IDEAS home Printed from https://ideas.repec.org/a/kap/theord/v93y2022i4d10.1007_s11238-021-09861-w.html
   My bibliography  Save this article

Representing preorders with injective monotones

Author

Listed:
  • Pedro Hack

    (Institute of Neural Information Processing, Ulm University)

  • Daniel A. Braun

    (Institute of Neural Information Processing, Ulm University)

  • Sebastian Gottwald

    (Institute of Neural Information Processing, Ulm University)

Abstract

We introduce a new class of real-valued monotones in preordered spaces, injective monotones. We show that the class of preorders for which they exist lies in between the class of preorders with strict monotones and preorders with countable multi-utilities, improving upon the known classification of preordered spaces through real-valued monotones. We extend several well-known results for strict monotones (Richter–Peleg functions) to injective monotones, we provide a construction of injective monotones from countable multi-utilities, and relate injective monotones to classic results concerning Debreu denseness and order separability. Along the way, we connect our results to Shannon entropy and the uncertainty preorder, obtaining new insights into how they are related. In particular, we show how injective monotones can be used to generalize some appealing properties of Jaynes’ maximum entropy principle, which is considered a basis for statistical inference and serves as a justification for many regularization techniques that appear throughout machine learning and decision theory.

Suggested Citation

  • Pedro Hack & Daniel A. Braun & Sebastian Gottwald, 2022. "Representing preorders with injective monotones," Theory and Decision, Springer, vol. 93(4), pages 663-690, November.
    • Handle: RePEc:kap:theord:v:93:y:2022:i:4:d:10.1007_s11238-021-09861-w
    DOI: 10.1007/s11238-021-09861-w
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11238-021-09861-w
    File Function: Abstract
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s11238-021-09861-w?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Fabio Maccheroni & Massimo Marinacci & Aldo Rustichini, 2006. "Ambiguity Aversion, Robustness, and the Variational Representation of Preferences," Econometrica, Econometric Society, vol. 74(6), pages 1447-1498, November.
    2. Herden, G., 1989. "On the existence of utility functions," Mathematical Social Sciences, Elsevier, vol. 17(3), pages 297-313, June.
    3. White, D. J., 1980. "Notes in decision theory: Optimality and efficiency II," European Journal of Operational Research, Elsevier, vol. 4(6), pages 426-427, June.
    4. José Carlos R. Alcantud & Gianni Bosi & Magalì Zuanon, 2016. "Richter–Peleg multi-utility representations of preorders," Theory and Decision, Springer, vol. 80(3), pages 443-450, March.
    5. Ok, Efe A., 2002. "Utility Representation of an Incomplete Preference Relation," Journal of Economic Theory, Elsevier, vol. 104(2), pages 429-449, June.
    6. White, D. J., 1980. "Optimality and efficiency I," European Journal of Operational Research, Elsevier, vol. 4(5), pages 346-355, May.
    7. Herden, G., 1989. "On the existence of utility functions ii," Mathematical Social Sciences, Elsevier, vol. 18(2), pages 107-117, October.
    8. Alcantud, José Carlos R. & Bosi, Gianni & Zuanon, Magalì, 2013. "Representations of preorders by strong multi-objective functions," MPRA Paper 52329, University Library of Munich, Germany.
    9. Peleg, Bezalel, 1970. "Utility Functions for Partially Ordered Topological Spaces," Econometrica, Econometric Society, vol. 38(1), pages 93-96, January.
    10. Gianni Bosi & Magalì E. Zuanon, 2017. "Maximal elements of quasi upper semicontinuous preorders on compact spaces," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 5(1), pages 109-117, April.
    11. Mehta, Ghanshyam, 1988. "Some general theorems on the existence of order-preserving functions," Mathematical Social Sciences, Elsevier, vol. 15(2), pages 135-143, April.
    12. Herden, Gerhard & Levin, Vladimir L., 2012. "Utility representation theorems for Debreu separable preorders," Journal of Mathematical Economics, Elsevier, vol. 48(3), pages 148-154.
    13. Kaminski, B., 2007. "On quasi-orderings and multi-objective functions," European Journal of Operational Research, Elsevier, vol. 177(3), pages 1591-1598, March.
    14. Mehta, Ghanshyam, 1977. "Topological Ordered Spaces and Utility Functions," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 18(3), pages 779-782, October.
