IDEAS home Printed from https://ideas.repec.org/a/eee/mateco/v99y2022ics0304406821001634.html
   My bibliography  Save this article

A unified view of the existence of maximals

Author

Listed:
  • Quartieri, Federico

Abstract

The paper examines the conditions for the existence of maximals of a relation on every nonempty compact subset of its ground set. A preliminary analysis shows that the existence of maximals of a Suzumura-consistent relation is implied by the existence of maximals of the right trace of its transitive closure. Building on this fact, various theorems of the literature are unified by identifying a common topological property of their assumptions that concerns the right trace of the transitive closure of the objective relation. Next, a generalization is provided so as to accommodate some cases of interest to economics. Finally, a necessary and sufficient condition is presented for the existence of maximals on every nonempty compact subset of the ground set of a relation.

Suggested Citation

  • Quartieri, Federico, 2022. "A unified view of the existence of maximals," Journal of Mathematical Economics, Elsevier, vol. 99(C).
    • Handle: RePEc:eee:mateco:v:99:y:2022:i:c:s0304406821001634
    DOI: 10.1016/j.jmateco.2021.102609
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304406821001634
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmateco.2021.102609?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. Kazuhiro Hara & Efe A. Ok & Gil Riella, 2019. "Coalitional Expected Multi‐Utility Theory," Econometrica, Econometric Society, vol. 87(3), pages 933-980, May.
    2. Walter Bossert, 2008. "Suzumura Consistency," Studies in Choice and Welfare, in: Prasanta K. Pattanaik & Koichi Tadenuma & Yongsheng Xu & Naoki Yoshihara (ed.), Rational Choice and Social Welfare, pages 159-179, Springer.
    3. Paola Manzini & Marco Mariotti, 2008. "On the Representation of Incomplete Preferences Over Risky Alternatives," Theory and Decision, Springer, vol. 65(4), pages 303-323, December.
    4. Denis Bouyssou & Jean-Paul Doignon, 2020. "Chain Representations of Nested Families of Biorders," Post-Print hal-02458975, HAL.
    5. Lehrer, Ehud & Teper, Roee, 2011. "Justifiable preferences," Journal of Economic Theory, Elsevier, vol. 146(2), pages 762-774, March.
    6. Duggan, John, 2011. "General conditions for the existence of maximal elements via the uncovered set," Journal of Mathematical Economics, Elsevier, vol. 47(6), pages 755-759.
    7. Schmeidler, David, 1969. "Competitive Equilibria in Markets with a Continuum of Traders and Incomplete Preferences," Econometrica, Econometric Society, vol. 37(4), pages 578-585, October.
    8. Amartya Sen, 1997. "Maximization and the Act of Choice," Econometrica, Econometric Society, vol. 65(4), pages 745-780, July.
    9. Juan Dubra & Fabio Maccheroni & Efe A. Ok, 2004. "Expected Utility Without the Completeness Axiom," Yale School of Management Working Papers ysm404, Yale School of Management.
    10. Smith, Tony E, 1974. "On the Existence of Most-Preferred Alternatives," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 15(1), pages 184-194, February.
    11. Campbell, Donald E. & Walker, Mark, 1990. "Maximal elements of weakly continuous relations," Journal of Economic Theory, Elsevier, vol. 50(2), pages 459-464, April.
    12. Paolo Ghirardato & Fabio Maccheroni & Massimo Marinacci & Marciano Siniscalchi, 2003. "A Subjective Spin on Roulette Wheels," Econometrica, Econometric Society, vol. 71(6), pages 1897-1908, November.
    13. Fuad Aleskerov & Denis Bouyssou & Bernard Monjardet, 2007. "Utility Maximization, Choice and Preference," Springer Books, Springer, edition 0, number 978-3-540-34183-3, December.
    14. Bergstrom, Theodore C., 1975. "Maximal elements of acyclic relations on compact sets," Journal of Economic Theory, Elsevier, vol. 10(3), pages 403-404, June.
    15. Simone Cerreia-Vioglio & Efe A. Ok, 2018. "The Rational Core of Preference Relations," Working Papers 632, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    16. Walker, Mark, 1977. "On the existence of maximal elements," Journal of Economic Theory, Elsevier, vol. 16(2), pages 470-474, December.
    17. Dubra, Juan & Maccheroni, Fabio & Ok, Efe A., 2004. "Expected utility theory without the completeness axiom," Journal of Economic Theory, Elsevier, vol. 115(1), pages 118-133, March.
    18. Denis Bouyssou & Thierry Marchant, 2011. "Biorders with Frontier," Post-Print hal-02359814, HAL.
    19. 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.
    20. Banks, Jeffrey S. & Duggan, John & Le Breton, Michel, 2006. "Social choice and electoral competition in the general spatial model," Journal of Economic Theory, Elsevier, vol. 126(1), pages 194-234, January.
    21. Duggan, John, 1999. "A General Extension Theorem for Binary Relations," Journal of Economic Theory, Elsevier, vol. 86(1), pages 1-16, May.
    22. Kukushkin, Nikolai S., 2008. "Maximizing an interval order on compact subsets of its domain," Mathematical Social Sciences, Elsevier, vol. 56(2), pages 195-206, September.
    23. Evren, Özgür, 2014. "Scalarization methods and expected multi-utility representations," Journal of Economic Theory, Elsevier, vol. 151(C), pages 30-63.
    24. Subiza, Begona & Peris, Josep E., 1997. "Numerical representation for lower quasi-continuous preferences," Mathematical Social Sciences, Elsevier, vol. 33(2), pages 149-156, April.
    25. Ok, Efe A., 2002. "Utility Representation of an Incomplete Preference Relation," Journal of Economic Theory, Elsevier, vol. 104(2), pages 429-449, June.
    26. Quartieri, Federico, 2021. "Existence of maximals via right traces," MPRA Paper 107189, University Library of Munich, Germany.
    27. Walter Bossert, 2018. "Suzumura†consistent relations: An overview," International Journal of Economic Theory, The International Society for Economic Theory, vol. 14(1), pages 21-34, March.
    28. 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.
    29. Truman F. Bewley, 1986. "Knightian Decision Theory: Part 1," Cowles Foundation Discussion Papers 807, Cowles Foundation for Research in Economics, Yale University.
    30. Galaabaatar, Tsogbadral & Karni, Edi, 2012. "Expected multi-utility representations," Mathematical Social Sciences, Elsevier, vol. 64(3), pages 242-246.
    31. Rader, Trout, 1972. "Theory of Microeconomics," Elsevier Monographs, Elsevier, edition 1, number 9780125750509.
    32. John Duggan, 2011. "General Conditions for Existence of Maximal Elements via the Uncovered Set," RCER Working Papers 563, University of Rochester - Center for Economic Research (RCER).
    33. Gerasimou, Georgios, 2018. "On the indifference relation in Bewley preferences," Economics Letters, Elsevier, vol. 164(C), pages 24-26.
    34. 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. Gorno, Leandro & Rivello, Alessandro T., 2023. "A maximum theorem for incomplete preferences," Journal of Mathematical Economics, Elsevier, vol. 106(C).
    2. 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.

