Skip to main content
SpringerLink
Log in

Restricting the domain allows for weaker independence

  • Original Paper
  • Published:
Social Choice and Welfare Aims and scope Submit manuscript

Abstract

Arrow’s classical axiom of independence of irrelevant alternatives may be more descriptively thought of as binary independence. This can then be weakened to ternary independence, quaternary independence, etc. It is known that under the full domain these are not real weakenings as they all collapse into binary independence (except for independence over the whole set of alternatives which is trivially satisfied). Here we investigate whether this still happens under restricted domains. We show that for different domains these different levels of independence may or may not be equivalent. We specify when and to what extent different versions of independence collapse into the same condition.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. Restricting the domain is a typical approach to impossibilities in social choice theory. Gaertner (2001, 2002) has produced elaborate surveys of the literature.

  2. Note \({\subset }\) is used for proper set inclusion; \(X\subset Y\) implies \(\left| {X} \right| < \left| {Y} \right| \).

  3. Note here, as is standard in the literature, “triple” means a subset of alternatives of size three. A triple is said to be free iff every possible ordering over this triple exists within the domain.

  4. Black (1948) provides the classic reference concerning single peaked domains.

  5. Preferences were not removed arbitrarily to create this domain. For the remaining preferences, note that the alternatives on either side of the “peak” are balanced; that these alternatives are interspersed as far as possible. Cf. the idea of equidistantly single-peaked domains described by Ozdemir and Sanver (2007).

  6. Pairwise majority determines the ranking over pairs of alternatives based upon which is preferred by more voters.

  7. A domain is single-crossing if its preferences can be listed \(R^1,R^2,\dots ,R^t\)—or placed on a line—such that, for all x and y, if \(xR^1y\) and \(yR^sx\), then \(xR^iy\) for \(s \le i\le t\). Gans and Smart (1996) provide some economic applications of this property. Rothstein (1991) shows that for any profile on a single-crossing domain there is a representative voter whose (strict) preferences coincide with the (strict) majority relation (though note that he does not use the term single-crossing).

  8. For all alternatives \(x\ne y\), the set of equivalence classes of \(\sim _4^{xy}\) on \(D_{\mathsf {u}}\) is the following refinement of \(\{D_{\mathsf {L}},D_{\mathsf {R}}\}: \qquad \{ \{ R^{{\mathsf {p}}1}, R^{{\mathsf {p}}2}, R^{{\mathsf {p}}3}, R^{{\mathsf {d}}2}, R^{{\mathsf {p}}3} \}, \{ R^{{\mathsf {p}}6}, R^{{\mathsf {p}}7}, R^{{\mathsf {p}}8}, R^{{\mathsf {d}}6}, R^{{\mathsf {p}}7} \}, \{ R^{{\mathsf {p}}4}\}, \{ R^{{\mathsf {p}}5}\}, \{ R^{{\mathsf {d}}4}\}, \{ R^{{\mathsf {d}}5}\} \} \).

References

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

    Article  Google Scholar 

  • Black D (1948) On the rationale of group decision-making. J Polit Econ 56(1):23–34

    Article  Google Scholar 

  • Blau JH (1957) The existence of social welfare functions. Econometrica 25(2):302–313

    Article  Google Scholar 

  • Blau JH (1971) Arrow’s theorem with weak independence. Economica 38(152):413–420

    Article  Google Scholar 

  • Gaertner W (2001) Domain Conditions in Social Choice Theory. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Gaertner W (2002) Domain restrictions. In: Handbook of Social Choice and Welfare, vol. 1. Elsevier, Amsterdam, pp. 131–170

    Chapter  Google Scholar 

  • Gans JS, Smart M (1996) Majority voting with single-crossing preferences. J Public Econ 59(2):219–237

    Article  Google Scholar 

  • May KO (1954) Intransitivity, utility, and the aggregation of preference patterns. Econometrica 22(1):1–13

    Article  Google Scholar 

  • Ozdemir U, Sanver MR (2007) Dictatorial domains in preference aggregation. Soc Choice Welf 28(1):61–76

    Article  Google Scholar 

  • Rothstein P (1991) Representative voter theorems. Public Choice 72(2–3):193–212

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243, LAMSADE, 75016, Paris, France

    Justin Kruger & M. Remzi Sanver

Authors
  1. Justin Kruger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Remzi Sanver
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Justin Kruger.

Additional information

Our work is partly supported by the Projects ANR-14-CE24-0007-01 CoCoRICo-CoDec and IDEX ANR-10-IDEX-0001-02 PSL* MIFID. We also wish to thank two anonymous referees for thoughtful reviews which greatly improved the presentation of this paper.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kruger, J., Sanver, M.R. Restricting the domain allows for weaker independence. Soc Choice Welf 51, 563–575 (2018). https://doi.org/10.1007/s00355-018-1129-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00355-018-1129-1

Search

Navigation