Nondictatorially independent pairs and Pareto

Suppose a social welfare function f has complete and transitive values and the set X of outcomes has size m>3. If the domain D of f has the free quadruple property then the fraction of pairs on which f satisfies nondictatorship, independence, and Pareto is less than or equal to . Received: 9 June 1999/Accepted: 30 May 2000     

Acyclic sets of linear orders: A progress report

Let f(n) be the maximum cardinality of an acyclic set of linear orders on {1, 2, … , n}. It is known that f(3)=4, f(4)=9, f(5)=20, and that all maximum-cardinality acyclic sets for n≤ 5 are constructed by an “alternating scheme”. We outline a proof that this scheme is optimal for n=6, where f (6)=45. It is known for large n that f (n) >(2.17)ⁿ and that no maximum-cardinality acyclic set conforms to the alternating scheme. Ran Raz recently proved that f (n)<c ⁿ for some c>0 and all n. We conjecture that f (n + m)≤f (n + 1) f (m + 1) for n , m≥ 1, which would imply f (n)<(2.591)^{n − 2} for all large n. Received: 12 April 2000/Accepted: 4 December 2000     

Nondictatorially independent pairs

Suppose that social welfare function f satisfies the Pareto condition and has complete and transitive values. On a domain satisfying the free quadruple property, if the outcome set X has m< outcomes, then the set of pairs of distinct alternatives on which f satisfies both independence and nondictatorship contains at most the fraction 2/m of all pairs. On a domain satisfying the free six-tuple property, if X is a subset of Euclidean space then the set of pairs on which f satisfies both independence and nondictatorship has Lebesgue measure zero.Campbell's research was financed by the National Science Foundation, grant SES 9209039.     

The number of weak orderings of a finite set

The number of Arrovian constitutions, when N agents are to rank n alternatives, is p(n) ^{p(n) N} , where p(n) is the number of weak orderings of n alternatives. For n≤15, p(n) is the nearest integer to n!/2(log2) ^{n +1}, the dominant term of a series derived by contour integration of the generating function. For large n, about n/17 additional terms in the series suffice to compute p(n) exactly. Received: 29 May 1995 / Accepted: 22 May 1997     

Actions of symmetry groups

This paper studies maps which are invariant under the action of the symmetry group S _k . The problem originates in social choice theory: there are k individuals each with a space of preferences X, and a social choice map :X ^k X which is anonymous i.e. invariant under the action of a group of symmetries. Theorem 1 proves that a full range map :X ^k X exists which is invariant under the action of S _k only if, for all i1, the elements of the homotopy group _i (X) have orders relatively prime with k. Theorem 2 derives a similar results for actions of subgroups of the group S _k . Theorem 3 proves necessary and sufficient condition for a parafinite CW complex X to admit full range invariant maps for any prime number k:X must be contractible.Hospitality and research support from the Standard Institute for Theoretical Economics during the summer of 1991 is gratefully acknowledged. This paper was presented at a Colloquium in the Department of Mathematics, Columbia University, February 6, 1991. I thank the participants of the Colloquium and Jerry Kelly for helpful comments.     

Dictatorial domains in preference aggregation

We call a domain of preference orderings “dictatorial” if there exists no Arrovian (Pareto optimal, IIA and non-dictatorial) social welfare function defined over that domain. In a finite world of alternatives where indifferences are ruled out, we identify a condition which implies the dictatoriality of a domain. This condition, to which we refer as “being essentially saturated”, is fairly weak. In fact, independent of the number of alternatives, there exists an essentially saturated (hence dictatorial) domain which consists of precisely six orderings. Moreover, this domain exhibits the superdictatoriality property, i.e., every superdomain of it is also dictatorial. Thus, given m alternatives, the ratio of the size of a superdictatorial domain to the size of the full domain may be as small as 6/m!, converging to zero as m increases.     

Intergenerational equity and an explicit construction of welfare criteria

Ranking infinite utility streams includes many impossibility results, most involving certain Pareto, anonymity, or continuity requirements. We introduce the concept of the future agreement extension, a method that explicitly extends orderings on finite time horizons to an infinite time horizon. The future agreement extension of the given orderings is quasi-transitive, complete, and pairwisely continuous. Furthermore, its asymmetric part is larger than that of any other pairwisely continuous extension of the orderings. In case of anonymous and strongly Paretian orderings, their future agreement extension is variable step anonymous and strongly Paretian. Characterizations of the future agreement extensions of the utilitarian and leximin orderings are obtained as applications.     

