Some issues related to the topological aggregation of preferences: addendum

 The purpose of this addendum is to correct some results published in our paper “Some issues related to the topological aggregation of preferences” (SCW (1992) 9: 213–227). Received: 28 November 1992/Accepted: 22 April 1996     

The topological approach to the aggregation of preferences

(1) A domain of preferences allows for topological aggregation for each number of people if and only if this domain of preferences is contractible. (2) The combination of continuity and the Pareto principle implies the existence of a unique manipulator. (3) Arrow’s theorem can be translated into the previous statement. All of these results are obtained via algebraic topology. This paper introduces some of the tools developed in topology and applies them upon the problem of preference aggregation. This paper is based upon a lecture given at the conference “Mathematical aspects of social choice”, CREM, Université de Caen and CNRS, November 8–10, 2004. I thank the organizers Maurice Salles and Vincent Merlin for their warm hospitality, and the participants for the stimulating discussion. I am indebted to Bart Capéau and Roeland Vervenne for helpful remarks. I thank the referee for combining speed and quality.     

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     

Manipulating an aggregation rule under ordinally fuzzy preferences

It is well known that many aggregation rules are manipulable through strategic behaviour. Typically, the aggregation rules considered in the literature are social choice correspondences. In this paper the aggregation rules of interest are social welfare functions (SWFs). We investigate the problem of constructing a SWF that is non-manipulable. In this context, individuals attempt to manipulate a social ordering as opposed to a social choice. Using techniques from an ordinal version of fuzzy set theory, we introduce a class of ordinally fuzzy binary relations of which exact binary relations are a special case. Operating within this family enables us to prove an impossibility theorem. This theorem states that all non-manipulable SWFs are dictatorial, provided that they are not constant. This theorem uses a weaker transitivity condition than the one in Perote-Peña and Piggins (J Math Econ 43:564–580, 2007), and the ordinal framework we employ is more general than the cardinal setting used there. We conclude by considering several ways of circumventing this impossibility theorem.     

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     

Bargaining over multiple issues with maximin and leximin preferences

Global bargaining problems over a finite number of different issues are formalized as cartesian products of classical bargaining problems. For maximin and leximin bargainers we characterize global bargaining solutions that are efficient and satisfy the requirement that bargaining separately or globally leads to equivalent outcomes. Global solutions in this class are constructed from the family of monotone path solutions for classical bargaining problems. We are indebted to two anonymous referees for comments. Financial support from CENTRA (EC014-2005), CREA-Barcelona Economics, and Ministerio de Educación y Ciencia (BEC2003-03111, SEJ2006-05441) is gratefully acknowledged.     

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     

Confounding issues related to determining sample size in qualitative research

The debate on determining sample size in qualitative research is confounded by four fundamental methodological issues: the exclusive focus on theme analysis; the diverse and imprecise use of ‘qualitative’; a reliance on only two logics of inquiry, induction and deduction, and the occasional confusion of abduction with induction; and a general lack of recognition of the importance of differences in ontological assumptions. Embedded in these issues is an unwarranted acceptance of limited associations between certain assumptions, logics, forms of data, and methods of data collection/generation and analysis. What is required is a reformulation of the problem and its discussion with reference to ontological assumptions and logics of inquiry.     

Some persistent issues of sociological polemics

The centrifugal forces and the jumble of polemics that have allegedly fragmented sociology today are shown to have existed also in the 1920s and the 1930s. The sharply contrasting visions of our field are illustrated in the debates among Ogburn, MacIver, Lundberg, and others. The logical analysis of certain persistent themes of intradisciplinary polemics shows them to be pseudo-issues—for example, differences in findings can be caused by differences in latent questions addressed by writers. From a cognitive point of view, the different approaches are frequently complementary rather than conflicting. The social conflict is caused by competition of various schools for scarce resources. This article proposes a two-pronged intellectual agenda that might clarify the complementarity of some orientations and strengthen the thrust towards theoretical integration.Paper read at the annual meeting of the Sociological Research Association, August 31, 1986.     

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     

Two-stage discrete aggregation: the Ostrogorski paradox and related phenomena

Motivated by certain paradoxa that have been discussed in the literature (Ostrogorski paradox), we prove an impossibility theorem for two-stage aggregation procedures for discrete data. We consider aggregation procedures of the following form: The whole population is partitioned into subgroups. First we aggregate over each subgroup, and in a second step we aggregate the subgroup aggregates to obtain a total aggregate. The data are either dichotomous (1 — 0; yes-no) or take values in a finite ordered set of possible attributes (e.g., exam grades A, B,...F). Examples are given by multistage voting procedures (indirect democracy, federalism), or by the forming of partial grades and overall grades in academic examinations and similar evaluation problems (sports competitions, consumer reports). It is well known from standard examples that the result of such a two-stage aggregation procedure depends, in general, not only on the distribution of attributes in the whole population, but also on how the attributes are distributed across the various subgroups (in other words: how the subgroups are defined). This dependence leads to certain paradoxa. The main result of the present paper is that these paradoxa are not due to the special aggregation rules employed in the examples, but are unavoidable in principle, provided the aggregators satisfy certain natural assumptions. More precisely: the only aggregator functions for which the result of a two-stage (a fortiori: multi-stage) aggregation does not depend on the partitioning are degenerate aggregators of the following form: there exists a partial order (dominance) on the set of possible attributes such that the aggregate over any collection of data is always equal to the supremum (w.r.t. dominance) of the attributes occurring in the data, regardless of the relative frequnencies of these occurrences. In the voting context, degeneracy corresponds to the unanimity principle. Our theorem is true for arbitrary partitionings of arbitrary (finite) sets and generalizes the results of Deb & Kelsey (for the matrix case with dichotomous variables and majority voting) to general two-stage aggregation procedures for attributes belonging to a finite ordered set. The general result is illustrated by some examples.This paper was completed during a visit to the University of Bielefeld. I am much indebted to the Faculty of Economics there for its hospitality; in particular I should like to thank Gerhard Schwödiauer and Walter Trockel for their support.     

