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.     

Constrained allocation problems with single-peaked preferences: An axiomatic analysis

We introduce a new class of problems that contains two existing classes: allocation problems with single-peaked preferences and bankruptcy problems. On this class, we analyze the implications of well-known properties such as Pareto optimality, strategy-proofness, resource-monotonicity, no-envy, equal treatment of equals, and two new properties we introduce, hierarchical no-envy and independence of nonbinding constraints. Unlike earlier literature, we consider rules that allow free-disposability. We present characterizations of a rule we introduce on this domain. We relate this rule to well-known rules on the aforementioned subdomains. Based on this relation, we present a characterization of a well-known bankruptcy rule called the constrained equal awards rule. Received: 22 June 2000/Accepted: 21 March 2002 This paper is based on the first chapter of my Ph.D. thesis submitted to the University of Rochester. I wish to thank my advisor, William Thomson, for helpful comments and suggestions.     

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.     

Judgment aggregation without full rationality

Several recent results on the aggregation of judgments over logically connected propositions show that, under certain conditions, dictatorships are the only propositionwise aggregation functions generating fully rational (i.e., complete and consistent) collective judgments. A frequently mentioned route to avoid dictatorships is to allow incomplete collective judgments. We show that this route does not lead very far: we obtain oligarchies rather than dictatorships if instead of full rationality we merely require that collective judgments be deductively closed, arguably a minimal condition of rationality, compatible even with empty judgment sets. We derive several characterizations of oligarchies and provide illustrative applications to Arrowian preference aggregation and Kasher and Rubinstein’s group identification problem. This paper was circulated in August 2006 and presented at the Yale workshop on Aggregation of Opinions, September 2006, at the Centre interuniversitaire de rechere en économie quantitative, Montreal, October 2006, and at the 1st International Workshop on Computational Social Choice, Amsterdam, December 2006. We are grateful to the participants at these occasions and the anonymous referees for comments. We also thank Ton Storcken for discussion. Elad Dokow and Ron Holzman have independently proved closely related results, which were also presented at the Yale workshop in September 2006, and circulated in the December 2006 paper (Dokow and Holzman 2006).     

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.     

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.     

Social choice and the closed convergence topology

This paper revisits the aggregation theorem of Chichilnisky (1980), replacing the original smooth topology by the closed convergence topology and responding to several comments (N. Baigent (1984, 1985, 1987, 1989), N. Baigent and P. Huang (1990) and M. LeBreton and J. Uriarte (1990a, b). Theorems 1 and 2 establish the contractibility of three spaces of preferences: the space of strictly quasiconcave preferences P _SCO, its subspace of smooth preferences P _infSCO ^supS , and a space P ₁ of smooth (not necessarily convex) preferences with a unique interior critical point (a maximum). The results are proven using both the closed convergence topology and the smooth topology. Because of their contractibility, these spaces satisfy the necessary and sufficient conditions of Chichilnisky and Heal (1983) for aggregation rules satisfying my axioms, which are valid in all topologies. Theorem 4 constructs a family of aggregation rules satisfying my axioms for these three spaces. What these spaces have in common is a unique maximum (or peak). This rather special property makes them contractible, and thus amenable to aggregation. However, these aggregation rules cannot be extended to the whole space of preferences P which is not contractible and therefore does not admit continuous aggregation rules satisfying anonymity and unanimity, Chichilnisky (1980, 1982). The results presented here clarify an erroneous example in LeBreton and Uriarte (1990a, b) and respond to Baigent (1984, 1985, 1987) and Baigent and Huang (1990) on the relative advantages of continuous and discrete approaches to Social Choice.Comments from Geoffrey M. Heal, Andreu Mas Colell, Jean Francois Mertens and Maurice Salles are gratefully acknowledged. Research support was provided by NSF SES 8409857.     

The retirement behavior of workers covered by union and nonunion pension plans

Previous authors have noted that there are significant differences between the provisions of union and nonunion pension plans. I present evidence that sheds light on two hypotheses. The first (Parsons, 1983) posits that union pensions should encourage earlier retirement because productivity falls as workers age, but union rules prohibit firms from lowering wages. The second (Freeman, 1985) argues that union pension plans reflect the preferences of older, more senior workers. I find some support for both hypotheses. I conducted some of the research for this paper as an economist with Unicon Research Corporation. I thank the National Institute on Aging for funding (grant number 5 RO1 AG06133-03). I also thank Fran Horvath, Mark Kennet, Mark Loewenstein, Bob McIntire, Tom Plewes, Bill Wiatrowski, and three anonymous referees for useful comments. All views and opinions expressed herein are the author's and do not necessarily reflect those of the Bureau of Labor Statistics, the National Institute on Aging, or Unicon Research Corporation.     

Two families of rules for the adjudication of conflicting claims

We define two families of rules to adjudicate conflicting claims. The first family contains the constrained equal awards, constrained equal losses, Talmud, and minimal overlap rules. The second family, which also contains the first two of these rules, is obtained from the first family by exchanging, for each claims problem, how well agents with relatively larger claims are treated as compared to agents with relatively smaller claims. In each case, we identify the subfamily of consistent rules. I gratefully acknowledge support from NSF under grant SES-0214691. I also thank Tarık Kara, Cori Vilella, and in particular Juan Moreno-Ternero, Rodrigo Velez, and a referee, for their comments. This paper grew out of a section of the Condorcet Lecture I delivered at the VI-th International Meeting of the Social Choice and Welfare Society, Alicante, July 2000.     

Measurement of inequality: An attempt at unification and generalization

This paper reviews and extends the theory of ethical inequality indices. It presents a novel axiom (strict separability of social welfare orderings in rank-ordered subspaces). This axiom allows to provide joint characterizations of the most important inequality measures (Atkinson family, Kolm-Pollak family and Generalized Ginis) and of some new more general classes of indices. The whole derivation is based on weak assumptions. In an ordinal framework only continuity of the underlying ordering is required and no cardinal properties are employed.I thank two anonymous referees for helpful comments and suggestions.     

The possibility of efficient provision of a public good in voluntary participation games

In this study, we provide the conditions for efficient provision of a public good in a participation game in which a non-negative integer number of units of the public good can be provided. In the case in which at most one unit of the public good can be provided, we provide refinements of Nash equilibria at which agents choose only a Nash equilibrium with an efficient allocation and provide sufficient conditions for cost-sharing rules that guarantee the existence of a Nash equilibrium with an efficient allocation. In the case of a multi-unit public good, we provide a necessary and sufficient condition for the existence of a Nash equilibrium with an efficient allocation and prove that Nash equilibria are less likely to support efficient allocations if the participation of many agents is needed for efficient provision of the public good in the case of identical agents. I would like to thank Koichi Tadenuma, Yukihiro Nishimura, Toshiyuki Fujita, John Weymark, Tatsuyoshi Saijo, Takehiko Yamato, Toshiji Miyakawa, Motohiro Sato, Takashi Shimizu, Nobue Suzuki, and Dirk T. G. Rübbelke for helpful comments and suggestions. I also thank the anonymous referees for detailed comments and suggestions. This research was supported by the Japanese Economic Research Foundation and the Grand-in-Aid for Young Scientists (Start-up) from the Japan Society for the Promotion of Science. Any remaining errors are my own.     

Non-collegial simple games and the nowhere denseness of the set of preference profiles having a core

In recent papers, somewhat conflicting results on the generic emptiness of the core have been proven. Rubinstein (1979) has shown that the core is generically empty in the Kannai topology even without a restriction on the dimensionality of the alternative space. Schofield (1980) finds the core generically empty in the Whitney-C topology, but requires a dimensionality condition. This note explores the apparent conflict by proving a generalization of Rubinstein's result for the Whitney-C ⁰ topology, and showing that the result does not extend to the Whitney-C ¹ topology without a dimensionality condition.I would like to thank Kim Border and Richard McKelvey for their helpful discussion and comments. This research was supported in part by the National Science Foundation under Grant no. SES 78-2478781     

A spatial study of teachers’ salaries in Pennsylvania school districts

I explore the relationship among teacher salaries across Pennsylvania school districts. Using techniques developed in spatial econometrics, I find that the error terms in a salary regression are spatially correlated, suggesting evidence of omitted labor market factors. I also find evidence that salaries in nearby, financially similar districts directly influence teacher salaries in a particular district, which is evidence of pattern bargaining or more informal social comparisons across districts. Econometric specifications that ignore these factors overstate the influence of own-district variables, such as economic indicators, on salary. I thank colleagues Linda Babcock and John Engberg for very valuable contributions and for use of their data. Thanks also to Wil Gorr, who helped with some of the GIS aspects of this research. A National Science Foundation grant supported data collection.     

Acyclicity and the dispersion of the veto power

Blair and Pollak (Econometrica (1982) 50: 931–943) prove that, if there are more alternatives than individuals, then, for every arrovian binary decision rule that is acyclic, there is at least one individual who has a veto power over a critical number of pairs of alternatives. If the number of individuals is larger than the number of alternatives, there need not be single vetoers but there could be small coalitions endowed with a similar power. Kelsey (Soc Choice Welfare (1985) 2: 131–137) states precise results in this respect. In this paper, we first give a new and much simpler proof of the main result of Blair and Pollak and complete proofs of the generalization of this result by Kelsey. Then we give a precise answer as to the minimum size of the coalitions that must have a veto power under any acyclic binary decision rule and the minimum number of pairs of alternatives on which these coalitions may exercise their power. We also show that, if the veto power of the coalitions of the minimal size attainable under the last objective is limited to the minimum number of pairs of alternatives, then all larger coalitions have a veto power on all pairs. All the results are obtained by appealing to an acyclicity condition found by Ferejohn and Fishburn (J Econ Theory (1979) 21: 28–45). In the case of symmetric and monotonic binary decision rules, proofs are even easier and illustrate clearly the reasons for the veto power.     

