A quantitative discursive dilemma

The typical judgment aggregation problem in economics and other fields is the following: a group of people has to judge/estimate the value of an uncertain variable y, which is a function of k other variables, i.e., y = D(x ₁, . . . , x _k ). We analyze when it is possible for the group to arrive at collective judgements on the variables that respect D. We consider aggregators that fulfill Arrow’s IIA-condition and neutrality. We show how possibility and impossibility depend on the functional form of D, and generalize Pettit’s (2001) binary discursive dilemma to quantitative judgements.     

An extension of a theorem on the aggregation of separable preferences

An Excess-Voting Function relative to a profile π assigns to each pair of alternatives (x,y), the number of voters who prefer x to y minus the number of voters who prefer y to x. It is shown that any non-binary separable Excess-Voting Function can be achieved from a preferences profile when individuals are endowed with separable preferences. This result is an extension of Hollard and Le Breton (1996). Received: 16 December 1996 / Accepted: 8 October 1997     

Comparison functions and choice correspondences

In this paper, we introduce the concept of a comparison function, which is a mapping g that assigns numbers to ordered pairs of alternatives (x,y) with the property that g(x,y)=−g(y,x). The paper discusses how some well-known choice correspondences on tournaments such as the uncovered set, the minimal covering set and the bipartisan set can be extended to this general framework. Axiomatic characterizations and properties are studied for these correspondences. Received: 4 November 1996/Accepted: 13 April 1998     

Egalitarian-equivalence and the Pareto principle for social preferences

The Pareto principle is often in conflict with the equity criteria in construction of social preferences: there exist two allocations x and y such that x Pareto dominates y, but y is an equitable allocation whereas x is not. The efficiency-first principle requires to rank an allocation x higher than y if (i) x Pareto dominates y or (ii) x and y are Pareto-noncomparable and x is equitable whereas y is not. The equity-first principle reverses the order of application of the two criteria. Adopting egalitarian-equivalence as the notion of equity, we examine rationality of the social preference functions based on the efficiency-first or the equity-first principle. The degrees of rationality vary widely depending on which principle is adopted, and depending on the range of egalitarian-reference bundles. We show several impossibility and possibility results as well as a characterization of the social preference function introduced by Pazner and Schmeidler (1978). We also identify the sets of maximal elements of the social preference relations in the set of feasible allocations. The results are contrasted with those in the case where no-envy is the notion of equity.This paper is a much extended version of a part of Tadenuma (1998). The author is grateful to Marc Fleurbaey, Serge-Chiristophe Kolm, Kotaro Suzumura, and William Thomson for helpful comments on the earlier version of the paper. ‘‘Financial support from the Ministry of Education, Culture, Sports, Science and Technology of Japan for the 21st Century Center of Excellence Project on the Normative Evaluation and Social Choice of Contemporary Economic Systems is gratefully acknowledged’’.     

Does the Borda rule provide more than a ranking?

Consider the following problem: a set of candidates {x, y, z} has to be ranked from best to worse by a committee. Each member of the committee provides his own ranking of the three candidates and you decide to use the Borda method to aggregate the rankings. The resulting scores are as follows: 107 for x, 106 for y and 51 for z. Would you conclude that x is better than y? Probably not, because the difference between the scores of x and y is small. The only conclusion you would draw is that z definitely is the worst candidate. But, is it meaningful to consider differences of Borda scores? We characterize the Borda method in this new framework and find conditions that are very close to those characterizing the classical Borda method. Throughout our paper, we consider a generalization of the Borda method designed to aggregate fuzzy relations. Received: 2 March 1998/Accepted: 5 May 1999     

Intermediate inequality and welfare

We introduce a new centrist or intermediate inequality concept, between the usual relative and absolute notions, which is shown to be a variant of the α-ray invariant inequality measures in Pfingsten and Seidl (1997). We say that distributions x and y have the same (x, π)-inequality if the total income difference between them is allocated among the individuals as follows: 100π% preserving income shares in x, and 100(1−π)% in equal absolute amounts. This notion can be made as operational as current standard methods in Shorrocks (1983). Received: 13 May 1998/Accepted: 25 January 1999     

Top-Pair and Top-Triple Monotonicity

We say that a social choice function (SCF) satisfies Top-k Monotonicity if the following holds. Suppose the outcome of the SCF at a preference profile is one of the top k-ranked alternatives for voter i. Let the set of these k alternatives be denoted by B. Suppose that i’s preference ordering changes in such a way that the set of first k-ranked alternatives remains the set B. Then the outcome at the new profile must belong to B. This definition of monotonicity arises naturally from considerations of set “improvements” and is weaker than the axioms of strong positive association and Maskin Monotonicity. Our main results are that if there are two voters then a SCF satisfies unanimity and Top-2 or Top-pair Monotonicity if and only if it is dictatorial. If there are more than two voters, then Top-pair Monotonicity must be replaced by Top-3 Monotonicity (or Top-triple Monotonicity) for the analogous result. Our results demonstrate that connection between dictatorship and “improvement” axioms is stronger than that suggested by the Muller–Satterthwaite result (Muller and Satterthwaite in J Econ Theory 14:412–418, 1977) and the Gibbard–Sattherthwaite theorem.     

Covering relations,closest orderings and hamiltonian bypaths in tournaments

We show that the Slater's set of a tournament, i.e. the set of the top elements of the closest orderings, is a subset of the top cycle of the uncovered set of the tournament. We also show that the covering relation is related to the hamiltonian bypaths of a strong tournament in that if x covers y, then there exists an hamiltonian bypath from x to y.We thank B. Monjardet and an anonymous editor for helpful suggestions.     

Positional information and preference aggregation

A new version of independence (I⁺) is proposed for social welfare functions based on the following notion of agreement. Two weak orders R and R’ on a finite set S agree on a pair {x,y}, denoted byif R|_{x,y} = R’|_{x,y} and [z R*x and z R*y for some z∈S] if and only if [z’ (R’)* x and z’(R’)*y for some z’∈S]. The last part says that x and y are strictly under z with respect to R exactly when x and y are strictly under z’ with respect to R’. Some examples and results on social welfare functions that satisfy (I⁺), Pareto, and nondictatorship are given.I am grateful for the comments and suggestions made by an anonymous referee on an earlier version of this paper.     

Aggregation of smooth preferences

Suppose p is a smooth preference profile (for a society, N) belonging to a domain P ^N . Let σ be a voting rule, and σ(p)(x) be the set of alternatives in the space, W, which is preferred to x. The equilibrium E(σ(p)) is the set {x∈W:σ(p)(x) is empty}. A sufficient condition for existence of E(σ(p)) when p is convex is that a “dual”, or generalized gradient, dσ(p)(x), is non-empty at all x. Under certain conditions the dual “field”, dσ(p), admits a “social gradient field”Γ(p). Γ is called an “aggregator” on the domain P ^N if Γ is continuous for all p in P ^N . It is shown here that the “minmax” voting rule, σ, admits an aggregator when P ^N is the set of smooth, convex preference profiles (on a compact, convex topological vector space, W) and P ^N is endowed with a C ¹-topology. An aggregator can also be constructed on a domain of smooth, non-convex preferences when W is the compact interval. The construction of an aggregator for a general political economy is also discussed. Some remarks are addressed to the relationship between these results and the Chichilnisky-Heal theorem on the non-existence of a preference aggregator when P ^N is not contractible. Received: 4 July 1995 / Accepted: 26 August 1996     

A generalised model of judgment aggregation

The new field of judgment aggregation aims to merge many individual sets of judgments on logically interconnected propositions into a single collective set of judgments on these propositions. Judgment aggregation has commonly been studied using classical propositional logic, with a limited expressive power and a problematic representation of conditional statements (“if P then Q ”) as material conditionals. In this methodological paper, I present a simple unified model of judgment aggregation in general logics. I show how many realistic decision problems can be represented in it. This includes decision problems expressed in languages of standard propositional logic, predicate logic (e.g. preference aggregation problems), modal or conditional logics, and some multi-valued or fuzzy logics. I provide a list of simple tools for working with general logics, and I prove impossibility results that generalise earlier theorems.     

Redistribution of wealth when knowledge is dispersed

The second theorem of welfare economics is generalized so that ethical redistribution of wealth can be carried out by the centre by means of an adjustment rule that depends only on prices and an exogenously given feasible allocation x. When equilibrium is achieved each household i will have at least as much utility as it would derive from its component of x.Mike Peters and a referee of this journal provided useful expositional comments. I am grateful for their help and to the Social Sciences and Humanities Research Council (Canada) for financial support. Any defects in the paper can be attributed to the author.     

