Economic Theory

, Volume 35, Issue 2, pp 321–331 | Cite as

Nash equilibria for games in capacities

  • Roman Kozhan
  • Michael Zarichnyi
Research Article


This paper provides a formal generalization of Nash equilibrium for games under Knightian uncertainty. The paper is devoted to counterparts of the results of Glycopantis and Muir (Econ Theory 13:743–751, l999, Econ Theory 16:239–244, 2000) for capacities. We prove that the expected payoff defined as the integral of a payoff function with respect to the tensor product of capacities on compact Hausdorff spaces of pure strategies is continuous if so is the payoff function. We prove also an approximation theorem for Nash equilibria when the expected utility payoff functions are defined on the space of capacities.


Knightian uncertainty Nash equilibrium Capacities 

JEL Classification Numbers

C72 D81 


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  1. Aliprantis D., Glycopantis D., Puzzello D. (2006) The joint continuity of the expected payoff functions. J Math Econ 42(2): 121–130CrossRefGoogle Scholar
  2. Barr M., Wells Ch. (1985) Toposes, triples and theories. Berlin, Springer-VerlagGoogle Scholar
  3. Bauer, C.: Products of convex measures: a Fubini theorem. Working paper No. prod-cap-2003-04, Department of Economics, Economics I, Bayreuth University (2003)Google Scholar
  4. Choquet G. (1953) Theory of capacities. Ann Inst Fourier 5: 131–295Google Scholar
  5. Denneberg D. (1994) Non-additive Measure and Integral. Dordrecht, Kluwer Academic PublishersGoogle Scholar
  6. Dow J., Werlang S. (1994) Nash equilibrium under Knightian uncertainty: breaking down backward induction. J Econ Theory 64: 205–224CrossRefGoogle Scholar
  7. Gilboa I., Schmeidler D. (1989) Maxmin expected utility with non-unique priors. J Math Econ 18: 141–153CrossRefGoogle Scholar
  8. Eichberger J., Kelsey D. (2000) Non-additive beliefs and strategic equilibria. Games Econ Behav 30: 183–215CrossRefGoogle Scholar
  9. Ellsberg D. (1961) Risk, ambiguity and savage axioms. Q J Econ 75: 643–669CrossRefGoogle Scholar
  10. Epstein L. (1999) The definition of uncertainty aversion. Rev Econ Stud 66(3): 579–608CrossRefGoogle Scholar
  11. Glicksberg I.L. (1952) A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points. Proc Am Math Soc 3:170–174CrossRefGoogle Scholar
  12. Glycopantis D., Muir A. (1999) Nash equilibria in ∞-dimensional spaces: an approximation theorem. Econ Theory 13:743–751CrossRefGoogle Scholar
  13. Glycopantis D., Muir A. (2000) Continuiuty of the payoff function. Econ Theory 16: 239–244CrossRefGoogle Scholar
  14. Glycopantis D., Muir, A.: Nash equilibria with Knightian uncertainty; the case of capacities. Preprint (2006)Google Scholar
  15. Nykyforchyn, O.: Probability measures, measurable mappings and convexity: categorical properties. Thesis, Lviv University (1996)Google Scholar
  16. Radul T. (1998) On the functor of order-preserving functionals. Comment Math Univ Carolin 39(3): 609–615Google Scholar
  17. Savage L. (1954) The Foundations of Statistics. New York, Dower PublicationsGoogle Scholar
  18. Schmeidler D. (1989) Subjective probability and expected utility without additivity. Econometrica 57(3): 571–587CrossRefGoogle Scholar
  19. Świrszcz T. (1984) Monadic functors and convexity. Bull Acad Polon Sci Sér Sci Math Astr Phys 22(1): 39–42Google Scholar
  20. Teleiko, A., Zarichnyi, M.: Categorical topology of compact Hausdorff spaces. Math. Studies, Monograph Series, vol 5. Lviv: VNTL Publisher (1999)Google Scholar
  21. Zarichnyi M. (2004) Continuity of the payoff function revisited. Econ Bull 3: 1–4Google Scholar
  22. Zarichnyi, M., Nykyforchyn, O.: Capacity functor in the category of compact spaces (in Russian). Preprint (2006)Google Scholar
  23. Zhou L. (1998) Integral representation of continuous comonotonically additive functionals. Trans Am Math Soc 350(5):1811–1822CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.WFRI, Warwick Business SchoolThe University of WarwickCoventryUK
  2. 2.Department of Mechanics and MathematicsLviv National UniversityLvivUkraine
  3. 3.Institute of MathematicsUniversity of RzeszówRzeszówPoland

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