article  MathSciNet  MATH  Google Scholar 

  • Wooders M.H. 1983. ‘The ε-Cores of a Large Replica Game’, Journal of Mathematical Economics, 11, pp. 277–300.

    Article  MathSciNet  MATH  Google Scholar 

  • Zimmermann H. J. 1985. Fuzzy Set Theory, and its Applications. Kluwer, Nijhoff Publishing, Boston.

    Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer Science+Business Media New York

About this chapter

Cite this chapter

Billot, A. (1998). Elements of Fuzzy Game Theory. In: Słowiński, R. (eds) Fuzzy Sets in Decision Analysis, Operations Research and Statistics. The Handbooks of Fuzzy Sets Series, vol 1. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5645-9_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4615-5645-9_5

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7583-8

  • Online ISBN: 978-1-4615-5645-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics