Skip to main content
SpringerLink
Log in

Geometry on the Utility Space

  • Conference paper
  • First Online:
Algorithmic Decision Theory (ADT 2015)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9346))

Included in the following conference series:

  • 1260 Accesses

Abstract

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.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Similar content being viewed by others

Notes

  1. 1.

    This is not always the case: for example, Gibbard [7] considers voters with expected utilities over the candidates.

  2. 2.

    The necessary and sufficient condition is that relation \(\le \) is complete, transitive, archimedean and independent of irrelevant alternatives.

  3. 3.

    Technically, this remark proves that \(\mathcal {U}_m\) (with its natural quotient topology) is not a \(T_1\) space [8].

References

  1. Audin, M.: Geometry. Universitext. Springer, Heidelberg (2003)

    Book  MATH  Google Scholar 

  2. Arrow, K.J.: A difficulty in the concept of social welfare. J. Polit. Econ. 58(4), 328–346 (1950)

    Article  Google Scholar 

  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. 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. Fishburn, P.C.: Utility Theory for Decision Making. Wiley, New York (1970)

    Book  Google Scholar 

  6. Fishburn, P.C.: Nonlinear Preference and Utility Theory. Johns Hopkins Series in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (1988)

    MATH  Google Scholar 

  7. Gibbard, A.: Straightforwardness of game forms with lotteries as outcomes. Econometrica 46(3), 595–614 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Guénard, F., Lelièvre, G.: Compléments d’analyse. Compléments d’analyse, E.N.S. 1 (1985)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  10. Kreps, D.M.: A Course in Microeconomic Theory. Princeton University Press, Princeton (1990)

    Book  Google Scholar 

  11. Mallows, C.L.: Non-null ranking models. Biometrika 44(1/2), 114–130 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  13. Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. III. Second edn. Publish or Perish Inc., Wilmington (1979)

    MATH  Google Scholar 

  14. Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. IV. Second edn. Publish or Perish Inc., Wilmington (1979)

    MATH  Google Scholar 

  15. Ulrich, G.: Computer generation of distributions on the m-sphere. Appl. Stat. 33(2), 158–163 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  17. Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)

    MATH  Google Scholar 

  18. Wood, A.T.A.: Simulation of the von mises fisher distribution. Commun. Stat. Simul. Comput. 23(1), 157–164 (1994)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The work presented in this paper has been partially carried out at LINCS (http://www.lincs.fr).

Author information

Authors and Affiliations

  1. Inria, Paris, France

    François Durand

  2. Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), UPEM, UPEC, CNRS, 94010, Créteil, France

    Benoît Kloeckner

  3. Alcatel-Lucent Bell Labs France, Nozay, France

    Fabien Mathieu & Ludovic Noirie

Authors
  1. François Durand
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Benoît Kloeckner
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fabien Mathieu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ludovic Noirie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to François Durand .

Editor information

Editors and Affiliations

  1. University of New South Wales, Kensington, Australia

    Toby Walsh

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Durand, F., Kloeckner, B., Mathieu, F., Noirie, L. (2015). Geometry on the Utility Space. In: Walsh, T. (eds) Algorithmic Decision Theory. ADT 2015. Lecture Notes in Computer Science(), vol 9346. Springer, Cham. https://doi.org/10.1007/978-3-319-23114-3_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-23114-3_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-23113-6

  • Online ISBN: 978-3-319-23114-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation