Nash equilibrium in games with incomplete preferences

  • Sophie Bade
Conference paper
Part of the Studies in Economic Theory book series (ECON.THEORY, volume 26)


This paper investigates Nash equilibrium under the possibility that preferences may be incomplete. I characterize the Nash-equilibrium-set of such a game as the union of the Nash-equilibrium-sets of certain derived games with complete preferences. These games with complete preferences can be derived from the original game by a simple linear procedure, provided that preferences admit a concave vector-representation. These theorems extend some results on finite games by Shapley and Aumann. The applicability of the theoretical results is illustrated with examples from oligopolistic theory, where firms are modelled to aim at maximizing both profits and sales (and thus have multiple objectives). Mixed strategy and trembling hand perfect equilibria are also discussed.

Keywords and Phrases

Incomplete preferences Nash equilibrium multiobjective programming Cournot Equilibrium 


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  1. Aumann, R.: Utility theory without the completeness axiom. Econometrica 30, 445–462 (1962)zbMATHCrossRefGoogle Scholar
  2. Bade, S.: Divergent platforms. Mimeo, New York University (2003)Google Scholar
  3. Baumol, W.: Business behavior, value and growth. New York: Macmillan 1959Google Scholar
  4. Bertrand, J.: Review of Cournot’s ‘Recherche sur la theorie mathematique de la richesse’. Journal des Savants, 499–508 (1883)Google Scholar
  5. Bewley, T.: Knightian utility theory: Part 1. Cowles Foundation Discussion Paper 807, 1986Google Scholar
  6. Danan, E.: Revealed cognitive preference theory. mimeo, Universite de Paris 1 (2003)Google Scholar
  7. Ding, X.: Existence of Pareto equilibria for constrained multiobjective games in H-space. Computers and Mathematics with Applications 39, 125–134 (2000)zbMATHGoogle Scholar
  8. Dubra, J., Ok, E.A., Maccheroni, F.: Expected utility theory without the completeness axiom. Journal of Economic Theory 115, 118–133 (2004)CrossRefMathSciNetGoogle Scholar
  9. Edgeworth, F.: Papers relating to political economy. London: Macmillan 1925Google Scholar
  10. Eliaz, K., Ok, E.A.: Indifference or indecisiveness? Choice theoretic foundations of incomplete preferences. Mimeo, New York University (2004)Google Scholar
  11. Fershtman, C., Judd, K.: Equilibrium incentives in oligopoly. American Economic Review 77, 927–940 (1987)Google Scholar
  12. Galbraith, J.: The new industrial state. Boston: Macmillan 1967Google Scholar
  13. Holmstrom, B.: Moral hazard in teams. Bell Journal of Economics 13, 324–340 (1982)CrossRefGoogle Scholar
  14. Kreps, D., Scheinkman, J.: Quantity precommitment and Bertrand competition yield Cournot outcomes. Bell Journal of Economics 14, 326–337 (1983)MathSciNetCrossRefGoogle Scholar
  15. Maskin, E.: The existence of equilibrium with price-setting firms. American Economic Review 76, 382–386 (1986)Google Scholar
  16. Mandler, M.: Compromises between cardinality and ordinality in preference theory and social choice theory. Cowles Foundation Discussion Paper 1322 (2001)Google Scholar
  17. Mandler, M.: Incomplete preferences and rational intransitivity of choice. Games and Economic Behavior (forthcoming)Google Scholar
  18. Marris, R.: The economic theory of managerial capitalism. New York: Macmillan 1964Google Scholar
  19. Ok, E.A.: Utility representation of an incomplete preference relation. Journal of Economic Theory 104, 429–449 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  20. Osborne, M., Pitchik, C.: Price competition in capacity constrained duopoly. Journal of Economic Theory 38, 238–260 (1986)CrossRefMathSciNetGoogle Scholar
  21. Roemer, J.: The democratic political economy of progressive income taxation. Econometrica 67, 1–19 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  22. Roemer, J.: Political competition, theory and applications. Boston: Harvard University Press 2001Google Scholar
  23. Sagi, J.: Anchored preference relations. Mimeo, UC-Berkeley (2003)Google Scholar
  24. Shafer, W., Sonnenschein, H.: Equilibrium in abstract economies without ordered preferences. Journal of Mathematical Economics 2, 345–348 (1975)CrossRefMathSciNetGoogle Scholar
  25. Shapley, L.: Equilibrium points in games with vector payoffs. Naval Research Logistics Quarterly 6, 57–61 (1959)MathSciNetCrossRefGoogle Scholar
  26. Simon, H.: On the concept of organizational goal. Administrative Science Quarterly 9, 1–21 (1964)CrossRefGoogle Scholar
  27. Sklivas, S.: The strategic choice of managerial incentives. Rand Journal of Economics 18, 452–458 (1987)CrossRefGoogle Scholar
  28. Szpilrajn, E.: Sur l’extension de l’ordre partiel. Fundamentae Mathematicae 16, 386–389 (1930)zbMATHGoogle Scholar
  29. Wang, S.: An existence theorem of a Pareto equilibrium. Applied Mathematics Letters 4, 61–63 (1991)zbMATHCrossRefGoogle Scholar
  30. Yu, J., Yuan, G.: The study of Pareto equilibria for multiobjective games by fixed point and Ky Fan mimimax inequality methods. Computers and Mathematics with Applications 35, 17–24 (1998)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Sophie Bade
    • 1
  1. 1.New York UniversityNew YorkUSA

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