Arrow's theorem for economic environments and effective social preferences

Arrow's impossibility theorem has been proved for the realm of private goods and economic preferences by Border and by Bordes and Le Breton. However, their proofs require the exclusion of the zero vector from the commodity space. This paper proves the impossibility theorem for the entire allocation space and the classical domain of economic preferences by adding effectiveness to Arrow's hypothesis. Social preference is effective if every nonempty compact set contains at least one socially optimal allocation.Financial support from the Social Sciences and Humanities Research Council (Canada) is gratefully acknowledged. The comments of an anonymous referee were much appreciated. The author assumes responsibility for any errors.     

More on independent decisiveness and Arrow's theorem

Denicolò [2, Theorem 1] strengthens Arrow's [1, p. 97] theorem by replacing the independence of irrelevant alternatives (IIA) condition by a strictly weaker one, relational independent decisiveness (RID). It is shown here that RID can be still substantially weakened. Yet, the new condition is equivalent to RID under the weak Pareto principle P and unrestricted domain U. In fact, any condition that can be put in place of IIA in Arrow's theorem must imply RID in the presence of P and U. Incidentally, it is argued that Denicolò's proof of his Theorem 1 contains an imprecision. Received: 7 March 2000/Accepted: 11 December 2000     

Abstention as an escape from Arrow's theorem

There are non-dictatorial social welfare functions satisfying the Pareto principle and Arrow's independence of irrelevant alternatives when voters can abstain. In particular, with just seven voters, the number of dictatorial social welfare functions satisfying Arrow's conditions could be deemed, relative to the total number of social welfare functions satisfying Arrow's conditions, negligible.     

Little and Bergson on Arrow's concept of social welfare

The paper revisits a debate regarding Arrow's concept of social welfare that took place shortly after the publication of his book, Social Choice and Individual Values. It is argued that several points made by Little and Bergson in that debate nearly fifty years ago may still be helpful in understanding Arrow's contribution. The paper interprets the aspect of social welfare emphasized by Little and Bergson and the aspect emphasized by Arrow in terms of two distinct phases of decision-making in a liberal democracy.     

Hansson’s theorem for generalized social welfare functions: an extension

 Hansson (1969) sets forth four conditions satisfied by no generalized social welfare function (GSWF), a mapping from profiles of individual preferences to arbitrary social preference relations. Though transitivity is not imposed on social preferences, one of Hansson’s conditions requires that socially maximal alternatives always exist. Of course, this condition is not satisfied by the majority GSWF. We prove a generalization of Hansson’s theorem that requires the existence of maximal alternatives only in very special cases. Our result applies to the majority GSWF and a large class of other GSWFs that sometimes produce no maximal alternatives. Received: 10 July 1995/Accepted: 4 March 1996     

Bergson social welfare functions in applied welfare analysis

Characterizations of the orderings induced on a set of alternatives by often-used Bergson Social Welfare Functions are provided. The characterizations are particularly useful in applied welfare analysis, because they are formulated entirely in terms of orderings of alternatives, rather than orderings of utilities, as is typically done.Helpful comments from Charles Blackorby, Ake Blomqvist, John Broome, Sam Bucovetsky, Mike Hoy, Glenn MacDonal, John Weymark and three referees are gratefully acknowledged, but the usual disclaimer applies.     

More on Harsanyi's utilitarian cardinal welfare theorem

If individuals and society both obey the expected utility hypothesis and social alternatives are uncertain, then the social utility must be a linear combination of the individual utilities, provided the society is indifferent when all its members are. This result was first proven by Harsanyi [4] who made implicit assumptions in the proof not actually needed for the result (see [5]). This note presents a straightforward proof of Harsanyi's theorem based on a separating hyperplane argument.I wish to thank Stephen Selinger for pointing out Resnick's argument to me and W. A. J. Luxemburg for a useful discussion which simplified the argument     

Generalized median social welfare functions

The article considers the construction of social welfare functions when the set of alternatives is the two-dimensional nonnegative orthant, as would be the case if there are two divisible public goods which can be consumed in any nonnegative quantities. With individual and social preferences required to be linear and strictly monotonic, but otherwise unrestricted, we characterize all of the social welfare functions which satisfy binary independence of irrelevant alternatives and anonymity and which satisfy binary independence of irrelevant alternatives, anonymity, and weak Pareto. These classes of social welfare functions are shown to be formally equivalent to the classes of social choice functions characterized by Moulin in his study of strategy-proof social choice with single-peaked preferences.We are grateful to Charles Blackorby, Hervé Moulin, and an anonymous referee for their comments.     

Social welfare functions and fairness

A definition of a social state is proposed that incorporates the notion of procedural fairness into Harsanyi's (1955) analytical framework. We show that, within the new framework, a Harsanyi-type social welfare function is immune to Diamond's (1967) criticism. Moreover, the resulting social welfare function embodies the notion of procedural fairness held by individual members of the society.     

Some neglected determinants of welfare functions

The welfare concept is central to economics and to some topical problems, requiring interdisciplinary research. Psychology, biology and pedagogics may be important contributors. Economists have overemphasized economic, and among these consumption, aspects. A learning process to reduce unhealthy consumption remains topical. On the production side the diversity of labor types has been neglected. Here a central problem is job choice by maximizing welfare. Variables needed are capabilities required by jobs and personality traits. We ignore the number of independent characteristics. Path analysis with inherited and learnable traits may be extended by data on grandparents and on non-cognitive capabilities. A more precise production function of education is badly lacking. Two methods of measuring welfare functions and the shape of the latter are discussed.     

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.     

