Complete characterization of functions satisfying the conditions of Arrow’s theorem

Arrow??s theorem implies that a social welfare function satisfying Transitivity, the Weak Pareto Principle (Unanimity), and Independence of Irrelevant Alternatives (IIA) must be dictatorial. When non-strict preferences are also allowed, a dictatorial social welfare function is defined as a function for which there exists a single voter whose strict preferences are followed. This definition allows for many different dictatorial functions, since non-strict preferences of the dictator are not necessarily followed. In particular, we construct examples of dictatorial functions which do not satisfy Transitivity and IIA. Thus Arrow??s theorem, in the case of non-strict preferences, does not provide a complete characterization of all social welfare functions satisfying Transitivity, the Weak Pareto Principle, and IIA. The main results of this article provide such a characterization for Arrow??s theorem, as well as for follow up results by Wilson. In particular, we strengthen Arrow??s and Wilson??s result by giving an exact if and only if condition for a function to satisfy Transitivity and IIA (and the Weak Pareto Principle). Additionally, we derive formulae for the number of functions satisfying these conditions.     

Public goods and Arrovian social choice

A social welfare function satisfying Arrow's independence axiom is constant or authoritarian if it generates continuous and transitive social preferences over the space of allocations of public and private goods, and individual preferences have the classical economic properties. The social welfare function will be oligarchial if it generates continuous and quasitransitive social preferences and satisfies a weak version of the Pareto criterion in addition to the independence axiom.This work was supported by the National Sciences Foundation grant no. SES 9007953. I am grateful to anonymous referees whose suggestions led to several substantial improvements.     

Utility, informed preference, or happiness: Following Harsanyi's argument to its logical conclusion

Harsanyi (1997) argues that, for normative issues, informed preferences should be used, instead of actual preferences or happiness (or welfare). Following his argument allowing him to move from actual to informed preferences to its logical conclusion forces us to use happiness instead. Where informed preferences differ from happiness due to a pure concern for the welfare of others, using the former involves multiple counting. This “concerning effect” (non-affective altruism) differs from and could be on top of the “minding effect” (affective altruism) of being happy seeing or helping others to be happy. The concerning/minding effect should be excluded/included in social decision. Non-affective altruism is shown to exist in a compelling hypothetical example. Just as actual preferences should be discounted due to the effects of ignorance and spurious preferences, informed preferences should also be discounted due to some inborn or acquired tendencies to be irrational, such as placing insufficient weights on the welfare of the future, maximizing our biological fitness instead of our welfare. Harsanyi's old result on utilitarianism is however defended against criticisms in the last decade. Harsanyi (1997) argues, among other things, that in welfare economics and ethics, what are important are people's informed preferences, rather than either their actual preferences (as emphasized by modern economists) or their happiness (as emphasized by early utilitarians). The main purpose of this paper is to argue that, pursuing Harsanyi's argument that allows him to move from actual to informed preferences to its logical conclusion forces us to happiness as the ultimately important thing. The early utilitarians were right after all! Since I personally approve of Harsanyi's basic argument, I regard myself as his follower who becomes more Catholic than the Pope. (It is not denied that, in practice, the practical difficulties and undesirable side-effects of the procedure of using happiness instead of preferences have to be taken into account. Thus, even if we ultimately wish to maximize the aggregate happiness of people, it may be best in practice to maximize their aggregate preferences in most instances. This important consideration will be largely ignored in this paper.) The secondary objective is to give a brief defence of Harsanyi's (1953, 1955) much earlier argument for utilitarianism (social welfare as a sum of individual utilities) that has received some criticisms in the last decade. The argument (e.g. Roemer 1996) that Harsanyi's result is irrelevant to utilitarianism is based on the point that the VNM (von Neumann-Morgenstern) utility is unrelated to the subjective and interpersonally comparable cardinal utility needed for a social welfare function. Harsanyi's position is defended by showing that the two types of utility are the same (apart from an indeterminate zero point for the former that is irrelevant for utilitarianism concerning the same set of people). Received: 29 May 1997 / Accepted: 3 November 1997     

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.     

Positional independence in preference aggregation

If, for strict preferences, a unique choice function (CF) is used to aggregate preferences position-wise then the resulting social welfare function (SWF) is dictatorial. This suggests that the task performed by non-dictatorial SWFs must be “more complex” than just selecting an alternative from a list using a single criterion. This is because the information required by non-dictatorial SWFs to aggregate preferences cannot be compressed into a CF. It is also shown that the attempt to reduce the working of a SWF to the working of a CF involves the adoption of certain positional requirements, whose relationship with the conditions in Arrow's theorem is established. Received: 28 May 2001/Accepted: 25 March 2002 My deepest gratitude to Donald G. Saari, who rescued this paper from the worst fate, and to the referee, who showed the escape route.     

