On multidimensional indices of poverty

The contribution of recent “multidimensional indices of poverty” may not be as obvious as one thinks. There are two issues in assessing that contribution: whether one believes that a single index can ever be a sufficient statistic of poverty, and whether one aggregates in the space of “attainments,” using prices when appropriate, or “deprivations,” using weights set by the analyst. The paper argues that we should aim for a credible set of multiple indices rather than a single multidimensional index. Partial aggregation will still be necessary, but ideally the weights should be consistent with well-informed choices by poor people.     

Quaternary dichotomous voting rules

In this article, we provide a general model of “quaternary” dichotomous voting rules (QVRs), namely, voting rules for making collective dichotomous decisions (to accept or reject a proposal), based on vote profiles in which four options are available to each voter: voting (“yes”, “no”, or “abstaining”) or staying home and not turning out. The model covers most of actual real-world dichotomus rules, where quorums are often required, and some of the extensions considered in the literature. In particular, we address and solve the question of the representability of QVRs by means of weighted rules and extend the notion of “dimension” of a rule.     

Anonymity in large societies

In a social choice model with an infinite number of agents, there may occur “equal size” coalitions that a preference aggregation rule should treat in the same manner. We introduce an axiom of equal treatment with respect to a measure of coalition size and explore its interaction with common axioms of social choice. We show that, provided the measure space is sufficiently rich in coalitions of the same measure, the new axiom is the natural extension of the concept of anonymity, and in particular plays a similar role in the characterization of preference aggregation rules.     

Argumentation in Multi-issue Debates

An axiomatic modeling approach to multi-issue debates is proposed. A debate is viewed as a decision procedure consisting of two stages: (1) an “argumentation rule” determines what arguments are admissible for each party, given the “raw data”, depending on the issue or set of issues under discussion; (2) a “persuasion rule” determines the strength of the admissible arguments and selects the winning party. Persuasion rules are characterized for various alternative specifications of the argumentation rule. These characterizations capture rhetorical effects that we sometimes encounter in real-life multi-issue debates.     

The replacement principle in economies with indivisible goods

 We consider the problem of allocating a list of indivisible goods and some amount of an infinitely divisible good among agents with equal rights on these resources, and investigate the implications of the following requirement on allocation rules: when the preferences of some of the agents change, all agents whose preferences are fixed should (weakly) gain, or they should all (weakly) lose. This condition is an application of a general principle of solidarity discussed in Thomson (1990b) under the name “replacement principle”. We look for selections from the no-envy solution satisfying this property. We show that in the general case, when the number of objects is arbitrary, there is no such selection. However, in the one-object case (a single prize), up to Pareto-indifference, there is only one selection from the no-envy solution satisfying the property. Such a solution always selects an envy-free allocation at which the winner of the prize is indifferent between his bundle and the losers’ common bundle. Received: 15 May 1995 / Accepted: 5 June 1996     

Pairwise Strategy-Proofness and Self-Enforcing Manipulation

“Strategy-proofness” is one of the axioms that are most frequently used in the recent literature on social choice theory. It requires that by misrepresenting his preferences, no agent can manipulate the outcome of the social choice rule in his favor. The stronger requirement of “group strategy-proofness” is also often employed to obtain clear characterization results of social choice rules. Group strategy-proofness requires that no group of agents can manipulate the outcome in their favors. In this paper, we advocate “effective pairwise strategy-proofness.” It is the requirement that the social choice rule should be immune to unilateral manipulation and “self-enforcing” pairwise manipulation in the sense that no agent of a pair has the incentive to betray his partner. We apply the axiom of effective pairwise strategy-proofness to three types of economies: public good economy, pure exchange economy, and allotment economy. Although effective pairwise strategy-proofness is seemingly a much weaker axiom than group strategy-proofness, effective pairwise strategy-proofness characterizes social choice rules that are analyzed by using different axioms in the literature.     

Characterizations of two power indices for voting games with r alternatives

In the first three sections of this paper we present a set of axioms which provide a characterization of an extension of the Banzhaf index to voting games with r alternatives, such as the United Nations Security Council where a nation can vote “yes”, “no”, or “abstain”. The fourth section presents a set of axioms which characterizes a power index based on winning sets instead of pivot sets. Received: 4 April 2000/Accepted: 30 April 2001     

Robustness against inefficient manipulation

This paper identifies a family of scoring rules that are robust against coalitional manipulations that result in inefficient outcomes. We discuss the robustness of a number of Condorcet consistent and “point runoff” voting rules against such inefficient manipulation and classify voting rules according to their potential vulnerability to inefficient manipulation.     

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.     

Locating libraries on a street

Hotelling (1929) studied two competing firms choosing their locations on a street. We consider instead a planner who builds two identical public facilities (e.g., libraries, parks, bridges, etc). We ask a normative question: Where should the planner build these facilities? We prove an axiomatic characterization of the efficient social choice rules that satisfy what is called the replacement-domination, which is a formulation of the idea of “solidarity” among the agents. Received: 26 November 1997/Accepted: 28 February 2000     

The power of a collectivity to act in weighted voting games with many small voters

We analyze the propensity to approve a random proposal of a large committee that makes decisions by weighted voting. The approach is a generalized version of James Coleman’s “power of a collectivity to act”. Throughout the paper it is assumed that the voters are of two kinds: a fixed (possibly empty) set of “major” (big) voters with fixed weights, and an ever-increasing number of “minor” (small) voters, whose total weight is also fixed, but where each individual’s weight becomes negligible. As our main result, we obtain that asymptotically many minor voters act like a modification of the quota for the vote among major voters. The paper estimates the rate of convergence which turns out to be very high if the weight distribution among the small voters is not too skewed. The results obtained are illustrated by evaluating the decision rules for the Council of Ministers of the EU for various scenarios of EU enlargement. I wish to thank Matthew Braham, Sidartha Gordon, Maurice Koster, Moshé Machover, Guillermo Owen and two anonymous referees for helpful comments.     

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     

Characterizing best–worst voting systems in the scoring context

An increasing body of theoretical and empirical work on discrete choice considers a choice design in which a person is asked to select both the best and the worst alternative in an available set of alternatives, in contrast to more traditional tasks, such as where the person is asked to: select the best alternative; select the worst alternative; rank the alternatives. Here we consider voting systems motivated by such “best–worst” choice; characterize a class of “best–worst” voting systems in terms of a set of axioms in the context of scoring rules; and discuss briefly possible extensions to approval–disapproval systems.     

Geometric models of consistent judgement aggregation

Given a set of propositions with unknown truth values, a ‘judgement aggregation function’ is a way to aggregate the personal truth-valuations of a group of voters into some ‘collective’ truth valuation. We introduce the class of ‘quasimajoritarian’ judgement aggregation functions, which includes majority vote, but also includes some functions which use different voting schemes to decide the truth of different propositions. We show that if the profile of individual beliefs satisfies a condition called ‘value restriction’, then the output of any quasimajoritarian function is logically consistent; this directly generalizes the recent work of Dietrich and List (Majority voting on restricted domains. Presented at SCW08; see , 2007b). We then provide two sufficient conditions for value-restriction, defined geometrically in terms of a lattice ordering or a metric structure on the set of individuals and propositions. Finally, we introduce another sufficient condition for consistent majoritarian judgement aggregation, called ‘convexity’. We show that convexity is not logically related to value-restriction.     

Aggregation of binary evaluations for truth-functional agendas

In the problem of judgment aggregation, a panel of judges has to evaluate each proposition in a given agenda as true or false, based on their individual evaluations and subject to the constraint of logical consistency. We elaborate on the relation between this and the problem of aggregating abstract binary evaluations. For the special case of truth-functional agendas we have the following main contributions: (1) a syntactical characterization of agendas for which the analogs of Arrow’s aggregation conditions force dictatorship; (2) a complete classification of all aggregators that satisfy those conditions; (3) an analysis of the effect of weakening the Pareto condition to surjectivity. This is a sequel to the paper “Aggregation of binary evaluations.” The contents of both papers were presented, under the title “An Arrovian impossibility theorem for social truth functions,” at the Second World Congress of the Game Theory Society, Marseille, July 2004. The first version of “Aggregation of binary evaluations” was completed in June 2005. That working paper was subsequently split into two parts, of which this is the second. The comments of an anonymous referee are gratefully acknowledged. Part of R. Holzman’s work was done while he was a Fellow of the Institute for Advanced Studies at the Hebrew University of Jerusalem.     

Altruistic Individualists: Motivations for International Volunteering Among Young Adults in Switzerland

Modernization theory posits a change from traditional or “collective” forms to modern or “reflective” forms of volunteering. In a research project using a combined qualitative–quantitative approach, the motivation of 118 young Swiss adults who showed an interest in international volunteering was investigated. Qualitative analysis revealed 12 different motives which could be categorized into three different groups: A first group called “Achieving something positive for others,” a second group named “Quest for the new,” and a third group of motives labeled “Quest for oneself.” Motivations of young Swiss adults for international volunteering clearly show the characteristics of “reflexive” volunteers. Most respondents displayed a combination of motives while for only 11% of them altruism (“Achieving something positive for others”) was the one and only driving force behind their interest in international volunteering. The inductively constructed typology of motives can be a useful planning device for organizations that run or intend to set up an international volunteering program for young adults.     

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     

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.     

Correction to: Divorce among European and Mexican immigrants in the U.S.

The original version of this article unfortunately contained few errors in tables. In Tables 2a, 2b, 3a and 3b, the “Percent of sample” and “Percent spouse absent” should be in decimal in order to be consistent. In Tables 4 and 5, the first row should read as “Age at marriage” instead of “Percent of sample” and the “Sample size” was incorrect.