Consistency between tastes and values: A universalization approach

The object of this paper is to propose a consistency test for an individual involved in collective choice process. Collective choice processes considered in the paper are those that transform individuals ‘tastes’– which reflect the self-interested view point of the individuals – into (social) ranking of alternatives. In addition to her tastes, an individual has values about the way by which collective decision should be made. We distinguish two categories of such values. First, there are end-values that restrict the class of social rankings that the individual considers ethically acceptable. Second there are aggregation-values that specify the way by which the social ranking should depend upon the individuals tastes. The consistency test stands on an hypothetical operation of universalization of the individual tastes to everyone. Five illustrations of the potential usefulness of our approach for interpreting social choice theory and welfare economics are proposed. These illustrations deal with utilitarian aggregation in the presence of income inequality aversion, the so-called ‘ethics of responsibility’ and the aggregation of individual ranking of opportunity sets based on their freedom of choice. A discussion of the relevance of the consistency test for addressing the problem of ‘laundering’ individual preferences is also provided. Received: 25 June 1998/Accepted: 16 March 1999     

Extended preferences and freedom of choice

The common choice theory in economics is based on the assumption that an individual is defined in terms of a binary preference relation. This preference relation is defined over alternatives without taking into account menu dependence and, in particular, freedom of choice or, more generally, the set that contains the alternatives. In this study we clarify the nature and the significance of freedom of choice which may positively or negatively affect the individual's welfare. Our proposed extended preference relation of the individual takes into account both the particular alternative and the opportunity set that he faces. This extended relation does not induce ranking of opportunity sets. Its restriction to a particular opportunity set is the paradigmatic preference relation and it can capture the dependence of preferences on freedom of choice. Our main result establishes the inconsistency between dependence of extended preferences on freedom of choice and the existence of a utility that represents the paradigmatic preference relation and any of its restrictions. Received: 30 December 1997/Accepted: 6 September 1999     

Freedom of choice in a social context: comparing game forms

In this paper, the set of outcomes of game forms is introduced as the relevant attribute for evaluating freedom of choice. These sets are defined as the cartesian product of every individual’s set of available options. It is argued that doing so is one way of taking into account social interactions when evaluating individual freedom. A set of axioms is introduced, each of which conveys some intuitions about how interactions affect freedom of choice, axioms by the mean of which two criteria, the Max and the MaxMin, are characterised for comparing game forms in terms of the freedom of choice they offer. These criteria are based, respectively, on the comparison of the best and the worst outcome the individual can reach in the game form.     

On the multi-preference approach to evaluating opportunities

The purpose of the paper is to provide a general framework for analyzing “preference for opportunities.” Based on two simple axioms a fundamental result due to Kreps is used in order to represent rankings of opportunity sets in terms of multiple preferences. The paper provides several refinements of the basic representation theorem. In particular, a condition of “closedness under compromise” is suggested in order to distinguish the flexibility interpretation of the model from normative interpretations which play a crucial role in justifying the intrinsic value of opportunities. Moreover, the paper clarifies the link between the multiple preference approach and the “choice function” approach to evaluating opportunities. In particular, it is shown how the well-known Aizerman/Malishevski result on rationalizability of choice functions can be obtained as a corollary from the more general multiple preference representation of a ranking of opportunity sets. Received: 3 September 1996 / Accepted: 18 August 1997     

Topological aggregation of preferences: the case of a continuum of agents

 This paper studies the topological approach to social choice theory initiated by G. Chichilnisky (1980), extending it to the case of a continuum of agents. The social choice rules are continuous anonymous maps defined on preference spaces which respect unanimity. We establish that a social choice rule exists for a continuum of agents if and only if the space of preferences is contractible. We provide also a topological characterization of such rules as generalized means or mathematical expectations of individual preferences. Received: 30 November 1994/Accepted: 22 April 1996     

Essential alternatives and freedom rankings

We study the problem of ranking opportunity sets in terms of freedom of choice. The analysis is based on the notion of essential alternatives introduced in Puppe (J Econ Theory 68:174–199, 1996). An alternative in an opportunity set is called essential if by deleting it, the reduced opportunity set offers less freedom than the original set. We provide an axiomatic characterization of the ranking according to which an opportunity set offers more freedom than another opportunity set if its share of essential elements in their union is larger.     