The Borda count and agenda manipulation

A standard objection to the Borda count, as an actual voting procedure, is that it is subject to agenda manipulation. The classical example is the introduction, in order to favour a candidate or option y, of a new option z ranked on every voter's preference scale immediately below y; y may as a result obtain the highest Borda count, although, if z had not been introduced, a different option would have done so. Strategic use of this device is not greatly to be feared, but it does point to a defect in the system: equating the distance on each voter's preference scale between any option and the next imposes a penalty for dissimilarity. The corrective is to vary the distances by introducing a measure of dissimilarity, as perceived by the voters and revealed by their preference scales. Two possible ways of doing this are described, yielding revised and adjusted Borda scores. It is shown by examples that these will often, but not always, undo the `agenda manipulation' effect, and questioned whether it is desirable to undo it altogether. Received: 22 August 1996/Accepted: 31 October 1996     

Hierarchy of players in swap robust voting games

Ordinarily, the process of decision making by a committee through voting is modeled by a monotonic game the range of whose characteristic function is restricted to {0, 1}. The decision rule that governs the collective action of a voting body induces a hierarchy in the set of players in terms of the a-priori influence that the players have over the decision making process. In order to determine this hierarchy in a swap robust game, one has to either evaluate a power index (e.g., the Shapley–Shubik index, the Banzhaf–Coleman index) for each player or conduct a pairwise comparison between players, whereby a player i is ranked higher than another player j if there exists a coalition in which i is more desirable as a coalition partner than j. In this paper, we outline an alternative mechanism to determine the ranking of players in terms of their a-priori power. This simple and elegant method uses only minimal winning coalitions, rather than the entire set of winning coalitions.     

The value of a player in n-person games

. The article decomposes the Shapley value into a value matrix which gives the value of every player to every other player in n-person games. Element Φ_ij(v) in the value matrix is positive, zero, or negative, dependent on whether row player i is beneficial, has no impact, or is not beneficial for column player j. The elements in each row and in each column of the value matrix sum up to the Shapley value of the respective player. The value matrix is illustrated by the voting procedure in the European Council of Ministers 1981–1995. Received: 9 September 1998/Accepted: 11 February 2000     

Where do classifications come from? The DSM-III,the transformation of American psychiatry,and the problem of origins in the sociology of knowledge

When something serves a function, it is easy to overlook its origins. The tendency is to proceed directly to function and retroactively construct a story about origin based on the function it fills. In this article, I address this problem of origins as it appears in the sociology of knowledge, using a case study of the publication of the 3rd edition of the Diagnostic and Statistical Manual of Mental Disorders (DSM-III) in 1980. The manual revolutionized American psychiatry and the treatment of mental illness, because it served the function of classification that had become critical to the field of mental health by this time. But this function must be bracketed in order to reveal the “extra-functional” origins of the DSM-III. Using field theory, I argue that the manual was necessary for reasons other than the function it filled as a classification. Specifically, its origin lies in a series of conflicts among psychiatrists, psychoanalysts, and clinical psychologists within the field of mental health, which followed in the wake of the collapse of psychoanalysis as the dominant treatment type for mental illness. I reveal the generative formula behind the production of the DSM-III, capturing a variety of social processes that influenced the format of the manual and made it a useful classification, but which are not reducible to function. In this way, I reproduce its raison d’etre in a manner similar to how the DSM-III appeared for the people who produced it. This focus on generative formulas offers the sociology of knowledge a way to capture the epistemic importance of a range of different social processes. Most importantly, it avoids the functional fallacy of reducing origin to function, and ignoring the idea that innovations might appear necessary even without clear recognition of their functional consequences.     

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     

On the Arrow and Wilson impossibility theorems

A social welfare function is Arrovian if it is transitive-valued and satisfies IIA. We examine the logical relation between the statements (A) If f is Arrovian with domain 𝒫 then it is dictatorial if it satisfies the Pareto criterion and (W) If f is Arrovian with domain 𝒫 then it is dictatorial or inversely dictatorial if it is non-null and satisfies non-imposition. We exhibit a class of domains on which W implies A and another on which A implies W. We also present examples of domains on which one of the statements is true and the other is false. Received: 23 October 2001/Accepted: 19 March 2002 We thank our two referees for their suggestions for improving the exposition.     

Improvement indices of well-being

In recent years, various classes of indices measuring the improvement of well-being have been suggested. Formally, an improvement index is a function Q* (x ₁, x ₂), where x _t is the level of some attribute of well-being, e.g., life expectancy, in period t. This paper extends previous works on improvement indices by introducing a class of multidimensional indices which aggregates the improvement of a group of attributes of well-being. We propose a set of axioms which leads inexorably to a multidimensional generalization of Kakwani's class of improvement indices of well-being (Kakwani 1993).Financial support from the Social Science and Education Panel of the Chinese University of Hong Kong (research grant account no: 220201620) is gratefully acknowledged