IDEAS home Printed from https://ideas.repec.org/a/eee/mateco/v45y2009i3-4p241-249.html
   My bibliography  Save this article

On the equivalence of the Arrow impossibility theorem and the Brouwer fixed point theorem when individual preferences are weak orders

Author

Listed:
  • Tanaka, Yasuhito

Abstract

We will show that in the case where there are two individuals and three alternatives (or under the assumption of the free-triple property), and individual preferences are weak orders (which may include indifference relations), the Arrow impossibility theorem [Arrow, K.J., 1963. Social Choice and Individual Values, second ed. Yale University Press] that there exists no binary social choice rule which satisfies the conditions of transitivity, Pareto principle, independence of irrelevant alternatives, and non-existence of dictator is equivalent to the Brouwer fixed point theorem on a 2-dimensional ball (circle). Our study is an application of ideas by Chichilnisky [Chichilnisky, G., 1979. On fixed points and social choice paradoxes. Economics Letters 3, 347-351] to a discrete social choice problem, and also it is in line with the work by Baryshnikov [Baryshnikov, Y., 1993. Unifying impossibility theorems: a topological approach. Advances in Applied Mathematics 14, 404-415].

Suggested Citation

  • Tanaka, Yasuhito, 2009. "On the equivalence of the Arrow impossibility theorem and the Brouwer fixed point theorem when individual preferences are weak orders," Journal of Mathematical Economics, Elsevier, vol. 45(3-4), pages 241-249, March.
    • Handle: RePEc:eee:mateco:v:45:y:2009:i:3-4:p:241-249
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4068(08)00094-3
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    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. Chichilnisky, Graciela, 1982. "The topological equivalence of the pareto condition and the existence of a dictator," Journal of Mathematical Economics, Elsevier, vol. 9(3), pages 223-233, March.
    2. Gleb Koshevoy, 1997. "Homotopy properties of Pareto aggregation rules," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 14(2), pages 295-302.
    3. Lauwers, Luc, 2000. "Topological social choice," Mathematical Social Sciences, Elsevier, vol. 40(1), pages 1-39, July.
    4. Luc Lauwers, 2004. "Topological manipulators form an ultrafilter," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 22(3), pages 437-445, June.
    5. Wilson, Robert, 1972. "Social choice theory without the Pareto Principle," Journal of Economic Theory, Elsevier, vol. 5(3), pages 478-486, December.
    6. Weinberger, Shmuel, 2004. "On the topological social choice model," Journal of Economic Theory, Elsevier, vol. 115(2), pages 377-384, April.
    7. Yuliy M. Baryshnikov, 1997. "Topological and discrete social choice: in a search of a theory," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 14(2), pages 199-209.
    8. Satterthwaite, Mark Allen, 1975. "Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions," Journal of Economic Theory, Elsevier, vol. 10(2), pages 187-217, April.
    9. Chichilnisky, Graciela, 1980. "Social choice and the topology of spaces of preferences," MPRA Paper 8006, University Library of Munich, Germany.
    10. Gibbard, Allan, 1973. "Manipulation of Voting Schemes: A General Result," Econometrica, Econometric Society, vol. 41(4), pages 587-601, July.
    11. Paras Mehta, 1997. "Topological methods in social choice: an overview," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 14(2), pages 233-243.
    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. Muto, Nozomu & Sato, Shin, 2016. "Bounded response of aggregated preferences," Journal of Mathematical Economics, Elsevier, vol. 65(C), pages 1-15.
    2. Guillaume Chèze, 2017. "Topological aggregation, the twin paradox and the No Show paradox," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 48(4), pages 707-715, April.
    3. Rajsbaum, Sergio & Raventós-Pujol, Armajac, 2022. "A Combinatorial Topology Approach to Arrow's Impossibility Theorem," MPRA Paper 112004, University Library of Munich, Germany.

    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. Yasuhito Tanaka, 2005. "A topological approach to the Arrow impossibility theorem when individual preferences are weak orders (forcoming in ``Applied Mathematics and Compuation''(Elsevier))," Public Economics 0506013, University Library of Munich, Germany, revised 17 Jun 2005.
    2. Yasuhito Tanaka, 2005. "A topological proof of Eliaz's unified theorem of social choice theory (forthcoming in "Applied Mathematics and Computation")," Public Economics 0510021, University Library of Munich, Germany, revised 26 Oct 2005.
    3. Tanaka, Yasuhito, 2007. "A topological approach to Wilson's impossibility theorem," Journal of Mathematical Economics, Elsevier, vol. 43(2), pages 184-191, February.
    4. Yasuhito Tanaka, 2005. "On the equivalence of the Arrow impossibility theorem and the Brouwer fixed point theorem (forthcoming in ``Applied Mathematics and Computation''(Elsevier))," Public Economics 0506012, University Library of Munich, Germany, revised 17 Jun 2005.
    5. Lauwers, Luc, 2000. "Topological social choice," Mathematical Social Sciences, Elsevier, vol. 40(1), pages 1-39, July.
    6. Kari Saukkonen, 2007. "Continuity of social choice functions with restricted coalition algebras," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 28(4), pages 637-647, June.
    7. Guillaume Chèze, 2017. "Topological aggregation, the twin paradox and the No Show paradox," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 48(4), pages 707-715, April.
    8. Chichilnisky, Graciela, 1983. "Social choice and game theory: recent results with a topological approach," MPRA Paper 8059, University Library of Munich, Germany.
    9. Miller, Michael K., 2009. "Social choice theory without Pareto: The pivotal voter approach," Mathematical Social Sciences, Elsevier, vol. 58(2), pages 251-255, September.
    10. Salvador Barberà & Dolors Berga & Bernardo Moreno, 2020. "Arrow on domain conditions: a fruitful road to travel," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 54(2), pages 237-258, March.
    11. Uuganbaatar Ninjbat, 2015. "Impossibility theorems are modified and unified," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 45(4), pages 849-866, December.
    12. Antoinette Baujard & Herrade Igersheim, 2007. "Expérimentation du vote par note et du vote par approbation lors de l'élection présidentielle française du 22 avril 2007," Post-Print halshs-00337290, HAL.
    13. Ning Yu, 2015. "A quest for fundamental theorems of social choice," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 44(3), pages 533-548, March.
    14. Elizabeth Maggie Penn, 2015. "Arrow’s Theorem and its descendants," Chapters, in: Jac C. Heckelman & Nicholas R. Miller (ed.), Handbook of Social Choice and Voting, chapter 14, pages 237-262, Edward Elgar Publishing.
    15. Stensholt, Eivind, 2020. "Anomalies of Instant Runoff Voting," Discussion Papers 2020/6, Norwegian School of Economics, Department of Business and Management Science.
    16. Kotaro Suzumura, 2002. "Introduction to social choice and welfare," Temi di discussione (Economic working papers) 442, Bank of Italy, Economic Research and International Relations Area.
    17. Salvador Barberà, 2003. "A Theorem on Preference Aggregation," Working Papers 166, Barcelona School of Economics.
    18. Ju, Biung-Ghi, 2004. "Continuous selections from the Pareto correspondence and non-manipulability in exchange economies," Journal of Mathematical Economics, Elsevier, vol. 40(5), pages 573-592, August.
    19. Stensholt, Eivind, 2019. "MMP-elections and the assembly size," Discussion Papers 2019/15, Norwegian School of Economics, Department of Business and Management Science.
    20. Donald Campbell & Jerry Kelly, 2014. "Universally beneficial manipulation: a characterization," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 43(2), pages 329-355, August.

    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:45:y:2009:i:3-4:p:241-249. 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.