Social welfare functions with a reference income

A foundation of social welfare functions is considered with a given reference income (or utility): relative and absolute invariance of the underlying welfare ordering are defined to hold for societies with either all members having incomes below the reference income or all members having incomes above the reference income. These conditions, alongside standard properties of a social preference relation, provide reference income dependent extensions of traditional classes of welfare functions. Dalton’s principle of positive transfers is incorporated, under which relative invariance leads to a class of piecewise (rank-)linear welfare functions, including the class of generalized Gini social welfare functions as a special case. To ensure quasi-concavity a new preference condition is proposed, which has the interpretation of aversion to income dropping below the reference income.     

An impossibility theorem for matching problems

This paper studies the possibility of strategy-proof rules yielding satisfactory solutions to matching problems. Alcalde and Barberá (Econ Theory 4:417–435, 1994) and Sönmez (Econ Des 1:365–380, 1994) show that efficient and individually rational matching rules are manipulable. We pursue the possibility of strategy-proof matching rules by relaxing efficiency to the weaker condition of respect for unanimity. First, we prove that a strategy-proof rule exists that is individually rational and respects unanimity. However, this rule is unreasonable in the sense that mutually best pairs of agents are matched on only rare occasions. In order to explore the possibility of better matching rules, we introduce a natural condition of “respect for 2-unanimity.” Respect for 2-unanimity states that a mutually best pair of agents should be matched, and an agent wishing to being unmatched should be unmatched. Our second result is negative. Secondly, we prove that no strategy-proof rule exists that respects 2-unanimity. This result implies Roth (Math Oper Res 7:617–628, 1962; J Econ Theory 36:277–288, 1985) showing that stable rules are manipulable.     

An international classification system for child welfare programs

A major challenge in child welfare is whether a program (or service) developed and successfully implemented in one jurisdiction, especially another country, will attain the same outcomes for children and families in another jurisdiction? This paper presents the “DCE Classification System” (Defining, Classifying, and Evaluating), a classification system that facilitates cross-jurisdiction comparisons of child and family services. The paper reviews the cross-national research literature in child and family services as well as literature on classification schemes and typologies. As an example of the issues that arise when importing a promising program, we briefly highlight the exporting and importing of family group conferencing. After tracing the history and development of the DCE Classification System, the paper describes the proposed classification scheme, and provides a brief example of how researchers and practitioners can use the classification system for cross-national comparisons of client outcomes and program costs. Finally, we discuss the strengths and weaknesses of this approach, as well as possible benefits for child and family practices.     

Arrow's paradox and mathematical theory of democracy

Two measures, the weight of coalitions and the probability of situations in decision making, are used to characterize the representativeness, i.e. the capability of individuals to represent the social preference. It is proved that there always exists an individual who represents a majority on average, and an individual who represents a majority in most cases. This result is applied to Arrow's social choice model. It follows that there always exists a dictator who is a representative of the society rather than a dictator in a proper sense. After the concept of dictator has been refined to a dictator in a proper sense, Arrow's axioms become consistent. The idea of optimal representation is extended to limited groups of representatives which make decisions on behalf of the whole society. We consider the cabinet (named by analogy with the cabinet of ministers) which consists of a few representatives with delimited domains of competence, and the council which makes decisions by means of voting. It is shown that the representativeness of optimal cabinets and councils tends to 100% of maximally possible values as the number of their members increases, independently of the size of the society. We suggest a geometric interpretation of optimal representatives, cabinets, and councils, based on approximation formulas for the indicators of representativeness derived for the model with a large number of independent individuals. Finally, for cabinets and councils we establish the consistency of different concepts of optimality with respect to different indicators of representativeness. Our consideration is applicable to multicriteria decision making. An appointment of a cabinet or a council corresponds to selecting a few partial criteria. Therefore, the obtained results can be used for reducing the set of partial criteria to a certain sufficient minimum. The concepts of dictator, cabinet, and council can be understood as models of president, government, and parliament, respectively. Thus our results justify reducing social choice to individual choice or small group choice. Although it is in use in all democratic systems, its acceptability is not evident at all. In other words, we justify the demoncraticity of such forms of political power as the president, the parliament, and the government.     

A pedagogical proof of Arrow's Impossibility Theorem

In this note I consider a simple proof of Arrow's Impossibility Theorem (Arrow 1963). I start with the case of three individuals who have preferences on three alternatives. In this special case there are 13³=2197 possible combinations of the three individuals' rational preferences. However, by considering the subset of linear preferences, and employing the full strength of the IIA axiom, I reduce the number of cases necessary to completely describe the SWF to a small number, allowing an elementary proof suitable for most undergraduate students.  This special case conveys the nature of Arrow's result. It is well known that the restriction to three options is not really limiting (any larger set of alternatives can be broken down into triplets, and any inconsistency within a triplet implies an inconsistency on the larger set). However, the general case of n≥3 individuals can be easily considered in this framework, by building on the proof of the simpler case. I hope that a motivated student, having mastered the simple case of three individuals, will find this extension approachable and rewarding.  This approach can be compared with the traditional simple proofs of Barberà (1980); Blau (1972); Denicolò (1996); Fishburn (1970); Kelly (1988); Mueller (1989); Riker and Ordeshook (1973); Sen (1979, 1986); Suzumura (1988), and Taylor (1995). Received: 5 January 1999/Accepted: 10 December 1999     

An overview of welfare reform in michigan

Welfare reform has swept across the nation. However, before Washington was able to develop a unified front, the state of Michigan was well on the way to reforming its own welfare system. As one of the forerunners in the welfare reform movement, Michigan is clearly a state worth watching. The author highlights major welfare reform efforts in Michigan throughout the 1990s and examines these efforts in light of concerns raised by child welfare advocates across the nation