From Arrow to cycles, instability, and chaos by untying alternatives

From remarkably general assumptions, Arrow's Theorem concludes that a social intransitivity must afflict some profile of transitive individual preferences. It need not be a cycle, but all others have ties. If we add a modest tie-limit, we get a chaotic cycle, one comprising all alternatives, and a tight one to boot: a short path connects any two alternatives. For this we need naught but (1) linear preference orderings devoid of infinite ascent, (2) profiles that unanimously order a set of all but two alternatives, and with a slightly fortified tie-limit, (3) profiles that deviate ever so little from singlepeakedness. With a weaker tie-limit but not (2) or (3), we still get a chaotic cycle, not necessarily tight. With an even weaker one, we still get a dominant cycle, not necessarily chaotic (every member beats every outside alternative), and with it global instability (every alternative beaten). That tie-limit is necessary for a cycle of any sort, and for global instability too (which does not require a cycle unless alternatives are finite in number). Earlier Arrovian cycle theorems are quite limited by comparison with these. Received: 31 July 1999/Accepted: 15 October 1999     

The structure of continuous-valued neutral monotonic social functions

This article consists of several results characterizing neutral monotonic social functions over alternative sets that are compact, convex subsets of Euclidean space. One major result is that a neutral monotonic social function is continuous-valued (i.e., does not make abrupt reversals) for all profiles of continuous-valued weak orderings if and only if it is simple (i.e., completely determined by a single set of decisive coalitions). A second major result is that a continuous-valued neutral monotonic social function will guarantee the existence of a socially undominated alternative for all profiles of continuousvalued, almost convex weak orderings if and only if the smallest empty intersection within the set of decisive coalitions is at least m+2 in size where m is the dimension of the alternative set.Associate Professor of Law, University of Southern California and Assistant Professor of Law and Economics, California Institute of Technology.This article is taken almost directly from a chapter of my Ph.D. dissertation (Strnad 1982). That chapter itself is substantially the same as an earlier paper, Strnad (1981). I have profited greatly from the suggestions, guidance and encouragement of Donald Brown in this work. Strnad (1981) was presented at the California Institute of Technology Theory Workshop in January 1982 and at the annual Public Choice Society Meetings in San Antonio, Texas on March 5–7, 1982. I am grateful to participants at both the Workshop and the Meetings for their comments. Comments by the referee and by Jerry Kramer, the editor, also were very helpful. Any remaining errors are solely my responsibility.Substantial financial support for this work has been provided by Summer Research Grants for 1981 and 1982 from the University of Southern California Law Center     

Extending the Condorcet Jury Theorem to a general dependent jury

We investigate necessary and sufficient conditions for the existence of Bayesian-Nash equilibria that satisfy the Condorcet Jury Theorem (CJT). In the Bayesian game G _n among n jurors, we allow for arbitrary distribution on the types of jurors. In particular, any kind of dependency is possible. If each juror i has a ??constant strategy??, ?? ⁱ (that is, a strategy that is independent of the size n ?? i of the jury), such that ?? = (?? ¹, ?? ², . . . , ?? ⁿ . . .) satisfies the CJT, then by McLennan (Am Political Sci Rev 92:413?C419, 1998) there exists a Bayesian-Nash equilibrium that also satisfies the CJT. We translate the CJT condition on sequences of constant strategies into the following problem:

(**) For a given sequence of binary random variables X?=?(X ¹, X ², . . . , X ⁿ , . . .) with joint distribution P, does the distribution P satisfy the asymptotic part of the CJT?

We provide sufficient conditions and two general (distinct) necessary conditions for (**). We give a complete solution to this problem when X is a sequence of exchangeable binary random variables.     

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.     

Global culture,learning style,and outcome: an interdisciplinary empirical study of international university students

The study examined 2500 business degree students from 21 countries, enrolled at an Australian university, using a survey to assess learning style, which was integrated into a global culture taxonomy. The research hypothesis was that academic outcome could be explained through an interdisciplinary model, by integrating proven theories from educational psychology (learning style) and anthropology (culture). The literature review included interdisciplinary and rival theories to make the research useful for global practitioners. Technical discussion was minimized, focusing on the pedagogical implications of a generalized least squares regression model, which was statistically significant (r ² = 0.87; X ² = 0, df = 0, p < 0.0001; n = 715). Four factors dominated the model in predicting higher multicultural student academic outcome: collectivist and risk‐taking cultures, along with visual‐input and active‐processing learning styles.

