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     

The structure of fuzzy preferences: Social choice implications

It has been shown that, with an alternative factorization of fuzzy weak preferences into symmetric and antisymmetric components, one can prove a fuzzy analogue of Arrow's Impossibility Theorem even when the transitivity requirements on individual and social preferences are very weak. It is demonstrated here that the use of this specification of strict preference, however, requires preferences to also be strongly connected. In the absence of strong connectedness, another factorization of fuzzy weak preferences is indicated, for which nondictatorial fuzzy aggregation rules satisfying the weak transitivity requirement can still be found. On the other hand, if strong connectedness is assumed, the fuzzy version of Arrow's Theorem still holds for a variety of weak preference factorizations, even if the transitivity condition is weakened to its absolute minimum. Since Arrow's Impossibility Theorem appeared nearly half a century ago, researchers have been attempting to avoid Arrow's negative result by relaxing various of his original assumptions. One approach has been to allow preferences – those of individuals and society or just those of society alone – to be “fuzzy.” In particular, Dutta [4] has shown that, to a limited extent, one can avoid the impossibility result (or, more precisely, the dictatorship result) by using fuzzy preferences, employing a particularly weak version of transitivity among the many plausible (but still distinct) definitions of transitivity that are available for fuzzy preferences. Another aspect of exact preferences for which the extension to the more general realm of fuzzy preferences is ambiguous is the factorization of a weak preference relation into a symmetric component (indifference) and an antisymmetric component (strict preference). There are several ways to do this for fuzzy weak preferences, all of them equivalent to the traditional factorization in the special case when preferences are exact, but quite different from each other when preferences are fuzzy (see, for example, [3]). A recent paper in this journal [1], by A. Banerjee, argues that the choice of definitions for indifference and strict preference, given a fuzzy weak preference, can also have “Arrovian” implications. In particular, [1] claims that Dutta's version of strict preference presents certain intuitive difficulties and recommends a different version, with its own axiomatic derivation, for which the dictatorship results reappear even with Dutta's weak version of transitivity. However, the conditions used to derive [1]'s version of strict preference imply a restriction on how fuzzy the original weak preference can be, namely, that the fuzzy weak preference relation must be strongly connected. Without this restriction, I will show that the rest of [1]'s conditions imply yet a third version of strict preference, for which Dutta's possibility result under weak transitivity still holds. On the other hand, if one accepts the strong connectedness required in order for it to be valid, I show that [1]'s dictatorship theorem can in fact be strengthened to cover any version of transitivity for fuzzy preferences, no matter how weak, and further, that this dictatorship result holds for any “regular” formulation of strict preference, including the one originally used by Dutta. Received: 13 May 1996 / Accepted: 13 January 1997     

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     

Equity, continuity, and myopia: A generalization of Diamond’s impossibility theorem

 We consider social preferences over infinite horizon intergenerational consumption paths. We use the Mackey topology to define continuity of social preferences. Our main objective is to generalize one of Diamond’s impossibility theorems. First, we show that the trivial preference relation is the only asymmetric social preference relation satisfying equity and continuity. Second, we compare Campbell’s impossibility theorem with ours. Finally we use an order-based notion of myopia and establish another impossibility result. Received: 26 August 1994/Accepted: 5 June 1996     

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.     

Toward a theory of choice: Socially embedded preference construction

We present a boundedly rational model of choice that makes room for individual values and social influences in a process of preference construction. It takes from the subjective expected utility model the notion that people assess their options in terms of expected outcomes, referenced to personal values, but it presumes that individuals assess sharply truncated lists of relevant options, outcomes and values and apply a classification logic rather than a calculative one. Such a model is consistent with the nature of evolved human cognitive abilities. The model treats cognitive heuristics and various forms of social influence as determinants of selection of the truncated lists and it treats moral norms as classification rules activated when certain actions and outcomes become salient. The model has implications for understanding how political rhetoric affects individual and social preferences and how “hot cognitions” fit into human choice and for developing improved methods of collective decision making.     

