Part II: Social choice correspondences
14. Single-peaked preferences and interval agenda do- mains
We now turn to restricted preference domains. In this section, we suppose that the set of alternatives X is a compact interval [x,x] of¯ R with x < x.¯ The agenda domain consists of all the nonempty closed intervals of X, which we denote by AI. For a preference domain, we consider the unrestricted single-peaked preference domain SPn introduced in Section 3. With these domain assumptions, the Arrow axioms with ND strengthened to ANON are consistent. We describe one class of social choice functions that satisfy all of these axioms, the class of generalized median social choice functions introduced in Moulin (1980), and we present an axiomatization of this class due to Moulin (1984).
We refer to the agenda domain considered in this section as a complete closed interval agenda domain.
Complete Closed Interval Agenda Domain. A =AI.
Note that AI is the set of all nonempty, compact, convex subsets of X.
For allR∈ SP, letπ(R) denote the peak of R. Consider anyA= [a, b]∈ AI and any R∈ SP. The restriction ofR toAis single-peaked with peak at PrAπ(R), the projection of π(R) onA. Thus, B(A, R) = π(R) if π(R) ∈A, B(A, R) =a if π(R)< a, and B(A, R) =b if π(R)> b.
Following Moulin (1984), we assume that choice is single-valued. One interpretation of our problem is that we are to locate a public facility on a street. Each individual has a preferred location, with preference declining monotonically from this ideal location. Not all locations may be feasible, but the ones that are form a closed interval.
Moulin (1984) required the alternative chosen to vary continuously with the endpoints of the feasible set for a given preference profile. This continuity axiom presupposes that the social choice correspondence C is single-valued.
Interval Continuity (IC). For all R ∈ D, the function C(·,R) is continuous at [a, b] with respect to a and b for all x≤a < b≤x.¯
The agenda domainAI is not closed under finite unions, so we cannot ap- peal to Hansson’s Theorem (Theorem 18) to conclude that ACA is equivalent to the existence of a social welfare function that rationalizes the social choice
correspondence C: AI × SPn → X. Nevertheless, Moulin (1984) has shown that this equivalence holds if C is single-valued and satisfies IC. Further, for each profile, the social preference that rationalizes the choice in each agenda can be chosen to be single-peaked.
Lemma 5. If the social choice correspondenceC: AI×SPn →X satisfies SV and IC, then C can be rationalized by a social welfare function F: SPn → SP
if and only if C satisfies ACA.
Proof. Suppose that C satisfies ACA. Consider any R ∈ SPn. Let β = C(X,R) and let Rβ ∈ SP have peak β. Now consider any [a, b] ∈ AI. We want to show that B([a, b], Rβ) = C([a, b],R). If β ∈ A, this follows immediately from ACA. If β ∈ [a, b], we may without loss of generality assume that β < a, which implies that B([a, b], Rβ) = a. Suppose that C([a, b],R) = c > a. By ACA, C([x, b],R) = β. Because C([x, b],R) = β and C([a, b],R) = c, by IC, there must exist an a ∈ (x, a) such that C([a, b],R) = a. ACA then implies that C([a, b],R) = a, a contradiction.
Hence, B([a, b], Rβ) =C([a, b],R) in this case as well.
The reverse implication is straightforward to verify.95
Moulin (1980) introduced the following class ofgeneralized median social choice functions for the agenda domain in which X is the only feasible set.
He extended his definition to the domain AI in Moulin (1984).
Generalized Median Social Choice Function. A social choice functionC: AI× SPn →Xis a generalized median social welfare function if there exists a profile RP = (RPn+1, . . . , RP2n−1)∈ SPn−1 such that for allA ∈ AI and allR∈ SPn,
C(A,R) = PrA median{π(R1), . . . , π(Rn), π(RPn+1), . . . , π(RP2n−1)}.96
As in the construction of a generalized median social welfare function in Example 1, fixed single-peaked preferences forn−1 phantom individuals are specified. For each profile R∈ SPn, the median of the peaks of the 2n−1 real and phantom individuals is determined. For any agenda A∈ AI, the choice
95Our proof of this lemma is based on the proof of Lemma 2.1 in Ehlers (2001). Note that the assumption that individual preferences are single-peaked is not used in the proof.
