Arrow's Weak Pareto Axiom requires that the social welfare function respects a unanimous strict ranking of pairs of alternatives. Arrow's theorem shows that it is impossible for a social welfare function to satisfy its three axioms if the preference domain is unrestricted.
Saturating preference domains
If each person has the same set of allowed preferences, the preference domain is shared. It implies that the structure of a social welfare function on A depends precisely on the restriction of the preference domain to A.
Examples of saturating preference domains
Each of the curves in I can be considered a subset of the set of alternatives X. 40This proof makes use of the fact that the set of alternatives is unbounded from above.
Topological domain restrictions for public goods
To complete the first part of the proof, we still need to show that this implies the existence of a dictator on O(x). The next part of the proof of Theorem 7 establishes that the same person is the local dictator for all x∈X.
Supersaturating preference domains
A preference domain D on a Cartesian set of alternatives X is selfish if a single preference domain Di on X is selfish for each i∈N. For example, the domain of classical economic preferences for public goods discussed in Example 8 is the domain of supersaturated preferences. The domain of preferences is the domain of classical preferences of private goods, denoted by Cpr.
The induced private preference domain Qi is formally the same as the public preference domain considered in Example 8. For a Cartesian set of alternatives X, if the preference domain D is both selfish and supersaturating, then it is also satiating.60.
Hypersaturating preference domains
An individual preference domain Di is hypersaturating if (a) Di is supersaturated and (b) all trivial pairs {x, y} with respect to Di in X for which ¬(xIiy) for all Ri ∈ Di are separable for i. For example, if xy= (0m,0m), {x, y} is a trivial pair for both individuals for the preference domain considered in example 18. To show that this preference domain is hypersaturating, it is sufficient with Lemma 4 to show that the induced domain of private preference.
The reasoning used to show that Example 19 is hypersaturating also shows that this preference domain is oversaturated.64 Border (1983) has shown that the preference domain in Example 20 is unstable with arrows. Because a hypersaturated preference domain is oversaturated, it follows from Theorem 9 that there exists an individual who is a dictator in trivial pairs.
Topological domain restrictions for private goods
With public goods, if the preference domain is bounded so that everyone always has identical preferences, then no pair or three alternatives are strictly free. However, with private goods, even if everyone has identical preferences for their own consumption, it is relatively easy to construct strict free triples, provided that the common domain of individual induced preferences exhibits sufficient preference diversity. A domain of selfish preferences for private goods D has a common domain of private preferences if Qi = Qj for all i, j ∈ N.
The domain of selfish preferences for private goods D shows the identity of preferences for own consumption if for all R ∈ D Qi = Qj for all i, j ∈ N. Redekop (1993a) showed that the domain of preferences is Arrow inconsistent if the domain has both properties and Q is an almost open subset of Ccsm.
Non-Cartesian sets of alternatives
For σ ∈ Σn, σCn is the set obtained by applying the permutation σ to each of the elements in Cn. For the same reason, preferences are unbounded on any general permutation of the alternatives in Cn. Similar arguments can be used to show that one of the two individuals is decisive for all ordered pairs of distinct alternatives in {x, y, z}.
However, our earlier argument made significant use of the assumption that the set of alternatives is unbounded from above. This example illustrates the importance of the assumption that the set of alternatives is Cartesian for Redekop's impossibility theorem.
Effective social welfare functions
Because the social welfare function is effective on compact sets, we therefore have B(A, R) = {x}, which contradicts the assumption that yRx. By strengthening WP to SP in Theorem 15, we obtain an impossibility theorem when the social welfare function is effectively required only on the compact, comprehensive subsets of X . Because the social welfare function satisfies WP, we can place our search for socially best alternatives for allocations in the Edgeworth box for this economy.
Assume that F is a social welfare function with a classical private goods preference domain that is efficient on E(ω) and satisfies IIA and WP. The requirement that the social welfare function be efficient on compact subsets of X is less demanding than the requirement that social preferences be continuous.
Social choice correspondences
A choice-theoretic version of Arrow’s theorem
For this agenda domain, Arrow (1959) has shown that a social choice correspondence C can be rationalized by a social welfare function if and only if C satisfies ACA. A social choice correspondence whose agenda domain is closed under finite unions can be rationalized by a social welfare function if and only if it satisfies ACA.83. A social choice correspondence is dictatorial if the choice set is always a subset of a single individual's best alternatives on the agenda.
Campbell assumes that a social choice correspondence C is generated from a social welfare function F such that C(A,R) is equal to the set of alternatives that are maximal in A for F(R). It is always clear from the context whether the correspondence of social choice or the axiom of the social welfare function is being used.