The implementation of social choice functions via social choice correspondences: A general formulation and a limit result

A corollary of Maskin's characterization theorem for Nash implementable social choice correspondences is that only trivial social choice functions can be implemented. This paper explores the consequences of implementing non-trivial social choice functions by extending them minimally to social choice correspondences which are implementable. The concept of asymptotic monotonicity is introduced. The main result states that it is not possible to find social choice rules satisfying a mild condition on its range, which is asymptotically monotonic. The implication of this result is that the multiplicity of equilibria problem which is at the heart of Nash implementation theory persists even in the limit as the number of individuals in society tends to infinity. This is true even though the opportunities for an individual to manipulate the outcome disappears in the limit.This paper is an extensively revised version of a chapter of my Ph.D. dissertation submitted to Princeton University in June 1987. I wish to thank my advisor Hugo Sonnenschein for his valuable advice and constant encouragement. I am also grateful to Andrew Caplin, Vijay Krishna, William Thomson, Jean-Luc Vila and two anonymous referees of this journal for their numerous suggestions. All remaining errors are my own responsibility.     

Cycling of simple rules in the spatial model

McKelvey [4] proved that for strong simple preference aggregation rules applied to multidimensional sets of alternatives, the typical situation is that either the core is nonempty or the top-cycle set includes all available alternatives. But the requirement that the rule be strong excludes, inter alia, all supermajority rules. In this note, we show that McKelvey's theorem further implies that the typical situation for any simple rule is that either the core is nonempty or the weak top-cycle set (equivalently, the core of the transitive closure of the rule) includes all available alternatives. Moreover, it is often the case that both of these statements obtain. Received: 13 October 1997/Accepted: 24 August 1998     

Arrow’s theorem in judgment aggregation

In response to recent work on the aggregation of individual judgments on logically connected propositions into collective judgments, it is often asked whether judgment aggregation is a special case of Arrowian preference aggregation. We argue for the converse claim. After proving two impossibility theorems on judgment aggregation (using “systematicity” and “independence” conditions, respectively), we construct an embedding of preference aggregation into judgment aggregation and prove Arrow’s theorem (stated for strict preferences) as a corollary of our second result. Although we thereby provide a new proof of Arrow’s theorem, our main aim is to identify the analogue of Arrow’s theorem in judgment aggregation, to clarify the relation between judgment and preference aggregation, and to illustrate the generality of the judgment aggregation model.     

Vetoer and tie-making group theorems for indifference-transitive aggregation rules

A binary relation is indifference-transitive if its symmetric part satisfies the transitivity axiom. We investigated the properties of Arrovian aggregation rules that generate acyclic and indifference-transitive social preferences. We proved that there exists unique vetoer in the rule if the number of alternatives is greater than or equal to four. We provided a classification of decisive structures in aggregation rules where the number of alternatives is equal to three. Furthermore, we showed that the coexistence of a vetoer and a tie-making group, which generates social indifference, is inevitable if the rule satisfies the indifference unanimity. The relationship between the vetoer and the tie-making group, i.e., whether the vetoer belongs to the tie-making group or not, determines the power structure of the rule.     

The Libertarian paradox: some further observations

It is shown that the weak Pareto principle consists of two parts: Pareto Neutrality and Weak Pareto Unanimity. It is the Pareto Neutrality which is responsible for the Paradox of the Paretian Libertarian. The libertarianism condition can also be factorized into two parts: the Libertarian Invariance and the Libertarian Non-Imposition. It is the Libertarian Invariance which is responsible or the Paradox of the Paretian Libertarian. Under conditions of Unrestricted Domain, Pareto Neutrality and Libertarian Invariance, if we require a social preference to be acyclic, then(1) neither can individuals' personal rights be respected, nor can they be reversed;(2) neither can unanimous group rights be respected, nor can they be reversed. Consequently, the Paradox of the Paretian Libertarian is due to the inconsistent use of information contained in the weak Pareto principle and the libertarianism condition.I would like to thank Nick Baigent, Prasanta Pattanaik, John Riley, Amartya Sen and Kotaro Suzumura for their comments. I would also like to thank two anonymous referees for their suggestions and comments on earlier drafts of the paper which led to a great improvement of the present version.