96For each j ∈ {n+ 1, . . . ,2n−1}, it is only necessary to specify the peak, and not the complete preference ordering. Generalized median social choice functions are called generalized Condorcet-winner social choice functions in Moulin (1984).
setC(A,R) is the projection of this median peak toA. Equivalently,C(A,R) is the best alternative in A for the the individual (either real or phantom) with the median peak. It is straightforward to verify that C(A,R) can also be determined by first projecting all 2n−1 peaks ontoAand then computing the median of these projected peaks.
Recall from Section 3 that the binary relation≤(resp. ≥) onX declares x to be weakly preferred to y if and only if x ≤ y (resp. x ≥ y). If n is odd, half of the phantoms have the preference ≤, and the other half have the preference ≥, then the phantom individuals are irrelevant and C(A,R) is obtained by maximizing the preference of the (real) individual with the median peak. If there are n−k phantoms with preference≤and k−1 with preference ≥, C(A,R) maximizes the preference of the individual with the kth smallest peak.
Suppose theCis a generalized median social choice function. Because the number of phantom individuals is less than the number of real individuals, the median peak in (R,RP) must lie in the interval defined by the smallest and largest peaks in R. As a consequence, C satisfies SP.97 For a fixed profile, the choices from different agendas are determined by maximizing the same preference, so C satisfies ACA. If the profiles R1 and R2 coincide on the agenda A, then the projections of the individual peaks coincide as well.
Hence, C satisfies IIF. It is clear that a generalized median social choice function also satisfies ANON and IC. Moulin (1984) has shown that these five axioms characterize the class of generalized median social choice functions.
To facilitate the comparison of this result with the other theorems in Part II, we state Moulin’s theorem as a theorem about social choice correspondences.
Theorem 25. For any X = [x,x]¯ ⊂R with x <x, if a social choice corre-¯ spondence has a complete closed interval agenda domain and an unrestricted single-peaked preference domain, then it satisfies SV, ACA, IIF, SP, ANON, and IC if and only if it is a generalized median social choice function.
By Lemma 5, SV, ACA, and IC imply that a social choice correspondence C: AI × SPn → X can be rationalized by a social welfare function F:SPn → SP. For k = 1, . . . , n−1, let Rk be the profile in which Rki = ≤ for i = 1, . . . , k and Rki = ≥ for i = k + 1, . . . , n and let βk = π(F(Rk)). The sufficiency part of the proof of Theorem 25 involves showing that C is the generalized median social choice function defined by the profile of phantom
97WP and SP are equivalent for the domainAI× SPn.
preferences ¯RP = (F(R1), . . . , F(Rn−1)). To do this, it is sufficient to show that for all R ∈ SPn, π(F(R)) = median{π(R1), . . . , π(Rn), β1, . . . , βn−1}. See Moulin (1984) for the details of the proof.
Moulin (1984) has also considered the domains of single-plateaued and quasiconcave preferences. A preference R onX = [x,x] is¯ single-plateaued if there exist β1, β2 ∈ X (not necessarily distinct) such that (i) xP y whenever β1 ≥x > y orβ2 ≤x < y and (ii) xIy whenever x, y ∈[β1, β2]. A preference R on X = [x,x] is¯ quasiconcave if there exists a β ∈ X such that (i) xRy whenever β ≥ x > y or β ≤ x < y.98 A single-peaked preference is single- plateaued and a single-plateaued preference is quasiconcave. Moulin has shown that a version of Theorem 25 holds for single-plateaued preferences and that his axioms are incompatible when the preference domain includes all profiles of quasiconcave preferences.
For the domain of Theorem 25, Ehlers (2001) has considered the problem of choosing exactlymalternatives from each agenda, wherem < n(so that it is not possible to always pick everyone’s preferred alternative). Thus, a social alternative consists of m (not necessarily distinct) points in X. Preferences need to be extended from X to the set of subsets of X of cardinality at most m. Ehlers assumes that each individual orders subsets by comparing his or her most-preferrred alternatives in these sets. For m = 2, he has shown that the only social choice correspondence satisfying ACA, IIF, SP, and IC is the extreme peaks social choice correspondence. For each profile, this solution identifies the individuals with the smallest and largest peaks and then maximizes their preferences on each agenda. For m >2, Ehlers has shown that SP and IC are incompatible.