Two impossibility results for social choice under individual indifference intransitivity

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.

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

$$\begin{aligned} \begin{aligned} x_1 \,I_1\, y_1 \,I_1\, z_1 \quad \wedge \quad x_1 \,P_1\, z_1, \\ x_2 \,I_2\, y_2 \,I_2\, z_2 \quad \wedge \quad x_2 \,P_2\, z_2. \end{aligned} \end{aligned}$$

(2)

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:

$$\begin{aligned} (\exists v\in X)[(x_1, y_2)\, P\, v \,I\, (y_1, x_2)], \end{aligned}$$

(3)

$$\begin{aligned} \lnot (\exists v\in X)[(x_1, y_2)\, P\, v \,I\, (y_1, x_2)]. \end{aligned}$$

(4)

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

$$\begin{aligned} (y_1, x_2)\, P\, (y_1, z_2). \end{aligned}$$

(5)

Combining (3) and (5) yields:

$$\begin{aligned} (\exists v\in X)\left[ (x_1, y_2)\, P\, v \,I\, (y_1, x_2)\, P\, (y_1, z_2)\right] . \end{aligned}$$

(6)

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

$$\begin{aligned} (x_1, y_2)\, P\, (z_1, y_2). \end{aligned}$$

(7)

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

$$\begin{aligned} \lnot [(x_1, y_2)\, P\, (z_1, y_2) \,I\, (y_1, x_2)]. \end{aligned}$$

(8)

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

$$\begin{aligned} \begin{aligned} \lnot [(x_1, y_2)\, I\,(y_1, z_2)] \quad \vee \quad \lnot [(z_1, y_2) \,I\, (y_1, x_2)]. \end{aligned} \end{aligned}$$

(9)

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

$$\begin{aligned} \begin{aligned} (x_1, y_2)\, I\,(y_1, y_2)\, I\,(y_1, z_2), \\ (z_1, y_2) \,I\, (y_1, y_2)\,I\, (y_1, x_2). \end{aligned} \end{aligned}$$

(10)

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

$$\begin{aligned} (x_1, x_2)\,P\,(z_1, x_2)\,P\,(z_1, z_2) \end{aligned}$$

(11)

$$\begin{aligned} (z_1, z_1)\,I\,(y_1, z_2)\,I\,(y_1, y_2)\,I\,(x_1, y_2)\,I\,(x_1, x_2). \end{aligned}$$

(12)

It follows from Indifference Agglomeration and (12) that

$$\begin{aligned} (z_1, z_1)\,I\,(y_1, y_2)\,I\,(x_1, x_2). \end{aligned}$$

(13)

Combining (11) and (13) yields:

$$\begin{aligned} (x_1, x_2)\,P\,(z_1, x_2)\,P\,(z_1, z_2)\,I\,(y_1, y_2)\,I\,(x_1, x_2). \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} (z_1, y_2) \,I\, (z_1, x_2), \\ (y_1, z_2) \,I\, (x_1, z_2). \end{aligned} \end{aligned}$$

(14)

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

$$\begin{aligned} \begin{aligned} (x_1, z_2) \,P\, (z_1, y_2), \\ (z_1, x_2) \,P\, (y_1, z_2). \end{aligned} \end{aligned}$$

(15)

Combining (14) and (15) yields:

$$\begin{aligned} (x_1, z_2) \,P\, (z_1, y_2) \,I\, (z_1, x_2) \,P\, (y_1, z_2) \,I\, (x_1, z_2). \end{aligned}$$

(16)

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

$$\begin{aligned} \begin{aligned} (y_1, z_2) \,I\, (y_1, y_2) \,I\, (x_1, y_2). \end{aligned} \end{aligned}$$

(17)

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

$$\begin{aligned} (x_1, y_2) \,P\, (z_1, x_2) \,P\, (y_1, z_2). \end{aligned}$$

(18)

Combining (17) and (18) yields:

$$\begin{aligned} (x_1, y_2) \,P\, (z_1, x_2) \,P\, (y_1, z_2) \,I\, (y_1, y_2) \,I\, (x_1, y_2). \end{aligned}$$

(19)

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 $n\ge 2$.■