Arrovian Social Welfare Functions Research Articles

Simple majority rule and integer programming

In this paper, we use the integer programming approach to mechanism design, first introduced by Sethuraman et al. (2003), and then systematized by Vohra (2011), to reformulate issues concerning the simple majority rule. Our main result consists in showing that, when the number of agents is even, a necessary and sufficient condition for the simple majority rule to be an Arrovian social welfare function is that it is defined on a domain which is echoic with antagonistic preferences. This result is an integer programming simplified version of Theorems 2, 3, and 4 in Inada (1969).

Kalai and Muller’s possibility theorem: a simplified integer programming version

We provide a respecification of an integer programming characterization of Arrovian social welfare functions introduced by Sethuraman et al. (Math Oper Res 28:309–326, 2003). By exploiting this respecification, we give a new and simpler proof of Theorem 2 in Kalai and Muller (J Econ Theory 16:457–469, 1977).

Working Draft of May Condorcet Arrow - Elaboration of Why SWFs with Polychotomous Balloting Procedures Fail to Overcome Condorcetian Objections

This paper elaborates why Arrovian social welfare functions that use polychotomous balloting procedures fail to overcome Condorcetian objections to the ability of SWFs to guarantee meaningful aggregation of individual votes. This paper clarifies a point made by Prasad in Working Draft of May, Condorcet, Arrow: A Social Choice Defense of the Meaningful Aggregation of Votes.

A Measure Of The Trade‐Off Between Responsiveness and Nondictatorship for Arrovian Social Welfare Functions

AbstractIf individuals are never indifferent between distinct alternatives then for any transitive‐valued social welfare function satisfying IIA, and any fraction t, either the set of pairs of alternatives that are socially ordered without consulting more than one individual's preferences comprises at least the fraction t of all pairs, or else the fraction of pairs that have their social ordering determined independently of everyone's s preferences exceeds, or is very close to, 1 −t. The Pareto criterion is not imposed. (There is also a version of this result for the domain of preferences that admit indifference.)

Product filters, acyclicity and Suzumura consistency

In a seminal contribution, Hansson (1976) demonstrates that the collection of decisive coalitions associated with an Arrovian social welfare function forms an ultrafilter. He goes on to show that if transitivity is weakened to quasi-transitivity as the coherence property imposed on a social relation, the set of decisive coalitions is a filter. We examine the notion of decisiveness with acyclical or Suzumura consistent social preferences and without the assumption that the social relation is reflexive and complete. This leads to a new set-theoretic concept applied to product spaces.

Generalized Binary Constitutions and the Whole Set of Arrovian Social Welfare Functions

Arrow's theorem [1963] states that a social welfare function (SWF) that simultaneously satis.es completeness, transitivity, independence of irrelevant alternatives (IIA) and Pareto principle is necessarily dictatorial in the sense that the social decision on any pair of candidates coincides with the strict preference of a fixed individual, the Arrow's dictator. When individual preferences are weak orders, no further description is provided on the social outcome as soon as the Arrow's dictator is indifferent on a pair of candidates. We provide in the present paper another proof of the Arrow's theorem using generalized binary constitutions. Moreover we completely characterize the set of Arrovian SWFs, those are complete and transitive SWFs that satisfy IIA and the Pareto principle.

A STUDY ON LINEAR INEQUALITY REPRESENTATION OF SOCIAL WELFARE FUNCTIONS(<Special Issue>Operations Research for Performance Evaluation)

This paper presents a study on the recently proposed linear inequality representation of Arrovian Social Welfare Functions (ASWFs). We correct and show several sufficient conditions on preference domains for the linear inequalities of the representation to form integral polytopes. We also show that a given probabilistic ASWF induces a real vector satisfying the inequalities.

Dictatorial domains in preference aggregation

We call a domain of preference orderings “dictatorial” if there exists no Arrovian (Pareto optimal, IIA and non-dictatorial) social welfare function defined over that domain. In a finite world of alternatives where indifferences are ruled out, we identify a condition which implies the dictatoriality of a domain. This condition, to which we refer as “being essentially saturated”, is fairly weak. In fact, independent of the number of alternatives, there exists an essentially saturated (hence dictatorial) domain which consists of precisely six orderings. Moreover, this domain exhibits the superdictatoriality property, i.e., every superdomain of it is also dictatorial. Thus, given m alternatives, the ratio of the size of a superdictatorial domain to the size of the full domain may be as small as 6/m!, converging to zero as m increases.

Is abstention an escape from Arrow’s theorem?

