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Arrow on domain conditions: a fruitful road to travel

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Abstract

We stress the importance that Arrow attached to studying the role of domain conditions in determining the validity of his impossibility theorem, a subject to which he devoted two chapters of Social Choice and Individual Values. Then we partially survey recent results about the role of domain conditions on the possibility to design satisfactory aggregation rules and social choice functions, as a proof of the continued vitality of this subject, that he pioneered, as he did with so many others.

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Notes

  1. Notice that many other variants of collective rules could be adopted when defining how individual preferences are combined. Domains could be enlarged by not requiring individual preferences to be transitive, and codomains could allow for different expressions of aggregation, including sets of alternatives, choice functions, lotteries, or different sorts of binary relations. We restrict attention to the three basic forms defined above, which are enough to make our points.

  2. We refer the reader to Le Breton and Weymark (2010) for definitions and results related to this section.

  3. Single crossing is equivalent to order restriction, a condition also used in different contexts. See Gans and Smart (1996).

  4. Here again this loose statement should be qualified depending on whether the number of alternatives is odd or even. The precise statement of Moulin (1980) will clarify any possible ambiguity. Similar remarks regarding strategy proofness of the median rule under single crossing preferences are contained in Saporiti and Tohmé (2006).

  5. For precise characterizations and properties of these rules in Cartesian domains, see Barberà et al. (1993), Le Breton and Sen (1995, 1999). The positive results in these papers must be qualified when some potential alternatives cannot be chosen and the range of the function is not a Cartesian product. See Barberà et al. (1998), Barberà et al. (1997b) and Barberà et al. (2005).

  6. In his classical book, Fishburn (1973 page 178) proposed a classification of different conditions that one may impose on social choice functions, and distinguished, among others, between intraprofile and interprofile conditions, depending on whether the requirements on the outcomes of a function could be expressed in reference to one profile at a time, or needed to identify several ones that were somewhat connected. The conditions we present are on domains of definition, rather than on a function’s outcome, and the classification does not directly apply, but there is a parallel. Single peakedness can be checked profile by profile, while the conditions we are about to present refer to combinations of profiles.

  7. A presentation of some of the properties that follow, directed to a computer science audience, is contained in Barberà et al. (2013).

  8. See Le Breton and Zaporozhets (2009) for a related domain condition.

  9. For details about this “almost necessity” result, see Barberà et al. (2010). The argument is not quite a necessity implication, and in that sense is reminiscent of previous definitions of “necessity” in social choice, like the one used in Sen and Pattanaik (1969).

  10. See for example Gibbard (1973), Myerson (1979), Dasgupta et al. (1979) and Harris and Townsend (1981).

  11. A recent attempt in that direction may be found in Barberà et al. (2018, 2019).

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Acknowledgements

We are thankful to the Journal Editor, Mark Fleurbaey, and to two extremely helpful referees, for comments and suggestions. We are also grateful to the editors of this special issue for allowing us to participate in this homage to Kenneth Arrow. S. Barberà acknowledges financial support through Grants ECO2014-53052-P and SGR2014-515, and Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0563). D. Berga and B. Moreno acknowledge the financial support from the Spanish Ministry of Science, Industry and Competitiveness through Grants ECO2016-76255-P and ECO2017-86245-P, respectively, and thank the MOMA network.

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Authors and Affiliations

  1. Departament d’Economia i d’Història Econòmica, MOVE, Universitat Autònoma de Barcelona, and Barcelona GSE, Edifici B, 08193, Bellaterra, Spain

    Salvador Barberà

  2. Departament d’Economia, Universitat de Girona, C/ Universitat de Girona 10, 17003, Girona, Spain

    Dolors Berga

  3. Departamento de Teoría e Historia Económica, Universidad de Málaga, Campus El Ejido, 29071, Málaga, Spain

    Bernardo Moreno

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Correspondence to Salvador Barberà.

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Barberà, S., Berga, D. & Moreno, B. Arrow on domain conditions: a fruitful road to travel. Soc Choice Welf 54, 237–258 (2020). https://doi.org/10.1007/s00355-019-01196-4

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