Harsanyi's Social Aggregation Theorem and the Weak Pareto Principle

Harsanyi's Social Aggregation Theorem is concerned with the aggregation of individual preferences defined on the set of lotteries generated from a finite set of basic prospects into a social preference. These preferences are assumed to satisfy the expected utility hypothesis and are represented by von Neumann-Morgenstern utility functions. Harsanyi's Theorem says that if Pareto Indifference is satisfied, then the social utility function must be an affine combination of the individual utility functions. This article considers the implications for Harsanyi's Theorem of replacing Pareto Indifference with Weak Pareto.I am grateful to Charles Blackorby, David Donaldson, Philippe Mongin, and an anonymous referee for their comments. The final version of this article was written while I was the Hinkley Visiting Professor at Johns Hopkins University. Research support was provided by the Social Sciences and Humanities Research Council of Canada.     

Further remarks on Harsanyi's social aggregation theorem and the weak Pareto principle

Weak Pareto versions of Harsanyi's Social Aggregation Theorem are established for mixture-preserving utility functions defined on a mixture set of alternatives.     

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     

Measurement theory and the foundations of utilitarianism

This article reconsiders the Harsanyi–Sen debate concerning whether Harsanyi is justified in interpreting his Aggregation and Impartial Observer Theorems as providing axiomatizations of utilitarianism. Sen's criticism and its formalization by Weymark are based on the claim that von Neumann–Morgenstern utility theory is ordinal, whereas Harsanyi's utilitarian conclusions require cardinal utility. Proposals for overcoming Sen's objection that appeal to formal measurement theory are considered. It is argued that one of these proposals due to Broome and Risse rightly points to a feature of expected utility theory that was ignored by Sen and Weymark, but that this proposal does not provide a normatively compelling justification for cardinal utility. The other proposal due to Broome is shown to make use of a strength of preference relation in addition to the axioms of expected utility theory.     

Generalized social welfare functionals: Welfarism,morality and liberty

The concept of a generalized social welfare functional (GSWFL) is introduced to permit the integration of appealing moral principles into the traditional Pareto-inclusive welfarist framework of utilitarianism and welfare economics. Attention is focused on a so-called liberal utilitarian GSWFL which incorporates libertarian rights within the utilitarian calculus in a fashion reminiscent of J. S. Mill's philosophy. It is suggested that welfarist social choice theory may be a far more powerful and flexible tool of political and economic analysis than is commonly thought.Revised version of a paper presented at Summer Public Choice Institute held under the auspices of the Council for Philosophical Studies, Dalhousie University, Halifax, Canada, July 29–August 19, 1984. The Author would particularly like to thank Nick Baigent, Bill Harper, Prasanta Pattanaik, John Rawls, Amartya Sen, Robert Sugden, John Weymark and an anonymous referee for helpful comments. Financial support in the form of Killam and SSHRC Fellowships is gratefully acknowledged     

Acyclic choice and group veto

The Arrow Impossibility Theorem assumes transitivity. Acyclicity is a more appropriate rationality condition for collective decision procedures. Many impossibility theorems for acyclic choice need to assume that there are more alternatives than individuals. In this paper we show that by considering circumstances under which groups (or coalitions) have veto it is possible to prove impossiblity theorems without making such an assumption. Our results extend recent theorems by Blau and Deb, and Blair and Pollak.I would like to thank my research supervisor Amartya Sen, the referee and the editor of this journal for their comments on earlier drafts of this paper.     

Distribution in fixed and variable number problems

Analogies are drawn between analyses of distribution where the number of individuals is fixed and the number of individuals varies (i.e., population theory); the theory of optimal population is fundamentally a problem of distribution. I examine the use of Harsanyi's veil of ignorance to evaluate different population patterns and whether ex ante preferences across uncertain outcomes are a useful source of moral information about outcomes. If we accept the premise that individuals should maximize expected welfare, Parfit's Repugnant Conclusion follows from very weak assumptions.The author wishes to thank John Broome, David Gordon, Gregory Kavka, Daniel Klein, Yew-Kwang Ng, Kenneth Small, Larry Temkin, Carole Uhlaner, and an anonymous referee for useful comments.     