In a society of variable size, Quesada (Soc Choice Welfare 25(1): 221–226, 2005) establishes the existence of Arrovian social welfare functions which are not dictatorial. We show that this escape from the Arrovian impossibility collapses when a very mild monotonicity condition is introduced.

A Study on Linear Inequality Representation of Social Welfare Functions

This paper presents a study on the recently proposed linear inequality representation of Arrovian Social Welfare Functions (ASWFs). We first give an alternative proof of the ASWF integer linear inequality representation theorem, and then show several sufficient conditions on preference domains for the linear inequalities of the representation to form integral polytopes. We also show that a given probabilistic ASWF induces a real vector satisfying the inequalities.

On continuity of Arrovian social welfare functions

We study continuity properties of Arrovian social welfare functions in the infinite population framework. We show that continuous welfare functions satisfying unanimity and independence of irrelevant alternatives are dictatorial. Weak anonymity is shown to be incompatible with continuity and unanimity: every continuous weakly anonymous social welfare function must be a constant function.

Anonymous monotonic social welfare functions

This paper presents two results about preference domain conditions that deepen our understanding of anonymous and monotonic Arrovian social welfare functions (ASWFs). We characterize the class of anonymous and monotonic ASWFs on domains without Condorcet triples. This extends and generalizes an earlier characterization (as Generalized Majority Rules) by Moulin (Axioms of Cooperative Decision Making, Cambridge University Press, New York, 1988) for single-peaked domains. We also describe a domain where anonymous and monotonic ASWFs exist only when there are an odd number of agents. This is a counter-example to a claim by Muller (Int. Econ. Rev. 23 (1982) 609), who asserted that the existence of 3-person anonymous and monotonic ASWFs guaranteed the existence of n-person anonymous and monotonic ASWFs for any n>3. Both results build upon the integer programming approach to the study of ASWFs introduced in Sethuraman et al. (Math. Oper. Res. 28 (2003) 309).

Integer Programming and Arrovian Social Welfare Functions

We characterize the class of Arrovian Social Welfare Functions (ASWFs) as integer solutions to a collection of linear inequalities. Many of the classical possibility, impossibility, and characterization results can be derived in a simple and unified way from this integer program. Among the new results we derive is a characterization of preference domains that admit a nondictatorial, neutral ASWF. We also give a polyhedral characterization of all ASWFs on single-peaked domains.

The Theory of Social Choice since 1951 / Die Theorie kollektiver Entscheidungen seit 1951

Summary Starting point for our analysis is Arrow’s famous impossibility result from 1951. Arrow had proved that within an ordinal framework no non-dictatorial social welfare function exists that satisfies the condition of unrestricted domain, the weak Pareto principle and the requirement of independence of irrelevant alternatives. In the first few sections, we discuss what kinds of positive results can be obtained once we relax one or the other of his axioms. The simple majority rule, for example, is an Arrow social welfare function under a domain restriction that is more general than single-peakedness. The Borda rule violates Arrow’s independence axiom but satisfies a weak independence requirement together with some other rather attractive properties. We next look at preference domains that admit the existence of Arrovian social welfare functions, i. e. aggregation rules fulfilling both the Pareto and the independence condition and being nondictatorial. We also characterize domains for non-manipulable voting procedures. The subsequent section analyzes social aggregation functions that allow for an interpersonal comparison of both utility levels and utility gains. Utilitarian rules and the Rawlsian lexicographic maximin principle are discussed in particular. We finally examine the exercise of individual rights within the social choice context, which means that non-utility information is considered as well.

Nondictatorial social welfare functions with different discrimination structures

In societies where agents distinguish imperfectly among the alternatives, a necessary and sufficient condition for the existence of an Arrovian Social Welfare Function is the presence of an agent who only distinguishes two groups of alternatives or of two agents with complementary discrimination patterns. This result is based on the observation that differences in discrimination structures may lead to the absence of free triples, thus providing a way to escape Arrow's impossibility result.

Individual rationality and the concept of social welfare

This paper, written in October 1974, deals with some game aspects of the social choice problem. The question asked is whether there exists a social decision rule satisfying the conditions imposed by Arrow over all the preference profiles that may logically arise under it (in the sense of being compatible with individual rationality). This question is answered in the affirmative. The meaning of this result is that if Arrow's condition of unrestricted domain is modified so as to exclude any profile which contradicts individual rationality, then an Arrovian social welfare function can be shown to exist (subject to the assumption that whenever the social outcome is in doubt, individuals use the maximin criterion in order to choose their voting strategy).