Part II: Social choice correspondences
15. Analytic preference domains
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.
the construction of social choice correspondences that satisfy the Arrovian axioms. We review their results in this section.
15.1. Euclidean spatial preferences
Recall that E denotes the set of all Euclidean spatial preferences on Rm+. In the spatial preference domain that we consider in this section, each individual is assumed to have a Euclidean spatial preference.
Euclidean Spatial Preference Domain. D ⊆ En.
The assumptions on the agenda domain are made precise in the following definition.
Full-Dimensional Compact Agenda Domain. For all A ∈ A, A is a compact set with a nonempty interior.
If we think of the preferences as belonging to legislators who must choose the quantities of various public goods, an agenda can be interpreted as the set of public goods allocations that are feasible given the resources at the legislators’
disposal. As these resources are varied, we obtain different agendas.
With these domain assumptions, IIF is vacuous.
Lemma 6. If X = Rm+ with m ≥ 2 and a social choice correspondence is defined on a full-dimensional compact agenda domain and a Euclidean spatial preference domain, then it satisfies IIF.
The basic idea of the proof is very simple. A Euclidean spatial preference is completely determined by the location of its ideal point. Further, the ideal point can be identified from a segment of an indifference contour by taking the point of intersection of the lines orthogonal to the indifference surface at two points in the segment. As a consequence, it is not possible to have two profiles coincide on an agenda unless they are identical.
With this result in hand, Le Breton and Weymark (2002) were able to establish the following possibility theorem.
Theorem 26. If X = Rm+ with m ≥ 2, on any full-dimensional compact agenda domain and any Euclidean spatial preference domain, there exist so- cial choice correspondences that satisfy ACA, IIF, SP, and ANON.
The following example was used to establish Theorem 26.
Example 25. ABergson–Samuelson social welfare function is a real-valued function defined onn-tuples of utilities. Let W: Rn →Rbe any continuous, symmetric, Bergson–Samuelson social welfare function, increasing in each of its arguments. For all R ∈ E, a continuous utility function UR is chosen to represent R. Using the Bergson–Samuelson social welfare function W and these representations of the individual preferences, the social welfare function F: D → R is defined by setting,
xF(R)y↔W[UR1(x), . . . , URn(x)]≥W[UR1(y), . . . , URn(y)],
for allR∈ Dand allx, y ∈X. The social choice correspondenceC: A×D → X is defined by letting C(A,R) be the set of best alternatives inAaccording to the social preference F(R). Formally, C(A,R) = B(A, F(R)) for all (A,R) ∈ A × D. Because W and URi, i ∈ N, are continuous functions, F(R) is a continuous ordering. Thus, C is well-defined because each agenda is compact.
By Lemma 6, C satisfies IIF. C satisfies SP because W is an increasing function. Because W is symmetric in its arguments and the same utility function URis used no matter who has the preference R,C satisfies ANON.
Because C is rationalized by the social welfare functionF, it satisfies ACA.
An attractive feature of Example 25 is that it provides a link between Ar- rovian social choice theory and traditional Bergson–Samuelson welfare eco- nomics.99 Because none of the binary agendas are feasible, the social welfare function used in this example does not satisfy IIA, thereby circumventing the social welfare function impossibility theorem for Euclidean spatial pref- erences discussed in Example 12.
15.2. Monotone analytic preferences
Le Breton and Weymark (2002) have also used the construction in Example 25 to establish a possibility theorem for monotone analytic preferences. Let M denote the set of all monotone analytic preferences preferences with no critical points onRm+. Le Breton and Weymark assumed that the preference domain is any subset ofMn. Thus, further restrictions, such as convexity of preferences, can be imposed on the preference domain.
99See also the related discussion in Pazner (1979).
Monotone Analytic Preference Domain. D ⊆ Mn.
As above, each agenda is assumed to be a nonempty compact set with a nonempty interior. An agenda can be interpreted as being the set of feasible allocations of public goods obtainable from the economy’s initial resources given the production possibility sets of the firms. Different agendas are ob- tained by varying the production technologies and/or the resource endow- ments. The compactness of an agenda follows from standard assumptions on firms’ technologies that imply that the aggregate production possibilities set is closed and that only finite amounts of goods may be produced with the economy’s resource endowment. Agendas can also be supposed to be convex and comprehensive, as would be the case if firms’ technologies are convex and exhibit free disposal. It is important that the agendas are compact.
On a noncompact, comprehensive agenda, the Pareto set is typically empty, making it impossible for a social choice correspondence to satisfy WP.
Consider two real-valued monotone analytic functions with no critical points defined on Rm++, with m ≥ 2. Le Breton and Weymark have shown that if these functions are ordinally equivalent on an open subset of Rm++, then they are ordinally equivalent on all of Rm++.100 Because the admissible agendas have nonempty interiors, it then follows that it is impossible for two distinct profiles in Mn to coincide on an agenda. Hence, IIF is vacuous.
Lemma 7. If X = Rm+ with m ≥ 2 and a social choice correspondence is defined on a full-dimensional compact agenda domain and a monotone analytic preference domain, then it satisfies IIF.
Using Lemma 7, the social choice correspondence in Example 25 is easily shown to satisfy all the axioms of Theorem 26 on a full-dimensional compact agenda domain and a monotone analytic preference domain.
Theorem 27. If X = Rm+ with m ≥ 2, on any full-dimensional compact agenda domain and any monotone analytic preference domain, there exist social choice correspondences that satisfy ACA, IIF, SP, and ANON.
As noted by Le Breton and Weymark, it is straightforward to construct a private goods version of Theorem 27. With private goods, this theorem is modified by (i) assuming that individuals are selfish and have preferences for own consumption that are monotone and analytic with no critical points and (ii) replacing ANON with PANON.
100This result is an ordinal version of the Analytic Continuation Principle for monotone analytic functions with no critical points.