Skip to main content
SpringerLink
Log in

Weakened Transitive Rationality: Invariance of Numerical Representations of Preferences

  • Chapter
  • First Online:
Trends in Mathematical Economics

  • 721 Accesses

Abstract

The ordinal invariance of utility functions representing the same preference is a fundamental issue for solving decision problems using mathematical techniques. As it is well known, this property holds when preferences are complete preorders. However, it is not necessarily verified for numerical representations of weaker concepts of preferences. Moreover, there are cases in which it is not possible to build order-preserving maps between two different numerical representations of the same preference. In this work, we characterize the classes of numerical representations that preserve the order introduced by a given preference in a set of alternatives.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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

References

  • Armstrong, W.E.: A note on the theory of consumers behavior. Oxf. Econ. Pap. 2, 119–122 (1950)

    Google Scholar 

  • Bridges, D.S.: Numerical representation of intransitive preferences on a countable set. J. Econ. Theory 30, 213–217 (1983)

    Article  Google Scholar 

  • Bridges, D.S.: Representating interval - orders by a single real - valued function. J. Econ. Theory. 36, 149–165 (1985)

    Article  Google Scholar 

  • Bridges, D.S., Mehta, G.B.: Representations of Preference Orderings. Lecture Notes in Economics and Mathematical Systems, vol. 422. Springer, Berlin (1995)

    Google Scholar 

  • Candeal, J., Indurain, E., Zudaire, M.: Numerical representability of semiorders. Math. Soc. Sci. 43, 61–77 (2002)

    Article  Google Scholar 

  • Cantor, G.: Contributions to the Founding of the Theory of Transfinite Numbers. Dover, New York (1955)

    Google Scholar 

  • Debreu, G.: Representation of preference ordering by a numerical function. Decis. Process. 3, 159–165 (1954)

    Google Scholar 

  • Estévez, M., Hervés, C.: On the existence of continuous preference orderings without utility representations. J. Math. Econ. 24, 305–309 (1995)

    Article  Google Scholar 

  • Fagin, R., Kumar, R., Mahdian, M., Sivakumar, D., Vee, E.: Comparing partial rank. SIAM J. Discret. Math. 20 (3), 628–648 (2006)

    Article  Google Scholar 

  • Fishburn, P.: Utility Theory of Decision Making. Wiley, New York (1970a)

    Google Scholar 

  • Fishburn, P.: Intransitive indifference with unequal - indifference intervals. J. Math. Psychol. 7, 144–149 (1970b)

    Google Scholar 

  • Fishburn, P.: Interval representation of interval orders and semiorders. J. Math. Psychol. 10, 91–105 (1973)

    Article  Google Scholar 

  • Fishburn, P.: Interval-Orders and Interval Graphs. Wiley, New York (1985)

    Google Scholar 

  • Fishburn, P.: Preference structures and their numerical representations. Theor. Comput. Sci. 217, 359–383 (1999)

    Article  Google Scholar 

  • Herrero, C., Subiza, B.: A characterization of acyclic preferences on countable sets. A Discussion WP - AD 91 - 01. Instituto Valenciano de Investigaciones Econmicas (1991)

    Google Scholar 

  • Jaminson, D.T., Lau, L.J.: Semiorders and the theory of choice. Econometrica 41, 901–912 (1973); A Correct. Econ. 43, 975–977

    Google Scholar 

  • Krantz, D.H., Luce, R.D., Suppes, T., Tversky, A.: Foundations of Measurement Volume 1 (Additive and Polynomial Representations). Academic, New York (1971)

    Google Scholar 

  • Luce, R.D.: Semiorders and a theory of utility discrimination. Econometrica 24, 178–191 (1956)

    Article  Google Scholar 

  • Monteiro, P.K.: Some results on the existence of utility functions on path connected spaces. J. Math. Econ. 16, 147–156 (1987)

    Article  Google Scholar 

  • Nakamura, Y.: Real interval representations. J. Math. Psychol. 46, 140–177 (2002)

    Article  Google Scholar 

  • Oloriz, E., Candeal, J.C., Indurain, E.: Representability of interval orders. J. Econ. Theory 78, 219–227 (1998)

    Article  Google Scholar 

  • Pirlot, M.: Minimal representation of a semiorder. Theory Decis. 28, 109–141 (1990)

    Article  Google Scholar 

  • Pirlot, M., Vinche, P.: Semiorders, Properties, Representations an Applications. Kluwer, Dordrecht (1997)

    Book  Google Scholar 

  • Plata, L.: A numerical representation of acyclic preferences when non comparability and indifference are concepts with different meaning. Econoquantum 1, 17–23 (2004)

    Article  Google Scholar 

  • Rodriguez-Palmero, C.: A representation of acyclic preferences. Econ. Lett. 54, 143–146 (1997)

    Article  Google Scholar 

  • Sen, A.K.: Collective Choice and Social Welfare. Holden Day, San Francisco (1970)

    Google Scholar 

  • Scott, D., Suppes, F.: Foundational aspects of theories of measurement. J. Symb. Log. 23, 113–128 (1958)

    Article  Google Scholar 

  • Subiza, B.: Numerical representations for acyclic preferences. J. Math. Psychol. 38 (4), 467–476 (1994)

    Article  Google Scholar 

  • Suppes, P., Krantz, D., Luce, R.D., Tversky, A.: Foundations of Measurement Volume 2 (Geometrical, Threshold and Probabilistic Representations). Academic, San Diego (1989)

    Google Scholar 

Download references

Acknowledgements

I wish to thank Elvio Accinelli, Erubiel Ordaz, and two anonymous referees for their helpful comments.

Author information

Authors and Affiliations

  1. Facultad de Economía, UASLP, Av. Pintores S/N, Colonia Burocratas del Estado, CP, San Luis Potos, S.L.P., 78263, México

    Leobardo Plata

Authors
  1. Leobardo Plata
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leobardo Plata .

Editor information

Editors and Affiliations

  1. Matemática, Universidade do Porto, Porto, Grande Porto, Portugal

    Alberto A. Pinto

  2. Economía, Universidad Autónoma de San Luis Potosi San Luis Potosí, San Luis Potosí, Mexico

    Elvio Accinelli Gamba

  3. Statistics, Athens University Economics Busines, Athens, Attiki, Greece

    Athanasios N. Yannacopoulos

  4. Ciencias Económicas y Empresariales, Universidade de Vigo, Vigo, Pontevedra, Spain

    Carlos Hervés-Beloso

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Plata, L. (2016). Weakened Transitive Rationality: Invariance of Numerical Representations of Preferences. In: Pinto, A., Accinelli Gamba, E., Yannacopoulos, A., Hervés-Beloso, C. (eds) Trends in Mathematical Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-32543-9_13

Download citation

Publish with us

Policies and ethics

Search

Navigation