Is a continuous rational social aggregation impossible on continuum spaces?

This paper provides a global topological setting for the social choice theory on continuum spaces of alternatives, in contrast to the local differentiable setting of Chichilnisky. Chichilnisky proved that a rational continuous social choice must be discontinuous in her setting. Our paper revisits her theorem to trace the source of this discontinuity. We find that the discontinuity is irrelevant to social aggregation, per se. The main theorem states that there exist a number of continuous social utility maps which are anonymous and satisfy the Pareto condition. As a corollary, we show that there exist corresponding continuous social welfare functions, if singularity is not separated from regular preferences in social preference topology. This extends the possibility result of Jonnes-Zhang-Simpson on linear preferences, to the general ones. The notion of singularity of preferences, relative to the given mathematical structure of an alternative space, is carefully studied.     

Risky social choice with incomplete or noisy interpersonal comparisons of well-being

In this study, we develop a model of social choice over lotteries, where people’s psychological characteristics are mutable, their preferences may be incomplete, and incomplete or noisy interpersonal comparisons of well-being are possible. Formally, we suppose individual preferences are described by a von Neumann-Morgenstern (vNM) preference order on a space of lotteries over psychophysical states; the social planner must construct a vNM preference order on lotteries over social states. First, we consider a model where the individual vNM preference order is incomplete (so not all interpersonal comparisons are possible). Then, we consider a model where the individual vNM preference order is complete, but unknown to the planner, and thus modelled by a random variable. In both cases, we obtain characterizations of a utilitarian social welfare function.     

Multi-profile intergenerational social choice

In an infinite-horizon setting, Ferejohn and Page showed that any social welfare function satisfying Arrow’s axioms and stationarity must be a dictatorship of the first generation. Packel strengthened this result by proving that no collective choice rule generating complete social preferences can satisfy unlimited domain, weak Pareto and stationarity. We prove that this impossibility survives under a domain restriction and without completeness. We propose an alternative stationarity axiom and show that a social welfare function on a specific domain satisfies this modified version and some standard social choice axioms if and only if it is a chronological dictatorship.     

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     

Index-number tests and the common-scaling social cost-of-living index

For a change in prices, the common-scaling social cost-of-living index is the equal scaling of each individual’s expenditure level needed to restore the level of social welfare to its pre-change value. This index does not, in general, satisfy two standard index-number tests. The reversal test requires the index value for the reverse change to be the reciprocal of the original index. And the circular test requires the product of index values for successive price changes to be equal to the index value for the whole change. We show that both tests are satisfied if and only if the Bergson–Samuelson indirect social-welfare function is homothetic in prices, a condition which does not require individual preferences to be homothetic. If individual preferences are homothetic, however, stronger conditions on the Bergson–Samuelson indirect must be satisfied. Given these results, we ask whether the restrictions are empirically reasonable and find, in the case that individual preferences are not homothetic, that they make little difference to estimates of the index.     

Social choice with analytic preferences

Arrow's axioms for social welfare functions are shown to be inconsistent when the set of alternatives is the nonnegative orthant in a multidimensional Euclidean space and preferences are assumed to be either the set of analytic classical economic preferences or the set of Euclidean spatial preferences. When either of these preference domains is combined with an agenda domain consisting of compact sets with nonempty interiors, strengthened versions of the Arrovian social choice correspondence axioms are shown to be consistent. To help establish the economic possibility theorem, an ordinal version of the Analytic Continuation Principle is developed. Received: 4 July 2000/Accepted: 2 April 2001     

A generalization of Campbell and Kelly’s trade-off theorem

This article considers social choice theory without the Pareto principle. We revisit the trade-off theorem developed by Campbell and Kelly (Econometrica 61:1355–1365, 1993) and generalize their result. By introducing an alternative measure of decisive structure, a dominance relation, we show that if a social welfare function dominates another social welfare function, then the number of pairs of alternatives which social ranking is independently of individual preferences under the former is not more than that under the latter. Moreover, we offer two applications of our main result.     

Social mobility and demand for redistribution in Europe: a comparative analysis

The literature on preferences for redistribution has paid little attention to the effect of social mobility on the demand for redistribution and no systematic test of the hypotheses connecting social mobility and preferences for redistribution has yet been done to date. We use the diagonal reference model to estimate the effect of origin and destination classes on preferences for redistribution in a large sample of European countries using data from the European Social Survey. Our findings are consistent with the logic of acculturation in the sense that newcomers tend to adapt their views to those of the destination class at early stages and that upward and downward mobility do not have distinctive effects on the formation of political preferences. However, even though social origins seem to have a limited impact on preferences for redistribution, the evidence does not support the hypothesis that mobile and non‐mobile individuals are alike. We also find that the effect of social origin on preferences varies largely across countries. The empirical evidence leads to the conclusion that the effect of social origin on preferences for redistribution increases in contexts of strong familism.     