Aggregation of coarse preferences

We consider weak preference orderings over a set A _n of n alternatives. An individual preference is of refinement?≤n if it first partitions A _n into ? subsets of `tied' alternatives, and then ranks these subsets within a linear ordering. When ?<n, preferences are coarse. It is shown that, if the refinement of preferences does not exceed ?, a super majority rule (within non-abstaining voters) with rate 1− 1/? is necessary and sufficient to rule out Condorcet cycles of any length. It is argued moreover how the coarser the individual preferences, (1) the smaller the rate of super majority necessary to rule out cycles `in probability'; (2) the more probable the pairwise comparisons of alternatives, for any given super majority rule. Received: 29 June 1999/Accepted: 25 February 2000     

Opportunity sets and individual well-being

 An opportunity set ranking rule assigns an ordering of opportunity sets to each individual utility function (defined on the universal set of alternatives) within the domain of this rule. Using an axiomatic approach, this paper characterizes a general class of opportunity set ranking rules which are based on the utilities associated with the elements of an opportunity set. It is argued that the addition of an alternative to a given opportunity set is not necessarily desirable in terms of overall well-being, and this position is reflected in replacing a commonly used monotonicity axiom with an alternative condition. Received: 15 May 1995/Accepted: 14 December 1995     

Acyclic sets of linear orders

 A set of linear orders on {1,2, ℕ, n} is acyclic if no three of its orders have an embedded permutation 3-cycle {abc, cab, bca}. Let f (n) be the maximum cardinality of an acyclic set of linear orders on {1,2, ℕ, n}. The problem of determining f (n) has interested social choice theorists for many years because it is the greatest number of linear orders on a set of n alternatives that guarantees transitivity of majority preferences when every voter in an arbitrary finite set has any one of those orders as his or her preference order. This paper gives improved lower and upper bounds for f (n). We note that f (5)=20 and that all maximum acyclic sets at n=4, 5 are generated by an “alternating scheme.” This procedure becomes suboptimal at least by n=16, where a “replacement scheme” overtakes it. The presently-best large-n lower bound is approximately f (n)≥(2.1708) ⁿ . Received: 5 April 1995/Accepted: 10 November 1995     

Choice rules with fuzzy preferences: Some characterizations

Consider an agent with fuzzy preferences. This agent, however, has to make exact choices when faced with different feasible sets of alternatives. What rule does he follow in making such choices? This paper provides an axiomatic characterization of a class of binary choice rules called the α satisfying rule. When α=1, this rule is the Orlovsky choice rule. On the other hand, for α≤1/2, the rule coincides with the M _α rule that has been extensively analyzed in the literature on fuzzy preferences. Received: 3 August 1995/Accepted: 19 November 1997     

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.     

The geometry of implementation: a necessary and sufficient condition for straightforward games

 We characterize games which induce truthful revelation of the players’ preferences, either as dominant strategies (straightforward games) or in Nash equilibria. Strategies are statements of individual preferences on R ⁿ . Outcomes are social preferences. Preferences over outcomes are defined by a distance from a bliss point. We prove that g is straightforward if and only if g is locally constant or dictatorial (LCD), i.e., coordinate-wise either a constant or a projection map locally for almost all strategy profiles. We also establish that: (i) If a game is straightforward and respects unanimity, then the map g must be continuous, (ii) Straightforwardness is a nowhere dense property, (iii) There exist differentiable straightforward games which are non-dictatorial. (iv) If a social choice rule is Nash implementable, then it is straightforward and locally constant or dictatorial. Received: 30 December 1994/Accepted: 22 April 1996     

Negatively interdependent preferences

We develop a theory of representation of interdependent preferences that reflect the widely acknowledged phenomenon of keeping up with the Joneses (i.e. of those preferences which maintain that well-being depend on “relative standing” in the society as well as on material consumption). The principal ingredient of our analysis is the assumption that individuals desire to occupy a (subjectively) better position than their peers. This is quite a primitive starting point in that it does not give any reference to what is actually regarded as “status” in the society. We call this basic postulate negative interdependence, and study its implications. In particular, combining this assumption with some other basic postulates that are widely used in a number of other branches of the theory of individual choice, we axiomatize the relative income hypothesis, and obtain an operational representation of interdependent preferences. Received: 7 December 1998/Accepted: 24 August 1999     

Equal factor equivalence in economies with multiple public goods

We reconsider the problem of provision and cost-sharing of multiple public goods. The efficient equal factor equivalent allocation rule makes every agent indifferent between what he receives and the opportunity of choosing the bundle of public goods subject to the constraint of paying r times its cost, where r is set as low as possible. We show that this rule is characterized in economies with a continuum of agents by efficiency, a natural upper bound on everyone's welfare, and a property of solidarity with respect to changes in population and preferences. Received: 3 August 1995 / Accepted : 29 April 1997     

Rationality,external norms,and the epistemic value of menus

Ever since Sen’s criticism of internal consistency properties of choice, there exists a perception that the standard rational choice approach has difficulties in coping with the existence of external norms, or the information a menu of choice might convey to a decision maker. This article provides a brief survey of possible responses to these criticisms of traditional rational choice theory. We review approaches to the internalization of external norms as well as ways to formulate norm-conditional rationalizability. Furthermore, we illustrate how the epistemic value of an opportunity set can be incorporated into rational choice theory.     

Strategy-proofness versus efficiency on restricted domains of exchange economies

 Strategy-proofness has been shown to be a strong property, particularly on large domains of preferences. We therefore examine the existence of strategy-proof and efficient solutions on restricted, 2-person domains of exchange economies. On the class of 2-person exchange economies in which agents have homothetic, strictly convex preferences we show, as Zhou (1991) did for a larger domain, that such a solution is necessarily dictatorial. As this proof requires preferences exhibiting high degrees of complementarity, our search continues to a class of linear preferences. Even on this “small” domain, the same negative result holds. These two results are extended to many superdomains, including Zhou’s. Received: 9 June 1995/Accepted: 8 January 1996     

The core of social choice problems with monotone preferences and a feasibility constraint

We consider collective choice with agents possessing strictly monotone, strictly convex and continuous preferences over a compact and convex constraint set contained in ₊^k . If it is non-empty the core will lie on the efficient boundary of the constraint set and any policy not in the core is beaten by some policy on the efficient boundary. It is possible to translate the collective choice problem on this efficient boundary to another social choice problem on a compact and convex subset of ₊^c (c<k) with strictly convex and continuous preferences. In this setting the dimensionality results in Banks (1995) and Saari (1997) apply to the dimensionality of the boundary of the constraint set (which is lower than the dimensionality of the choice space by at least one). If the constraint set is not convex then the translated lower dimensional problem does not necessarily involve strict convexity of preferences but the dimensionality of the problem is still lower. Broadly, the results show that the homogeneity afforded by strict monotonicity of preferences and a compact constraint set makes generic core emptyness slightly less common. One example of the results is that if preferences are strictly monotone and convex on ² then choice on a compact and convex constraint exhibits a version of the median voter theorem.I thank Donald Saari for helpful comments on an earlier version of this paper.     

Coalitionally strategyproof functions depend only on the most-preferred alternatives

In a framework allowing infinitely many individuals, I prove that coalitionally strategyproof social choice functions satisfy “tops only.” That is, they depend only on which alternative each individual prefers the most, not on which alternative she prefers the second most, the third,…, or the least. The functions are defined on the domain of profiles measurable with respect to a Boolean algebra of coalitions. The unrestricted domain of profiles is an example of such a domain. I also prove an extension theorem. Received: 10 August 1998/Accepted: 3 May 1999     

Strategy-proof resolute social choice correspondences

We qualify a social choice correspondence as resolute when its set valued outcomes are interpreted as mutually compatible alternatives which are altogether chosen. We refer to such sets as “committees” and analyze the manipulability of resolute social choice correspondences which pick fixed size committees. When the domain of preferences over committees is unrestricted, the Gibbard–Satterthwaite theorem—naturally—applies. We show that in case we wish to “reasonably” relate preferences over committees to preferences over committee members, there is no domain restriction which allows escaping Gibbard–Satterthwaite type of impossibilities. We also consider a more general model where the range of the social choice rule is determined by imposing a lower and an upper bound on the cardinalities of the committees. The results are again of the Gibbard–Satterthwaite taste, though under more restrictive extension axioms.     

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