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     

An informationally parsimonious impartial observer theorem

In Harsanyi's impartial observer theorem, an impartial observer determines a social ordering of the lotteries on the set of social alternatives based on a sympathetic but impartial concern for all individuals in society. This ordering is derived from a more primitive ordering on the set of all extended lotteries. An extended lottery is a lottery which determines both the observer's personal identity and the social alternative. We establish a version of Harsanyi's theorem in which the observer is only required to have preferences on the extended lotteries in which there is an equal chance of being any person in society. Received: 19 June 1996 / Accepted: 30 December 1996     

Nash Implementation with Lottery Mechanisms

Consider the problem of exact Nash Implementation of social choice correspondences. Define a lottery mechanism as a mechanism in which the planner can randomize on alternatives out of equilibrium while pure alternatives are always chosen in equilibrium. When preferences over alternatives are strict, we show that Maskin monotonicity (Maskin in Rev Econ stud 66: 23–38, 1999) is both necessary and sufficient for a social choice correspondence to be Nash implementable. We discuss how to relax the assumption of strict preferences. Next, we examine social choice correspondences with private components. Finally, we apply our method to the issue of voluntary implementation (Jackon and Palfrey in J Econ Theory 98: 1–25, 2001).I thank Toyo Sakai for his comments on a previous draft. I also thank two anonymous referees and an editor of this journal for helpful comments that improved this paper. A previous version circulated as “A note on Maskin monotonicity”. After the results presented here were obtained, I became aware of a new unpublished paper by Benoit and Ok (2004). The result of Theorem 2 and the discussion that follows is partially similar to their Theorem 1.     

The relation between monotonicity and strategy-proofness

The Muller–Satterthwaite Theorem (J Econ Theory 14:412–418, 1977) establishes the equivalence between Maskin monotonicity and strategy-proofness, two cornerstone conditions for the decentralization of social choice rules. We consider a general model that covers public goods economies as in Muller–Satterthwaite (J Econ Theory 14:412–418, 1977) as well as private goods economies. For private goods economies, we use a weaker condition than Maskin monotonicity that we call unilateral monotonicity. We introduce two easy-to-check preference domain conditions which separately guarantee that (i) unilateral/Maskin monotonicity implies strategy-proofness (Theorem 1) and (ii) strategy-proofness implies unilateral/Maskin monotonicity (Theorem 2). We introduce and discuss various classical single-peaked preference domains and show which of the domain conditions they satisfy (see Propositions 1 and 2 and an overview in Table 1). As a by-product of our analysis, we obtain some extensions of the Muller–Satterthwaite Theorem as summarized in Theorem 3. We also discuss some new “Muller–Satterthwaite preference domains” (e.g., Proposition 3).     

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     

Nonpaternalistic altruism and functional interdependence of social preferences

Utility functions embodying nonpaternalistic altruism can be regarded as being generated through social interactions among altruistic individuals. As such, they show an important interdependence. Assuming linear altruism, the paper obtains the following results. First, nonpaternalistic altruism has a tendency to unify utility functions. Second, by linearly extrapolating a given structure of altruistic interactions, one can ascertain the extent to which the unification can proceed and its direction. Third, the unification implies narrowing down of the range of conflicting choices but the direction of unification lacks ethical principles. Fourth, the conditions utilitarianism imposes on the structure of altruistic interactions are characterized. The author would like to thank Marc Fleurbaey and three anonymous referees for insightful comments.     

On metrically conservative societies

In this paper, social choice theory is considered from the standpoint of social change. Various metrics (in a discrete setting) are introduced to measure changes in individual and collective preferences, and a society is said to be metrically conservative if social change does not exceed total individual changes. Arrow's IIA Axiom is found to be intimately related to a very restrictive metrical condition called metrical ultraconservatism. Strong characterization theorems are proved for metrically ultraconservative societies. A natural relaxation is the condition of metrical conservatism. We show that metrically conservative societies exist, and the number of possibilities can in fact grow exponentially with the population. But when the metrical condition is placed into the more specific socio-economic context of strict preference orderings, normative restrictions appear. One is the constitutional protection against the election of a dictator; another is the nonexistence of metrically conservative stable matchings, in the sense of Gale-Shapley. Some similar questions have been raised in continuous social choice theory, but the conclusions are quite different. We also consider the effect of an increasing population on the average rate of social change.The author is grateful to Dr. T.M. Tang for first drawing his attention to Arrow's General Possibility Theorem, to Professors Robert M. Anderson, Kenneth J. Arrow, and an anonymous referee for many valuable suggestions, and to Dr. W.Y. Poon for pointing out an important reference. This work was done when the author was at the University of California at Berkeley and formed part of his Berkeley Ph.D. dissertation. The views expressed here are the author's, and not necessarily those of AT&T Bell Laboratories.     