Social Welfare of the Aged Comments on some Overseas developments

I must first make a brief comment on this very broad title. Such a vast topic could not be reviewed in one paper, even if I had at my fingertips all the knowledge it would require. What in fact I aim to do is simply to comment on trends in programmes of social welfare for the aged which I became aware of while on study leave in 1966. My primary object was to observe developments in the field of medical social work, but since the practice of social work in hospitals is concerned to a large and ever-increasing extent with the social needs of the aged, and community endeavours to provide for those needs, the welfare of the aged always has a very central place in the medical social worker's interest.     

Cost-benefit analysis: the mystique versus the reality.

Cost-benefit analysis is promoted as a method for making social policy decisions more rational. A review of its goals, procedures, assumptions, and recent applications exposes technological shortcomings and implicit value preferences. If the method is to be useful, it must be balanced by the judgment of decision makers. Consequently, the power of cost-benefit analysis to improve policy decisions depends on the political context. In a conservative environment this analysis provides a rationalization for disinvesting in social welfare. At such times social reformers would be wise to engage in political action and to challenge cost-benefit analysis.     

The direct social preference relation in the theory of public expenditure

It is shown that if there is a finite number of private goods, a single public good, and the individual preferences satisfy certain monotonicity and separability conditions then there is a unique and complete social preference relation defined on the set of allocations by the requirement that the relation is individualistic in terms of the individual ordinal preferences. This relation, called the direct social preference relation, is derived without imposing interpersonal welfare comparisons and all its ordinal properties are inherited directly from those of the individual preferences. However, an allocation which is maximal in terms of the direct social preference relation on the set of feasible allocations is in general not Pareto optimal and the relation may therefore not be suitable as a guide to optimizing social choice.This version of the paper has benefitted from J. Weymark's insightful and detailed comments. In addition, valuable suggestions have been received from G. C. Archibald, C. Blackorby, D. Donaldson and D. Primont. All remaining errors are the sole responsibility of the author     

Inheritance of Welfare Recipiency: An Intergenerational Study of Social Assistance Recipiency in Postwar Sweden

Intergenerational transmission of welfare dependency has received increasing attention among social scientists, especially in the United States, as greater availability of longitudinal data has shed new light on this issue. It remains unclear, however, to what extent the intergenerational correlation of welfare recipiency observed in the United States reflects or interacts with unobserved variables, the racial composition of the population, and the institutional structure of social policies. This study focuses on Sweden, a country with an ethnically homogenous population and institutional social policy structures that differ from those in the United States. It utilizes an internationally unique longitudinal data set to test hypotheses on the inheritance of welfare benefit recipiency as indicated by reliance on means‐tested social assistance. A clear intergenerational effect is observed. This effect, however, reflects a combination of social assistance in the family of origin, children's school adjustment, and parental criminality. Children who lack this combination of problems do not show signs of intergenerational welfare dependency.     

The flight from universalism

Up to the beginning of the 1990s Sweden had been considered a paragon welfare state in its realisation of universalist principles and an institutional welfare model. This seems to be changing rapidly. Mass unemployment, welfare expenditure cuts and institutional transformation have introduced several selective mechanisms into the Swedish welfare system, adding up to a retreat from universalism. New forms of selectivity can be seen in all layers of the welfare system, both transfer benefits and social security, public personal social services and the relation between state and voluntary organisations. The shifting of burdens from universal social security and insurance-based welfare onto local means tested systems has already meant a restigmatisation of unemployment, as the unemployed lose eligibility for the insurance-based systems, and an increase in the proportion of people who have to rely on poor relief instead of rights-based welfare, and when unemployment has gone up, so have work requirements for benefits. A rising proportion of labour market programmes are now municipally organised obligations instead of state administered rights. Conditioning the right to day care, appraising needs-tested services for the elderly, like home help and care, make personal social services change in the same directions. This may endanger the classical alliance between women and the welfare state.     

The political consequences of contemporary immigration

This article synthesizes research on political outcomes associated with increasing immigration, with an emphasis on cross‐national studies of European countries, where immigration is a relatively newer phenomenon compared to the United States and other traditional immigrant destinations. We begin with explanations of and research on anti‐immigrant sentiment, not a political phenomenon in itself but considered an important precursor to other relevant political attitudes. Next, we review scholarship on the relationship between immigration and support for the welfare state, as well as exclusionary attitudes regarding immigrants' rights to welfare benefits. Then, we review research on immigration and political party preferences, in particular radical right parties, whose platforms often combine anti‐immigration and welfare chauvinistic positions. We conclude by discussing how these processes may ultimately shape social policies, which may in turn influence immigration itself.