L’apporte a examiné le degré d’affaires 2500 étudiants de 21 pays, inscrits dans une université australienne, au moyen d’une enquête visant à évaluer le style d’apprentissage, qui a été intégré à une culture mondiale en matière de taxonomie. La recherche a été hypothèse que études résultat pourrait être expliqué par le biais d’un modèle interdisciplinaire, en intégrant les théories éprouvées de psychologie de l’éducation (style d’apprentissage) et de l’anthropologie (la culture). La revue de la littérature inclus interdisciplinaire et théories rivales à faire de la recherche utiles pour les praticiens mondial. Discussion technique a été réduit au minimum, axée sur les implications pédagogiques d’un généralisée des moindres carrés modèle de régression, qui a été statistiquement significative (r ² = 0.87; X ² = 0, df = 0, p < 0.0001; n = 715). Quatre facteurs dominent le modèle supérieur à la prévision multiculturelle des étudiants universitaires résultat: collectiviste et de prendre des risques cultures, ainsi que les visuels d’entrée et de processus actif styles d’apprentissage.     

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     

Determinants of approval of the work of truth commissions in the southern cone: a comparative study / Determinantes de la aprobación del trabajo realizado por las Comisiones de Verdad y Reconciliación en el cono sur: un estudio comparativo

The socio-demographic and psychosocial approval determinants of the work conducted by the truth and reconciliation commissions (TRC) are analysed based on original survey data from Chilean, Argentinian and Peruvian samples (N = 2,947). A linear multiple regression analysis was carried out (R² between .28 and .44; f² between .45 and .78) which reveals the positive effect of commission functions perceived achievement, mainly the knowledge of truth (β = .28), justice (β = .15) and the contribution in creating a common history (β = .15). An analysis by country reveals in Chile salient variables such as institutional trust, political ideas, institutional apologies and hope and sadness emotions. In Argentina, together with TRC functions, the degree of information about the work of the commission and the perception of a negative social climate appear as relevant predictors. As regards Peru, the degree of exposure to violence appears as a good predictor of TRC’s work support. The results show the relevance of TRC functions’ fulfillment, as well as the role of institutional variables in the approval of the work they have done.     

Uncovering bias in order assignment

To mitigate sequencing effects in decision-making, many situations require a set of items to be considered in a random order. When such orderings are repeated, one can test whether randomization indeed obtains, or whether some orderings have been manipulated in order to achieve a favorable result. This paper articulates the key features of this problem and presents three general tests for randomness. These methods are used to analyze the order in which lottery numbers are drawn in Powerball, contestants perform on American Idol, and candidates are placed on election ballots. This last application features frequent manipulation, with potentially serious consequences.     

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.     

Boolean analysis of questionnaire data

The principle of Boolean analysis is to look for entailments between variables in a questionnaire. If all the respondents answering “yes” to the question X answers “yes” to the question Y, we note the entailment X → Y. Boolean algebra generalizes this approach. A questionnaire of n binary variables generates a set of 2ⁿ possible patterns of response. These patterns generate a Boolean algebra. If only some of the possible patterns are present, the bipartition in this set defines a Boolean equation, the solutions of which are entailments. Usually we use approximations of this structure to analyse a questionnaire. A presentation of the method is given here with application to sociological data.     

X-inefficiency in the use of income to attain economic satisfaction

This investigation uses a subjective well-being approach to provide a novel empirical answer to an old normative debate in economic literature: whether consumers use efficiently their income. Based on a large database from Mexico, the paper shows that there exists substantial X-inefficiency in the use of income; even when a relaxed criterion to define the thick frontier is followed. X-inefficiency in the use of income can emerge from personal errors and from social-organization deficiencies. Sustainable development concerns make it critical to focus on reducing X-inefficiency as an alternative way to increase economic well-being.     

Poverty orderings and welfare dominance

This paper examines the partial orderings of discrete distributions derived from various poverty indices and sets of welfare functions. The poverty ordering with respect to some indexP is the ordering obtained whenP ranks consistently over a range of admissible poverty lines. The poverty orderings derived from the headcount ratio, the per-capita income gap and another distribution-sensitive index are characterized in some detail when the poverty standard is allowed to take any positive value, and these orderings are shown to coincide with the natural interpretation of first, second and third degree welfare dominance, respectively. Additional results are then obtained for the situation in which the admissible poverty lines cannot exceed some finite upper bound.We are particularly indebted to Nick Kiefer for initial insights into the issues addressed in this paper. We have also had helpful comments from Amartya Sen, from an anomymous referee, and from many seminar audiences. Financial support from the Krannert Endowment Trust and the Leverhulme Trust is gratefully acknowledged.