Harsanyi's impartial observer theorem with a restricted domain

In this paper we extend Harsanyi's impartial observer theorem by showing that the large domain of social lotteries can be significantly restricted – it is sufficient that the domain consists only of constant extended lotteries. Received: 27 December 2000/Accepted: 12 September 2001 We thank Simon Grant, Edi Karni, Atsushi Kajii, Ben Polak, Yemima Thompson and John Weymark for their comments and the Israel Institute of Business Research for financial support.     

Paretian efficiency,rawlsian justice and the Nozick theory of rights

In this paper the circumstances under which a Nozickian libertarian claim can produce self-consistent results are studied. These circumstances are shown to consist of technological separability of social alternatives and partial separability of individual preferences. Under these conditions some restricted versions of the Pareto principle and of the Rawlsian maximin can be reformulated consistently with the Nozickian libertarian claim. Also, in absence of external utility, Paretian efficiency, Rawlsian distributive justice and the Nozickian libertarian claim are consistent and can be together satisfied by a choice procedure. Moreover, the Nozick libertarian claim is shown to be satisfied by a perfectly competitive economy. A parallelism between a general equilibrium market system and a Nozickian rights system is pointed out.For helpful comments on an earlier version I am very grateful to C.J. Bliss, G. De Fraja, A. Sen, and an anonymous referee     

John Harsanyi's contributions to social choice and welfare economics

This article provides an overview of John Harsanyi's contributions to Social Choice and Welfare Economics.This article is a slightly expanded version of remarks I made on July 9, 1994 when I introduced John Harsanyi's address (which follows in this same issue) to the meetings of the Society for Social Choice and Welfare in Rochester, New York. In preparing my remarks for publication I have tried to keep the conversational tone of the original presentation but have added references for the reader's convenience. At the time of the meetings I had not anticipated publishing my introduction. When a few months later Maurice Salles asked if I would write up my remarks, I had to do so from memory, not having kept the notes I used to make my presentation. Subsequent to the Rochester meetings, Professor Harsanyi was awarded the Nobel Memorial Prize in Economic Sciences and I am pleased to add my congratulations to him for this honour in the form of this introduction to his research in Social Choice and Welfare Economics. Research support from the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged.     

The ignorant observer

We propose an extension of Harsanyi’s Impartial Observer Theorem based on the representation of ignorance as the set of all possible probability distributions over individuals. We obtain a characterization of the observer’s preferences that, under our most restrictive conditions, is a convex combination of Harsanyi’s utilitarian and Rawls’ egalitarian criteria. This representation is ethically meaningful, in the sense that individuals’ utilities are cardinally measurable and fully comparable. This allows us to conclude that the impartiality requirement cannot be used to decide between Rawls’ and Harsanyi’s positions. We thank D. Bouyssou, A. Chateauneuf, M. Cohen, M. Fleurbaey, E. Karni, J.-F. Laslier, P. Mongin, J. Moreno-Ternero and especially J. Weymark, as well as seminar audiences at University Pompeu Fabra, University of Cergy-Pontoise, the Roy Seminar and RUD 2006 for useful comments. Comments by two anonymous referees have been extremely useful to improve the paper. Financial support from an ACI grant by the French Ministry of Research is gratefully acknowledged.     

Normative population theory

Conclusion We have shown, contrary to the claims of Cowen, that average and critical-level utilitarianism, which do not suffer from the repugnant conclusion, do not recommend the killing of people with low but positive utilities. We have shown, in addition, that Methuselah's paradox and the repugnant conclusion do not stem from preferences that are represented by additive utility functions. Further, we have shown that Cowen's ideal participant method suffers from the repugnant conclusion.We believe that Parfit's [4] criticism of classical utilitarianism — that it satisfies the repugnant conclusion — should be taken seriously. That suggests that the ideal participant method should be rejected (along with classical utilitarianism) as a reasonable solution to the optimal population problem.We have benefited from private correspondence with T. Cowen, and the remarks of T. Hurka and an anonymous referee.     

Rights as alternative game forms

The paper analyzes the problem of modeling rights using extensive game forms with perfect information. Three dimensions of rights – protocol, autonomy and power – are identified and it is shown, that under a certain condition, the effectivity function captures completely all rights relevant aspects of a game form. The condition under which this is possible is interpretable both as a particular formulation of individual liberty and as a rejection of consequentialism. I wish to express my debt to Professor A.K. Sen and P.K. Pattanaik for several helpful discussions on rights as game forms. I also wish to thank professors N. Balke, M. Dasgupta, M. Davis, J. Hadar, A. Sengupta, and W.M. Thomson for their comments.