Top

Social Choice and Welfare

Published in:

22-05-2018 | Original Paper

Restricting the domain allows for weaker independence

Authors: Justin Kruger, M. Remzi Sanver

Published in: Social Choice and Welfare | Issue 3/2018

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

previous article Increasing discriminatory power in well-being analysis using convex stochastic dominance

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

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

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

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.

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

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).

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

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).

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

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

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

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

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

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

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

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

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

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

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

Title: Restricting the domain allows for weaker independence
Authors: Justin Kruger
M. Remzi Sanver
Publication date: 22-05-2018
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 3/2018
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-018-1129-1

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 3/2018

Multiwinner analogues of the plurality rule: axiomatic and algorithmic perspectives

Cost asymmetry and incomplete information in a volunteer’s dilemma experiment

Income inequality measurement: a fresh look at two old issues

Increasing discriminatory power in well-being analysis using convex stochastic dominance

The role of aggregate information in a binary threshold game

Working time regulation, unequal lifetimes and fairness