Unrestricted preference domains
If |X| ≥ 3, there is no social choice correspondence with the complete finite agenda domain and the unbounded preference domain corresponding to ACA, IIF, WP, and ND. If 3≤ |X|<∞, there is no social choice correspondence with an unbounded preference domain corresponding to KSF, ACA, IIF, WP, and ND. Therefore, by Hansson's Theorem (Theorem 18), the ACA implies that the social choice correspondence can be rationalized by a social welfare function.
Their proof strategy starts by assuming that the social choice correspondence satisfies all the assumptions of Theorem 21 except SND. The non-uniqueness of the rationalizing social welfare function makes it a challenging problem to characterize all correspondences of social choices that satisfy Arrow's axioms in a complete domain of bounded agenda ¯x.
Single-peaked preferences and interval agenda do- mains
Moulin (1980) introduced the following class of generalized median social choice functions for the agenda domain in which X is the only feasible set. A social choice functionC: AI× SPn →Xis a generalized median social welfare function if there exists a profile RP = (RPn+1,. Generalized median social choice functions are called generalized Condorcet winner social choice functions in Moulin (1984).
Moulin (1984) has shown that these five axioms characterize the class of generalized median social choice functions. The sufficiency part of the proof of Theorem 25 involves showing that C is the generalized median social choice function defined by the phantom profile.
Analytic preference domains
For m = 2 he has shown that the only social choice correspondence that satisfies ACA, IIF, SP and IC is the social choice correspondence with extreme peaks. The social choice correspondenceC: A×D → An attractive feature of Example 25 is that it draws a link between Arrovian social choice theory and traditional Bergson-Samuelson welfare economics.99 Because neither of the binary agendas is feasible, the social welfare function used in this example does not meet the demands of society. IIA, which circumvents the impossibility theorem of the social welfare function for Euclidean spatial preferences discussed in Example 12.
On a non-compact, comprehensive agenda, the Pareto set is typically empty, making it impossible for a social choice correspondence to satisfy the WP. 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 monotonic analytic preference domain.
Classical domains of spatial and economic prefer- ences
Donaldson and Weymark (1988) claimed that the Walrasian competitive equilibrium correspondence when the endowment is shared equally satisfies all the Arrow axioms with ND strengthened to PANON on an Edgeworth Box agenda domain and a classical private goods preference domain. This is not possible if the equilibrium allocation is inside the Edgeworth box. This correspondence satisfies all the axioms of Theorem 28 about an Edgeworth box agenda domain and a classical private goods preference domain.
105 The finiteness of the set of alternatives ensures that the Pareto sets in Grether and Plott's construction are not empty. Duggan (1996) has shown that the arrow axioms are unstable when the preference domain includes all spatial preference profiles and the agenda domain is the set of compact convex subsets of the universal set of alternatives X when X is a multidimensional convex subset of a Euclidean space.
Independence of Pareto Irrelevant Alternatives
With the Independence of Pareto Inadequate Alternatives, each set of choices depends only on the restriction of the preference profile to the Pareto set. Rather, suppose that there exists a social correspondence C: A×D →X that satisfies all the assumptions of the theorem. This social welfare function is then shown to satisfy all the axioms of the Arrow social welfare function.
Donaldson and Weymark (1988) and Duggan (1996) investigated the consistency of the axioms in Theorem 32 in convex production economies. This observation was used by Donaldson and Weymark to produce the following example that satisfies all the assumptions of Theorem 34.
Concluding remarks
The critical assumption that allows the application of the local approach is Independence of irrelevant alternatives. In such cases, it is very easy to construct social choice correspondences that satisfy all the other axioms of Arrow. Of Arrow's four axioms of social choice correspondence, Arrow's axiom of choice is the most controversial.
When Arrow's axiom of choice is viewed as a constraint on the consistency of choices made across different agendas, the normative property of choice correspondence is compelling. See also Fleurbaey, Suzumura, and Tadenuma's (2002b) discussion of the relative strength of independence axioms in social welfare function and social choice correspondence frameworks.
The literature reviewed here suggests that pursuing a similar strategy in Arrovian social choice theory may be quite promising. Thus, despite the significant progress that has been made, there is much more to be discovered about Arrovian social choice theory in economic domains. The informative basis for the theory of equitable distribution. 122, Project on Intergenerational Equity, Institute of Economic Research, Hitotsubashi University.
This pour le Doctorat d''Etat en Sciences 'Economiques, Uni- versit'e de Rennes 1. Arrovian social choice over economic domains. Social Welfare Functions for Economics, Unpublished Manuscript, Darwin College, Cambridge University and Department of Economics, Harvard University.
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