    15. Sebastian Gottwald & Daniel A Braun, 2020. "The two kinds of free energy and the Bayesian revolution," PLOS Computational Biology, Public Library of Science, vol. 16(12), pages 1-32, December.
    16. Paolo Bevilacqua & Gianni Bosi & Magalì Zuanon, 2018. "Multiobjective Optimization, Scalarization, and Maximal Elements of Preorders," Abstract and Applied Analysis, Hindawi, vol. 2018, pages 1-6, January.
    17. Evren, Özgür & Ok, Efe A., 2011. "On the multi-utility representation of preference relations," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 554-563.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Pedro Hack & Daniel A. Braun & Sebastian Gottwald, 2023. "The classification of preordered spaces in terms of monotones: complexity and optimization," Theory and Decision, Springer, vol. 94(4), pages 693-720, May.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Pedro Hack & Daniel A. Braun & Sebastian Gottwald, 2023. "The classification of preordered spaces in terms of monotones: complexity and optimization," Theory and Decision, Springer, vol. 94(4), pages 693-720, May.
    2. Bosi, Gianni & Herden, Gerhard, 2012. "Continuous multi-utility representations of preorders," Journal of Mathematical Economics, Elsevier, vol. 48(4), pages 212-218.
    3. Yann Rébillé, 2019. "Continuous utility on connected separable topological spaces," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 7(1), pages 147-153, May.
    4. Pivato, Marcus, 2009. "Social choice with approximate interpersonal comparisons of well-being," MPRA Paper 17060, University Library of Munich, Germany.
    5. Alcantud, José Carlos R. & Bosi, Gianni & Zuanon, Magalì, 2013. "Representations of preorders by strong multi-objective functions," MPRA Paper 52329, University Library of Munich, Germany.
    6. Pivato, Marcus, 2010. "Approximate interpersonal comparisons of well-being," MPRA Paper 25224, University Library of Munich, Germany.
    7. Pedro Hack & Daniel A. Braun & Sebastian Gottwald, 2022. "The classification of preordered spaces in terms of monotones: complexity and optimization," Papers 2202.12106, arXiv.org, revised Aug 2022.
    8. Bosi, G. & Mehta, G. B., 2002. "Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof," Journal of Mathematical Economics, Elsevier, vol. 38(3), pages 311-328, November.
    9. José Carlos R. Alcantud & Gianni Bosi & Magalì Zuanon, 2016. "Richter–Peleg multi-utility representations of preorders," Theory and Decision, Springer, vol. 80(3), pages 443-450, March.
    10. Asier Estevan & Roberto Maura & Óscar Valero, 2023. "Quasi-Metrics for Possibility Results: Intergenerational Preferences and Continuity," Mathematics, MDPI, vol. 11(2), pages 1-19, January.
    11. Bosi, Gianni & Isler, Romano, 1995. "Representing preferences with nontransitive indifference by a single real-valued function," Journal of Mathematical Economics, Elsevier, vol. 24(7), pages 621-631.
    12. Gianni Bosi & Magalì Zuanon, 2019. "Upper Semicontinuous Representability of Maximal Elements for Nontransitive Preferences," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 758-765, June.
    13. Evren, Özgür & Ok, Efe A., 2011. "On the multi-utility representation of preference relations," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 554-563.
    14. Gianni Bosi & Asier Estevan & Armajac Raventós-Pujol, 2020. "Topologies for semicontinuous Richter–Peleg multi-utilities," Theory and Decision, Springer, vol. 88(3), pages 457-470, April.
    15. Giarlotta, Alfio & Greco, Salvatore, 2013. "Necessary and possible preference structures," Journal of Mathematical Economics, Elsevier, vol. 49(2), pages 163-172.
    16. Herden, G., 1995. "On some equivalent approaches to Mathematical Utility Theory," Mathematical Social Sciences, Elsevier, vol. 29(1), pages 19-31, February.
    17. Herden, Gerhard & Pallack, Andreas, 2002. "On the continuous analogue of the Szpilrajn Theorem I," Mathematical Social Sciences, Elsevier, vol. 43(2), pages 115-134, March.
    18. Cosimo Munari, 2020. "Multi-utility representations of incomplete preferences induced by set-valued risk measures," Papers 2009.04151, arXiv.org.
    19. Federico Quartieri, 2022. "On the Existence of Greatest Elements and Maximizers," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 375-389, November.
    20. Gerasímou, Georgios, 2010. "Consumer theory with bounded rational preferences," Journal of Mathematical Economics, Elsevier, vol. 46(5), pages 708-714, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:kap:theord:v:93:y:2022:i:4:d:10.1007_s11238-021-09861-w. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.