Performance measurement: Some conceptual issues

Measures of how well a system is operating are clearly of interest to a wide range of users, from organizational researchers to system managers. The current literatures in such areas as “performance measurement, ” “organizational effectiveness, ” and the like show little consistency in their definitions of terms or in their methods for generating measures. The present paper attempts a clarification of these issues in the form of a conceptual minimalist position which requires only three basic definitions, and leaves, as far as possible, all remaining issues open to empirical investigation.We first review the literature on “organizational effectiveness” contrasting the organizational goals and systems paradigms, and note the lack of either theoretical or empirical convergence between the two. An examination of the nature of effectiveness statements suggests that this failure of convergence flows mainly from the different criterion sets generated by the two paradigms — and, importantly, that one should not expect convergence on a single measure or set of measures which uniquely define how well a system is performing. One's view of how well a given system is performing is a function of where one stands (either theoretically or in relationship to the system), and pursuit of the one true set of performance measures is a futile exercise. Instead, we propose to redirect attention to the identification of the various individuals and groups (“constituencies”) with an interest in system performance, and to the investigation of those items of system relevant information (their “performance measure sets”) which do, in fact, change their evaluations of how well the system is performing. This perspective will, we hope, redirect effort from futile theoretical debate to empirical investigations of what measures are used, by whom, and to what effect, in specific settings.     

Some hypotheses and evidence on tax knowledge and preferences

The article comments on a national quota sample of 900 adults in the UK. Questions were asked concerning beliefs as to the sources of government revenue to pay for services; whether taxes should he increased to improve services; and whether any extra tax levied should go on incomes or the goods people buy. The overall results showed widespread ignorance of government sources of revenue. The majority preferred to keep the level of taxation and expenditure at present levels, but if taxes had to he increased these should he levied on goods rather than incomes. Predictions of the overall and disaggregated results from economic theory and past survey research met with varying success, the implications of which are discussed.     

A quantitative theory of preferences: Some results on transition functions

We investigate a general theory of combining individual preferences into collective choice. The preferences are treated quantitatively, by means of preference functions (a,b), where 0(a,b) expresses the degree of preference of a to b. A transition function is a function (x,y) which computes (a,c) from (a,b) and (b,c), namely (a,c)=((a,b),(b,c)). We prove that given certain (reasonable) conditions on how individual preferences are aggregated, there is only one transition function that satisfies these conditions, namely the function (x,y)=x·y (multiplication of odds). We also formulate a property of transition functions called invariance, and prove that there is no invariant transition function; this impossibility theorem shows limitations of the quantitative method.Research supported in part by the National Science Foundation.     

Some issues for cooperative learning and intercultural education

This article connects the two fields of cooperative learning and intercultural education. We argue that cooperative learning strategies need to be equipped with intercultural understandings. Two key points that are raised here are: (1) that issues of competitiveness amongst learners and students must be dealt with head on rather than treating it from the sidelines or by brushing them aside; and (2) for learning to take place in a truly cooperative manner, there must be an emphasis on an intercultural focus within the curriculum; the content of knowledge within the curriculum needs to be non-centric. This article emphasizes that cooperative learning strategies are effective when the curricular knowledge taught in the school is drawn from all groups (dominant, subordinate or minority groups).     

Evidence from the field: Findings on issues related to planning,implementing and evaluating gender-based programs

An Initiative of the United States Department of Health and Human Services’ Office on Women's Health (OWH), Coalition for a Healthier Community (CHC), supports ten grantees across the U.S. in the implementation of gender-based health interventions targeting women and girls. A national evaluation is assessing whether gender-focused public health systems approaches are sustainable and cost effective in addressing health disparities in women and girls. To inform the evaluation, a systematic examination was conducted of literature in both the public and private sector designed to track, assess, understand, and improve women's health, public health systems approaches, and the cost-effectiveness and sustainability of gender-based programs. A two-person team assured the quality of the results following the review of abstracts and full-text articles. Of 123 articles meeting eligibility criteria (See inclusion criteria described in Section 2.2 below), only 18 met inclusion criteria specific to a focus on a systems approach, cost-effectiveness and/or sustainability. Studies assessing systems approaches suggested their effectiveness in changing perceptions and increasing knowledge within a community; increasing involvement of local decision-makers and other community leaders in women's health issues; and increasing community capacity to address women and girls’ health. Further evaluation of the cost-effectiveness and sustainability of gender-based approaches is needed.