Geometry on the Utility Space

  • François DurandEmail author
  • Benoît Kloeckner
  • Fabien Mathieu
  • Ludovic Noirie
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9346)


We study the geometrical properties of the utility space (the space of expected utilities over a finite set of options), which is commonly used to model the preferences of an agent in a situation of uncertainty. We focus on the case where the model is neutral with respect to the available options, i.e. treats them, a priori, as being symmetrical from one another. Specifically, we prove that the only Riemannian metric that respects the geometrical properties and the natural symmetries of the utility space is the round metric. This canonical metric allows to define a uniform probability over the utility space and to naturally generalize the Impartial Culture to a model with expected utilities.


Utility theory Geometry Impartial culture Voting 



The work presented in this paper has been partially carried out at LINCS (


  1. 1.
    Audin, M.: Geometry. Universitext. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Arrow, K.J.: A difficulty in the concept of social welfare. J. Polit. Econ. 58(4), 328–346 (1950)zbMATHCrossRefGoogle Scholar
  3. 3.
    Beltrami, E.: Résolution du problème de reporter les points d’une surface sur un plan, de manière que les lignes géodésiques soient représentée par des lignes droites. Annali di Matematica (1866)Google Scholar
  4. 4.
    Beltrami, E.: Essai d’interprétation de la géométrie noneuclidéenne. Trad. par J. Hoüel. Ann. Sci. École Norm. Sup. 6, 251–288 (1869)Google Scholar
  5. 5.
    Fishburn, P.C.: Utility Theory for Decision Making. Wiley, New York (1970)zbMATHGoogle Scholar
  6. 6.
    Fishburn, P.C.: Nonlinear Preference and Utility Theory. Johns Hopkins Series in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (1988)zbMATHGoogle Scholar
  7. 7.
    Gibbard, A.: Straightforwardness of game forms with lotteries as outcomes. Econometrica 46(3), 595–614 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Guénard, F., Lelièvre, G.: Compléments d’analyse. Compléments d’analyse, E.N.S. 1 (1985)Google Scholar
  9. 9.
    Hammond, P.J.: Interpersonal comparisons of utility: why and how they are and should be made. In: Elster, J., Roemer, J.E. (eds.) Interpersonal Comparisons of Well-Being, pp. 200–254. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar
  10. 10.
    Kreps, D.M.: A Course in Microeconomic Theory. Princeton University Press, Princeton (1990)Google Scholar
  11. 11.
    Mallows, C.L.: Non-null ranking models. Biometrika 44(1/2), 114–130 (1957)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford University Press, Oxford (1995)zbMATHGoogle Scholar
  13. 13.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. III. Second edn. Publish or Perish Inc., Wilmington (1979)zbMATHGoogle Scholar
  14. 14.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. IV. Second edn. Publish or Perish Inc., Wilmington (1979)Google Scholar
  15. 15.
    Ulrich, G.: Computer generation of distributions on the m-sphere. Appl. Stat. 33(2), 158–163 (1984)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Von Neumann, J., Morgenstern, O., Kuhn, H.W., Rubinstein, A.: Theory of Games and Economic Behavior. Commemorative edn. Princeton University Press, Princeton, Princeton Classic Editions (2007)zbMATHGoogle Scholar
  17. 17.
    Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)zbMATHGoogle Scholar
  18. 18.
    Wood, A.T.A.: Simulation of the von mises fisher distribution. Commun. Stat. Simul. Comput. 23(1), 157–164 (1994)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • François Durand
    • 1
    Email author
  • Benoît Kloeckner
    • 2
  • Fabien Mathieu
    • 3
  • Ludovic Noirie
    • 3
  1. 1.InriaParisFrance
  2. 2.Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), UPEM, UPEC, CNRSCréteilFrance
  3. 3.Alcatel-Lucent Bell Labs FranceNozayFrance

Personalised recommendations