## 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 Maj**ority 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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## Notes

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.

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).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).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).

The interval-order property was introduced by Fishburn (1970).

The semi-order property was introduced by Luce (1956).

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.

In contrast, given that societal preferences satisfy strict preference transitivity, Consistency and Weak Majority Preference are individually strictly stronger than

*Weak Pareto.*This is for instance trivially implied by Ng's (1975) summation theorem (p. 554).

I owe this point to an anonymous reviewer for

*Social Choice and Welfare.*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).

As it clear from Theorem 1, this is an implication of Consistency whenever societal preferences satisfy the interval-order property or semi-transitivity.

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.

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## Acknowledgements

I am grateful to Rossella Argenziano, Frank Cowell, Loren Fryxell, John Firth, Itzhak Gilboa, Johan Gustafsson, Julian Jamison, Luís Mota Freitas, Yew-Kwang Ng, Rossa O’Keeffe-O’Donovan, Paolo Piacquadio, Maxime Cugnon de Sévricourt, Benjamin Tereick, Teru Thomas, Elliott Thornley, and Mattie Toma for helpful discussion and/or comments on earlier drafts of this paper. I am also greatly indebted to an anonymous reviewer for very detailed and constructive comments. The final version of this paper was compiled at Holywell Cemetery in Oxford, where F.Y. Edgeworth - key contributor to the literature on JND approach - is buried.

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## Appendix: proofs

### Appendix: proofs

### 1.1 Proposition 2

The proofs of (a) and (c) are immediate. To prove (b), suppose that Strong Pareto holds and that \(R_i\) is complete for every \(i\in N\). Moreover, suppose that the social preferences satisfy \((z_{-ij}, x_i, x_j)\, I\, (z_{-ij}, y_i, x_j)\, I\, (z_{-ij}, y_i, y_j)\). Given this, Strong Pareto implies that *i* does not have a strict preference between \(x_i\) and \(y_i\) and that *j* does not have a strict preference between \(x_j\) and \(y_j\). By completeness of \(R_i\), it follows that \(x_i\,I_i\,y_i\) and \(x_j\,I_j\,y_j\). By Strong Pareto, this in turn implies that \((z_{-ij}, x_i, x_j)\, I\, (z_{-ij}, y_i, y_j)\). Thus, Strong Pareto implies Indifference Agglomeration. To prove (d) consider the preference profile described by (1). Since neither Consistency, Weak Majority Preference nor Indifference Agglomeration imply anything about the societal preferences over *a*, *b*, *c* and *d*, whereas Strong Pareto implies that \(a\, P\, b\, P\, c\, P\, a\), it follows that the former three axioms do not imply Strong Pareto.■

### 1.2 Theorem 1 and Theorem 2

Before venturing into any of the specific proofs, it is useful to note that if \(N=\{1, 2\}\) and the preferences of each \(i\in \{1, 2\}\) satisfy mild indifference intransitivity, then there exist some \(x_1, y_1, z_1 \in X_1\) and some \(x_2, y_2, z_2 \in X_2\) such that

The individual preference profile described by (2) is useful to invoke in the proofs of each of the theorems below.

#### 1.2.1 Proof of Theorem 1(A)

Suppose that Consistency and Indifference Agglomeration hold and that the societal preferences satisfy the interval-order properties. Moreover, suppose for now that \(N=\{1, 2\}\) and that the preferences of each \(i\in \{1, 2\}\) satisfy mild indifference intransitivity, so that we can say that (2) holds without loss of generality. Note that exactly one of the following two statements must hold:

Suppose first that (3) holds. It follows from Consistency and (2) that

By the interval-order properties of the societal preferences, it follows from (6) that \(\lnot [(x_1, y_2)\, I\,(y_1, z_2)]\).

Now, suppose instead that (4) holds. It follows from Consistency and (2) that

Since (4) is assumed to hold, there is no \(v\in X\) such that society both strictly prefers \((x_1, y_2)\) to *v* and is indifferent between *v* and \((y_1, x_2)\). In the special case where \(v=(z_1, y_2)\), this implies that

It follows from (7) and (8) that \(\lnot [(z_1, y_2) \,I\, (y_1, x_2)]\).