Approval Voting on Dichotomous Preferences

The aim of this paper is to find normative foundations of Approval Voting when individuals have dichotomous preferences. We show that a social choice function is anonymous, neutral, strategy-proof and strictly monotone if and only if it is Approval Voting and interpret this result as an extension of May’s theorem (Econometrica 20:680–684, 1952). Then, we show that Approval Voting is the only strictly symmetric, neutral and efficient social choice function. This result is related to a characterization of Baigent and Xu (Math Soc Sci 21:21–29, 1991).     

Using elections to represent preferences

This paper takes voting theory out of the realm of mechanism design and studies elections as tools for representing preferences: every preference relation on a set of n elements is the outcome of pairwise voting by approximately 2 log₂ n voters with transitive preferences. Results like this one provide representation for preference relations not representable by utility functions. They also motivate definitions of the levels of intransitivity, nonlinearity and nonrepresentability (by utility function) of a preference relation. Received: 25 March 1999/Accepted: 19 June 2000     

Rational choice and revealed preference without binariness

 This paper attempts to provide a unified account of the rationalization of possibly non-binary choice-functions by “Extended Preference Relations” (relations between sets and elements). The analysis focuses on transitive EPRs for which three choice-functional characterizations are given, two of them based on novel axioms. Transitive EPRs are shown to be rationalizable by sets of orderings that are “closed under compromise”; this novel requirement is argued to be the key to establish a canonical relationship between sets of orderings and choice-functions. The traditional assumption of “binariness” on preference relations or choice functions is shown to be analytically unhelpful and normatively unfounded; non-binariness may arise from “unresolvedness of preference”, a previously unrecognized aspect of preference incompleteness. Received: 28 August 1995/Accepted: 14 February 1996     

A general concept of scoring rules: general definitions,statistical inference,and empirical illustrations

We develop the general conceptual, mathematical and statistical foundations of behavioral social choice for scoring rules. Traditional scoring rules are difficult to assess empirically because one rarely observes the deterministic complete linear orders that they require as input. We provide a general concept of scoring rules in terms of a broad range of mathematical representations of preference or utility, namely arbitrary finite binary relations, probability distributions over such relations, real valued multi-criteria utility vectors and real valued random utility representations. We extend Regenwetter et al.’s (Behavioral social choice. Cambridge University Press, Cambridge, 2006) statistical framework to a more general setting. We illustrate the general modeling and statistical tools by applying them to four well known sets of survey data. We illustrate two potential problems that have previously received little attention and that deserve systematic study in the future: (1) Scoring rule outcomes can suffer from model dependence in that the social welfare functions computed from ballot, survey, or hypothetical data may depend on implicit or explicit modeling assumptions. (2) Scoring rule outcomes may suffer from low statistical confidence in that the correct assessment of social orders from empirical data can be far from certain. We also illustrate the empirical congruence among conceptually competing social choice methods.     

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     

Aggregation of fuzzy preferences: Some rules of the mean

This paper studies by means of reciprocal fuzzy binary relations the aggregation of preferences when individuals show their preferences gradually. We have characterized neutral aggregation rules through functions from powers of the unit interval in the unit interval. Furthermore, we have determined the neutral aggregation rules that are decomposable and anonymous. In this class of rules, the collective intensity of preference is the arithmetic mean of the values assigned by a function to the individual intensities of preference. We have also considered the neutral aggregation rules based on quasiarithmetic means. We have established that this class of rules generalizes the simple majority, when individuals have ordinary preferences and collective preferences are reciprocal. Received: 23 April 1999/Accepted: 25 September 1999     

Characterizing uncertainty aversion through preference for mixtures

Uncertainty aversion is often modelled as (strict) quasi-concavity of preferences over uncertain acts. A theory of uncertainty aversion may be characterized by the pairs of acts for which strict preference for a mixture between them is permitted. This paper provides such a characterization for two leading representations of uncertainty averse preferences; those of Schmeidler [24] (Choquet expected utility or CEU) and of Gilboa and Schmeidler [16] (maxmin expected utility with a non-unique prior or MMEU). This characterization clarifies the relation between the two theories. Received: 20 February 1998/Accepted: 25 March 1999     

