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Published in: Social Choice and Welfare 4/2023

28-07-2023 | Original Paper

Two impossibility results for social choice under individual indifference intransitivity

Author: Gustav Alexandrie

Published in: Social Choice and Welfare | Issue 4/2023

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Abstract

Due to the imperfect ability of individuals to discriminate between sufficiently similar alternatives, individual indifferences may fail to be transitive. I prove two impossibility theorems for social choice under indifference intransitivity, using axioms that are strictly weaker than Strong Pareto and that have been endorsed (sometimes jointly) in prior work on social choice under indifference intransitivity. The key axiom is Consistency, which states that if bundles are held constant for all but one individual, then society’s preferences must align with those of that individual. Theorem 1 combines Consistency with Indifference Agglomeration, which states that society must be indifferent to combined changes in the bundles of two individuals if it is indifferent to the same changes happening to each individual separately. Theorem 2 combines Consistency with Weak Majority Preference, which states that society must prefer whatever the majority prefers if no one has a preference to the contrary. Given that indifference intransitivity is a necessary condition for the just-noticeable difference (JND) approach to interpersonal utility comparisons, a key implication of the theorems is that any attempt use the JND approach to derive societal preferences must violate at least one of these three axioms.

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Appendix
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Footnotes
1
A substantial literature in psychology shows that stimulus intensity must exceed a threshold, known as the just-noticeable difference, in order to produce detectable variations in sensory experience (see Stern and Johnson (2010); for a brief overview). See Dziewulski (2016) for a recent discussion of methods for eliciting just-noticeable differences.
 
2
Interval orders and semi-transitivity are defined in Sect. 2. For now, it is sufficient to note that both of these properties allow for indifference intransitivity. Note that Theorem 1 and 2 hold whenever societal preferences are interval-orders or semi-transitive (or both).
 
3
Blau (1979); Blair and Pollak (1979); Cato (2015) and Pirlot and Vincke (2013) extend versions of Arrow’s theorem to semi-orders, whereas Salles (2008) extends Sen’s theorem.
 
4
Arrow’s theorem is multi-profiled, but it should be noted that there are single-profiled versions thereof (Kemp and Ng (1976); Parks (1976)).
 
5
Since non-welfarists agree that individual welfare is one determinant of social welfare, my axioms may appeal to non-welfarists as well, at least if a ceteris paribus-clause is added. For discussion of the merits of welfarism, see Sen (1979, 1981) and Ng (1981, 1985), or Kaplow and Shavell (2001, 2004), Fleurbaey et al. (2003) and Weymark (2017).
 
6
Critical assessments of the JND approach are found in e.g., Arrow (1963), Luce and Raiffa (1989) and Sen (2017), whereas supportive assessments are found in e.g., Edgeworth (1881); Goodman and Markowitz (1952); Waldner (1974); Ng (1975); Tännsjö (1989) and Argenziano and Gilboa (2019). For broader surveys of the literature, see Hammond (1991) and Fleurbaey and Hammond (2004).
 
7
Bentham wrote that the “the degree of intensity possessed by that pleasure which is the faintest of any that can be distinguished to be pleasure, may be represented by unity” (quoted in Goldworth 1979). Borda (1781) is also often cited as an early discussion of JNDs.
 
8
Ng (2019) endorses Consistency given that individuals only have preferences about their own bundles (which is what I assume in this paper). My impossibility results also trivially generalize to the variant of Consistency, called Consistency’, which Ng (2019) accepts unconditionally.
 
9
The interval-order property was introduced by Fishburn (1970).
 
10
The semi-order property was introduced by Luce (1956).
 
11
The first representation theorem of this kind was proved by Scott and Suppes (1958). See Svensson (1985) and Ellingsen (1994) for perceptive discussions of the sense in which the utility representation of semi-orders are ‘approximately cardinal’.
 
12
Gilboa and Lapson (1990) and Argenziano and Gilboa (2019) define Consistency as \(x_i\, P_i\, y_i \,\Leftrightarrow \, (z_{-i}, x_i)\,P\,(z_{-i}, y_i)\) for every \(z \in X\) and every \(x_i, y_i \in X_i\). The two formulations are equivalent under completeness (which both they and I assume). I stick to my formulation because it facilitates discussion of the strict preference part, (i), and the indifference part, (ii), of Consistency separately.
 
13
In contrast, given that societal preferences satisfy strict preference transitivity, Consistency and Weak Majority Preference are individually strictly stronger than Weak Pareto.
 
14
This is for instance trivially implied by Ng's (1975) summation theorem (p. 554).
 
15
I owe this point to an anonymous reviewer for Social Choice and Welfare.
 
16
The term “underlying preference” is used by Ng (1975), but the idea is very closely related to Luce's (1956) concept of “induced weak orders”. For an overview of different ways of obtaining induced weak orders from semi-orders, see Lehrer and Wagner (1985).
 
17
Basing social choice on interpersonally comparable utilities violates Arrow's (1950) original Independence axiom, but arguably in a non-problematic way. Note that allowing for interpersonal utility comparisons is not necessarily the only reasonable way out of Arrow’s paradox (see e.g., Fleurbaey and Mongin (2005) and Baccelli (2023), for useful discussion).
 
18
The idea that only noticeable differences matter for well-being has been discussed in the philosophical literature. For example, Bergström (1982) suggests that a change in well-being has “to be noticeable to count for something” (p. 310). See also the similar remarks in Tännsjö (1989).
 
19
As it clear from Theorem 1, this is an implication of Consistency whenever societal preferences satisfy the interval-order property or semi-transitivity.
 
20
Argenziano and Gilboa (2019) also propose an alternative characterization that does not rely on Consistency. However, that characterization requires a very strong anonymity condition that is unlikely to be intuitively appealing for non-utilitarians.
 
21
This perspective coheres well with the literature on methods for inferring welfare from choice data that reveal violations of standard rationality axioms (e.g., Nishimura 2018; Apesteguia and Ballester 2015).
 
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Metadata
Title
Two impossibility results for social choice under individual indifference intransitivity
Author
Gustav Alexandrie
Publication date
28-07-2023
Publisher
Springer Berlin Heidelberg
Published in
Social Choice and Welfare / Issue 4/2023
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI
https://doi.org/10.1007/s00355-023-01478-y

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