    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. Quartieri, Federico, 2021. "Existence of maximals via right traces," MPRA Paper 107189, University Library of Munich, Germany.
    2. McCarthy, David & Mikkola, Kalle & Thomas, Teruji, 2021. "Expected utility theory on mixture spaces without the completeness axiom," Journal of Mathematical Economics, Elsevier, vol. 97(C).
    3. Galaabaatar, Tsogbadral & Khan, M. Ali & Uyanık, Metin, 2019. "Completeness and transitivity of preferences on mixture sets," Mathematical Social Sciences, Elsevier, vol. 99(C), pages 49-62.
    4. Gorno, Leandro & Rivello, Alessandro T., 2023. "A maximum theorem for incomplete preferences," Journal of Mathematical Economics, Elsevier, vol. 106(C).
    5. Evren, Özgür, 2014. "Scalarization methods and expected multi-utility representations," Journal of Economic Theory, Elsevier, vol. 151(C), pages 30-63.
    6. 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.
    7. Giarlotta, Alfio & Greco, Salvatore, 2013. "Necessary and possible preference structures," Journal of Mathematical Economics, Elsevier, vol. 49(2), pages 163-172.
    8. McCarthy, David & Mikkola, Kalle & Thomas, Teruji, 2020. "Utilitarianism with and without expected utility," Journal of Mathematical Economics, Elsevier, vol. 87(C), pages 77-113.
    9. David McCarthy & Kalle Mikkola & Teruji Thomas, 2019. "Aggregation for potentially infinite populations without continuity or completeness," Papers 1911.00872, arXiv.org.
    10. M. Ali Khan & Metin Uyanık, 2021. "Topological connectedness and behavioral assumptions on preferences: a two-way relationship," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(2), pages 411-460, March.
    11. Eric Danan, 2010. "Randomization vs. Selection: How to Choose in the Absence of Preference?," Management Science, INFORMS, vol. 56(3), pages 503-518, March.
    12. McCarthy, David & Mikkola, Kalle, 2018. "Continuity and completeness of strongly independent preorders," Mathematical Social Sciences, Elsevier, vol. 93(C), pages 141-145.
    13. Pivato, Marcus, 2013. "Multiutility representations for incomplete difference preorders," Mathematical Social Sciences, Elsevier, vol. 66(3), pages 196-220.
    14. Özgür Evren, 2012. "Scalarization Methods and Expected Multi-Utility Representations," Working Papers w0174, Center for Economic and Financial Research (CEFIR).
    15. Leandro Gorno, 2018. "The structure of incomplete preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 66(1), pages 159-185, July.
    16. Weixuan Xia, 2023. "Optimal Consumption--Investment Problems under Time-Varying Incomplete Preferences," Papers 2312.00266, arXiv.org.
    17. Duggan, John, 2011. "General conditions for the existence of maximal elements via the uncovered set," Journal of Mathematical Economics, Elsevier, vol. 47(6), pages 755-759.
    18. Cosimo Munari, 2020. "Multi-utility representations of incomplete preferences induced by set-valued risk measures," Papers 2009.04151, arXiv.org.
    19. Susumu Cato, 2013. "Quasi-decisiveness, quasi-ultrafilter, and social quasi-orderings," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(1), pages 169-202, June.
    20. Marcus Pivato, 2020. "Subjective expected utility with a spectral state space," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 69(2), pages 249-313, March.

    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:eee:mateco:v:99:y:2022:i:c:s0304406821001634. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/jmateco .

    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.