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Social Choice and Welfare

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13-06-2019 | Original Paper

Arrow on domain conditions: a fruitful road to travel

Authors: Salvador Barberà, Dolors Berga, Bernardo Moreno

Published in: Social Choice and Welfare | Issue 2-3/2020

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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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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.

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

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

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).

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).

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.

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

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

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).

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

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

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Title: Arrow on domain conditions: a fruitful road to travel
Authors: Salvador Barberà
Dolors Berga
Bernardo Moreno
Publication date: 13-06-2019
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 2-3/2020
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-019-01196-4

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