As noted above, either (3) or (4) holds. We have shown that the former disjunct implies that \(\lnot [(x_1, y_2)\, I\,(y_1, z_2)]\), whereas the latter disjunct implies that \(\lnot [(z_1, y_2) \,I\, (y_1, x_2)]\). Thus, we have shown that

Now, our assumptions also imply that (9) does not hold, yielding a contradiction. To see this, note that Consistency and (2) imply that

It follows from Indifference Agglomeration and (10) that both \((x_1, y_2)\, I\,(y_1, z_2)\) and \((z_1, y_2) \,I\, (y_1, x_2)\) hold. However, this contradicts (9). This establishes that Theorem 1(A) holds for \(n=2\). The proof trivially generalizes to any finite \(n\ge 2\).■

#### 1.2.2 Proof of Theorem 1(B)

Suppose that Consistency and Indifference Agglomeration hold and that the societal preferences satisfy the semi-transitivity properties. Moreover, suppose for now that \(N=\{1, 2\}\) and that the preferences of each \(i\in \{1, 2\}\) satisfy mild indifference intransitivity, so that we can say that (2) holds without loss of generality. It follows from Consistency and (2) that

It follows from Indifference Agglomeration and (12) that

Combining (11) and (13) yields:

By the semi-transitivity property, this implies that \((x_1, x_2)\,P\,(y_1, y_2)\,I\,(x_1, x_2)\), which is a contradiction. This establishes that Theorem 1(B) holds for \(n=2\). The proof trivially generalizes to any finite \(n\ge 2\).■

#### 1.2.3 Proof of Theorem 2(A)

Suppose that Consistency and Weak Majority Preference hold and that the societal preferences satisfy the interval-order properties. Moreover, suppose for now that \(N=\{1, 2\}\) and that the preferences of each \(i\in \{1, 2\}\) satisfy mild indifference intransitivity, so that we can say that (2) holds without loss of generality. It follows from Consistency and (2) that

It follows from (2) that individual 1 strictly prefers their bundle in \((x_1, z_2)\) over their bundle in \((z_1, y_2)\), but that individual 2 is indifferent. Therefore, Weak Majority Preference dictates that \((x_1, z_2)\, P\, (z_1, y_2)\). Analogously, it follows from (2) that individual 2 strictly prefers their bundle in \((z_1, x_2)\) over their bundle in \((y_1, z_2)\), but that individual 1 is indifferent. Therefore, Weak Majority Preference dictates that \((z_1, x_2) \,P\, (y_1, z_2)\). This establishes that

Combining (14) and (15) yields:

By the interval-order property, (16) implies that \((x_1, z_2) \,P\, (y_1, z_2) \,I\, (x_1, z_2)\), which is a contradiction. This establishes that Theorem 2(A) holds for \(n=2\). The proof trivially generalizes to any finite \(n\ge 2\).■

#### 1.2.4 Proof of Theorem 2(B)

The proof of Theorem 2(B) is very similar to that of Theorem 2(A). Suppose that Consistency and Weak Majority Preference hold and that the societal preferences satisfy the semi-transitivity properties. Moreover, suppose for now that \(N=\{1, 2\}\) and that the preferences of each \(i\in \{1, 2\}\) satisfy mild indifference intransitivity, so that we can say that (2) holds without loss of generality. It follows from Consistency and (2) that

It follows from (2) that individual 1 strictly prefers their bundle in \((x_1, y_2)\) over their bundle in \((z_1, x_2)\), but that individual 2 is indifferent. Therefore, Weak Majority Preference dictates that \((x_1, y_2) \,P\, (z_1, x_2)\). Analogously, it follows from (2) that individual 2 strictly prefers their bundle in \((z_1, x_2)\) over their bundle in \((y_1, z_2)\), but that individual 1 is indifferent. Therefore, Weak Majority Preference dictates that \((z_1, x_2) \,P\, (y_1, z_2)\). This establishes that

Combining (17) and (18) yields:

By the semi-transitivity property, (19) implies that \((x_1, y_2) \,P\, (y_1, y_2) \,I\, (x_1, y_2)\), which is a contradiction. This establishes that Theorem 2(B) holds for \(n=2\). The proof trivially generalizes to any finite \(n\ge 2\).■

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Alexandrie, G. Two impossibility results for social choice under individual indifference intransitivity.
*Soc Choice Welf* **61**, 919–936 (2023). https://doi.org/10.1007/s00355-023-01478-y

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DOI: https://doi.org/10.1007/s00355-023-01478-y