When the Players Are Not Expectation Maximizers

  • Amos Fiat
  • Christos Papadimitriou
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6386)


Much of Game Theory, including the Nash equilibrium concept, is based on the assumption that players are expectation maximizers. It is known that if players are risk averse, games may no longer have Nash equilibria [11,6]. We show that

  1. 1

    Under risk aversion (convex risk valuations), and for almost all games, there are no mixed Nash equilibria, and thus either there is a pure equilibrium or there are no equilibria at all, and,

  2. 1

    For a variety of important valuations other than expectation, it is NP-complete to determine if games between such players have a Nash equilibrium.



Nash Equilibrium Expectation Maximizer Mixed Strategy Pure Strategy Prospect Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Allais, M.: Le comportement de l’homme rationnel devant le risque: Critique des postulats et axiomes de l’ecole americaine. Econometrica 21(4), 503–546 (1953), MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bernoulli, D.: Exposition of a new theory on the measurement of risk. Commentaries of the Imperial Academy of Science of Saint Petersburg (1738); Republished Econometrica 22(1), 23–36 (1954),
  3. 3.
    Borel, E.: The theory of play and integral equations with skew symmetric kernels. In: Translated by Leonard J. Savage, University of Chicago, from Comptes Rendus de l’Academie des Sciences, December 19, vol. 173, pp. 1304–1308 (1921); Reprinted in Econometrica 21(1), 97–100 (1953),
  4. 4.
    Chen, X., Deng, X.: Settling the complexity of two-player nash equilibrium. In: FOCS 2006: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 261–272. IEEE Computer Society, Washington (2006)CrossRefGoogle Scholar
  5. 5.
    Conitzer, V., Sandholm, T.: New complexity results about nash equilibria. Games and Economic Behavior 63(2), 621–641 (2008), MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Crawford, V.P.: Equilibrium without independence. Journal of Economic Theory 50(1), 127–154 (1990), MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a nash equilibrium. ACM Commun. 52(2), 89–97 (2009)CrossRefzbMATHGoogle Scholar
  8. 8.
    Friedman, M., Savage, L.J.: The utility analysis of choices involving risk. Journal of Political Economy 56 (1948),
  9. 9.
    Chen, H.-C.: Pure-strategy equilibria with non-expected utility players. Theory and Decision 46, 201–212 (1999), MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kahneman, D., Tversky, A.: Prospect theory: An analysis of decision under risk. Econometrica 47(2), 263–291 (1979), CrossRefzbMATHGoogle Scholar
  11. 11.
    Karni, E., Safra, Z.: Dynamic consistency, revelations in auctions and the structure of preferences. Review of Economic Studies 56(3), 421–433 (1989), MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Machina, M.J.: Choice under uncertainty: Problems solved and unsolved. The Journal of Economic Perspectives 1(1), 121–154 (1987), CrossRefGoogle Scholar
  13. 13.
    Machina, M.J.: Dynamic consistency and non-expected utility models of choice under uncertainty. Journal of Economic Literature 27(4), 1622–1668 (1989), Google Scholar
  14. 14.
    Markowitz, H.: Portfolio Selection: Efficient Diversification in Investments. John Wiley, Chichester (1959)Google Scholar
  15. 15.
    Markowitz, H.: Portfolio selection. The Journal of Finance 7(1), 77–91 (1952), Google Scholar
  16. 16.
    Marschak, J.: Rational behavior, uncertain prospects, and measurable utility. Econometrica 18(2), 111–141 (1950), MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Myerson, R.B., Myerson, R.B.: Nash equilibrium and the history of economic theory. Journal of Economic Literature 37, 1067–1082 (1999)CrossRefGoogle Scholar
  18. 18.
    Nash, J.: Non-cooperative games. The Annals of Mathematics 54(2), 286–295 (1951), MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Neumann, J.v.: Zur Theorie der Gesellschaftsspiele. Math. Ann. 100(1), 295–320 (1928), MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Neumann, J.V., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944), zbMATHGoogle Scholar
  21. 21.
    Papadimitriou, C.H.: On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci. 48(3), 498–532 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Translated by Richard J. Pulskamp, N.B.: Correspondence of nicolas bernoulli concerning the st. petersburg game (1713-1732) (1999),
  23. 23.
    Ritzberger, K.: On games under expected utility with rank dependent probabilities. Theory and Decision 40(1), 1–27 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sharpe, W.F.: A simplified model for portfolio analysis. Management Science 9(2), 277–293 (1963), CrossRefGoogle Scholar
  25. 25.
    Stanford, W.: The limit distribution of pure strategy nash equilibria in symmetric bimatrix games. Mathematics of Operations Research 21(3), 726–733 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Starmer, C.: Developments in non-expected utility theory: The hunt for a descriptive theory of choice under risk. Journal of Economic Literature 38(2), 332–382 (2000), CrossRefGoogle Scholar
  27. 27.
    Varian, H.: A portfolio of nobel laureates: Markowitz, miller and sharpe. The Journal of Economic Perspectives 7(1), 159–169 (1993), CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Amos Fiat
    • 1
  • Christos Papadimitriou
    • 2
  1. 1.School of Computer ScienceTel Aviv UniversityIsrael
  2. 2.Computer Science DivisionUniversity of California at BerkeleyUSA

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