Unlikelihood of Condorcet’s paradox in a large society

We provide intuitive, formal, and computational evidence that in a large society Condorcet’s paradox (the intransitivity of social preference obtained by pairwise vote) can hardly occur. For that purpose, we compare two models of social choice, one based on voting and another one based on summing individual cardinal utilities, expressed either in reals, or integers. We show that in a probabilistic model with a large number of independent individuals both models, almost surely, provide the same decision results. This implies that Condorcet’s and Borda’s methods tend to give the same decisions as the number of voters increases. Therefore, in the model with a large number of voters, the transitivity of the Borda preference is inherent in a majority preference as well. Received: 26 June 1998/Accepted: 16 April 1999     

On the probabilities of correct or incorrect majority preference relations

While majority cycles may pose a threat to democratic decision making, actual decisions based inadvertently upon an incorrect majority preference relation may be far more expensive to society. We study majority rule both in a statistical sampling and a Bayesian inference framework. Based on any given paired comparison probabilities or ranking probabilities in a population (i.e., culture) of reference, we derive upper and lower bounds on the probability of a correct or incorrect majority social welfare relation in a random sample (with replacement). We also present upper and lower bounds on the probabilities of majority preference relations in the population given a sample, using Bayesian updating. These bounds permit to map quite precisely the entire picture of possible majority preference relations as well as their probabilities. We illustrate our results using survey data. Received: 13 November 2000/Accepted: 19 March 2002 This collaborative work was carried out while Regenwetter was a faculty member at the Fuqua School of Business, Duke University. We thank Fuqua for sponsoring our collaboration and the National Science Foundation for grant SBR-97-30076 to Michel Regenwetter. We are indebted to the editor and the referees, as well as to Jim Adams, Bob Clemen, Bernie Grofman, Bob Nau, Saša Pekeč, Jim Smith and Bob Winkler for helpful comments and suggestions.     

Axiomatic foundations for fairness-motivated preferences

Much work in social choice theory takes individual preferences as uninvestigated inputs into aggregation functions designed to reflect considerations of fairness. Advances in experimental and behavioural economics show that fairness can also be an important motivation in the preferences of individuals themselves. A proper characterisation of how fairness concerns enter such preferences can enrich the informational basis of many social choice exercises. This paper proposes axiomatic foundations for individual fairness-motivated preferences that cover most of the models developed to rationalise observed behaviour in experiments. These models fall into two classes: Outcome-based models, which see preferences as defined only over distributive outcomes, and context-dependent models, which allow rankings over distributive outcomes to change systematically with non-outcome factors. I accommodate outcome-based and context-sensitive fairness concerns by modelling fairness-motivated preferences as a reference-dependent preference structure. I first present a set of axioms and two theorems that generate commonly used outcome-based models as special cases. I then generalise the axiomatic basis to allow for reference-dependence, and derive a simple functional form in which the weight on each person’s payoff depends on a reference vector of how much each person deserves.     

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     

A preference foundation for Fehr and Schmidt’s model of inequity aversion

Fehr and Schmidt (FS) introduced an influential social utility function for individuals in interpersonal contexts that captures self-centered inequity aversion. The value of this social utility function lies in its exceptionally good balance between parsimony and fit. This paper provides a preference foundation for exactly the model of FS with preference conditions that exactly capture the exceptionally good balance of FS. Remarkably, FS is a special case of Schmeidler’s rank-dependent utility for decision under uncertainty.     

A Social Choice Lemma on Voting Over Lotteries with Applications to a Class of Dynamic Games

We prove a lemma characterizing majority preferences over lotteries on a subset of Euclidean space. Assuming voters have quadratic von Neumann–Morgenstern utility representations, and assuming existence of a majority undominated (or “core”) point, the core voter is decisive: one lottery is majority-preferred to another if and only if this is the preference of the core voter. Several applications of this result to dynamic voting games are discussed.This paper was completed after Jeff Banks’s death. John Duggan is deeply indebted to him for his friendship and his collaboration on this and many other projects.