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     

Subcyclic sets of linear orders

A set T of linear orders of [n] = {1,2, . . . ,n} is subcyclic if every 3-set in [n] has one order that appears in no order in T. With g(n) the maximum cardinality of a subcyclic set for n, Raz [8] proved that g(n)<cⁿ for some c>0, thus resolving a longstanding conjecture of a similar upper bound for maximum acyclic sets. The present paper proves that g(4)=14 and g(5)=42 with both maxima attained only by subcyclic sets with a particular structure for all quadruples in [n]. We conjecture a similar result for larger n, in which case g(n) is the n^th Catalan number and min(c)=4 for Raz’s bound.     

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     

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     

Ensuring every candidate wins under positional voting

Given a fixed set of voter preferences, different candidates may win outright given different scoring rules. We investigate how many voters are able to allow all n candidates to win for some scoring rule. We will say that these voters impose a disordering on these candidates. The minimum number of voters it takes to impose a disordering on three candidates is nine. For four candidates, six voters are necessary, for five candidates, four voters are necessary, and it takes only three voters to disorder nine candidates. In general, we prove that m voters can disorder n candidates when m and n are both greater than or equal to three, except when m = 3 and n ≤ 8, when n = 3 and m ≤ 8, and when n = 4 and m = 4 or 5.     

Stability of voting games

This paper generalizes the result of Le Breton and Salles (1990) about stable set (far-sighted core of order 1) for voting games to far-sighted core of arbitrary order. Let m be the number of alternatives, n be the number of voters and G(n,k) be a proper symmetric simple game in which the size of a winning coalition is greater or equal to k. It is shown that the far-sighted core of order d for G(n,k) is nonempty for all preference profiles and for all n and k with n/(n–k)=v ¹ iff m(d+1)(v–1).This paper is part of my dissertation. I am grateful to my thesis advisor Leonid Hurwicz for his guidance and encouragement. I would like to thank Edward Green, Lu Hong, James Jordan, Andrew McLennan, Herve Moulin and Marcel Richter for their very helpful suggestions. Especially a referee and Maurice Salles made many good comments. Of course, any errors that remain are the sole responsibility of the author.     

Existence of a multicameral core

When a single group uses majority rule to select a set of policies from an n-dimensional compact and convex set, a core generally exists if and only if n = 1. Finding analogous conditions for core existence when an n-dimensional action requires agreement from m groups has been an open problem. This paper provides a solution to this problem by establishing sufficient conditions for core existence and characterizing the location and dimensionality of the core for settings in which voters have Euclidean preferences. The conditions establish that a core may exist in any number of dimensions whenever n ≤ m as long as there is sufficient preference homogeneity within groups and heterogeneity between groups. With m > 1 the core is however generically empty for . These results provide a generalization of the median voter theorem and of non-existence results for contexts of concern to students of multiparty negotiation, comparative politics and international relations.     

Acyclic sets of linear orders via the Bruhat orders

We describe Abello’s acyclic sets of linear orders [SIAM J Discr Math 4(1):1–16, 1991] as the permutations visited by commuting equivalence classes of maximal reduced decompositions. This allows us to strengthen Abello’s structural result: we show that acyclic sets arising from this construction are distributive sublattices of the weak Bruhat order. This, in turn, shows that Abello’s acyclic sets are, in fact, the same as Chameni-Nembua’s distributive covering sublattices (S.T.D.C s). Fishburn’s alternating scheme is shown to be a special case of the Abello/Chameni-Nembua acyclic sets. Any acyclic set that arises in this way can be represented by an arrangement of pseudolines, and we use this representation to derive a simple closed form for the cardinality of the alternating scheme. The higher Bruhat orders prove to be a natural mathematical framework for this approach to the acyclic sets problem. We would like to thank the Editor, Professor Bernard Monjardet, and two anonymous referees for their comments and additional references.     

The C1 topology on the space of smooth preference profiles

This paper defines a fine C ¹-topology for smooth preferences on a “policy space”, W, and shows that the set of convex preference profiles contains open sets in this topology.  It follows that if the dimension(W)≤v(?)−2 (where v(?) is the Nakamura number of the voting rule, ?), then the core of ? cannot be generically empty. For higher dimensions, an “extension” of the voting core, called the heart of ?, is proposed. The heart is a generalization of the “uncovered set”. It is shown to be non-empty and closed in general. On the C ¹-space of convex preference profiles, the heart is Paretian. Moreover, the heart correspondence is lower hemi-continuous and admits a continuous selection. Thus the heart converges to the core when the latter exists. Using this, an aggregator, compatible with ?, can be defined and shown to be continuous on the C ¹-space of smooth convex preference profiles. Received: 3 April 1995/Accepted: 8 April 1998     

Powers of subgroups in voting bodies

This paper discusses the power p _n of an n-member subgroup B _n of an N-member voting body, N odd and 1 n N. In contrast to bloc voting, we assume that the members vote independently with equal probability for and against a given issue. Power p _n is defined as the probability that the outcome of a vote changes if all members of B _n reverse their votes. Theorems: p _{n + 1} = _n for odd n < N; p _n + p _{N – n} = 1; P _m + p _n > p _{m + n} if m + n < N; p _{n + 1}/p _n (n + 1)/n as N for fixed even n; for rational 0 > > 1, p _N 2^–1 sin^{–1 1/2} as N . A simple summation formula is given for p _n .     

The sensitivity of weight selection for scoring rules to profile proximity to single-peaked preferences

Niemi (Am Polit Sci Rev 63:488–497, 1969) proposed a simple measure of the cohesiveness of a group of n voters’ preferences that reflects the proximity of their preferences to single-peakedness. For three-candidate elections, this measure, k, reduces to the minimum number of voters who rank one of the candidates as being least preferred. The current study develops closed form representations for the conditional probability, PASW(n,IAC|k), that all weighted scoring rules will elect the Condorcet winner in an election, given a specified value of k. Results show a very strong relationship between PASW(n,IAC|k) and k, such that the determination of the voting rule to be used in an election becomes significantly less critical relative to the likelihood of electing the Condorcet winner as voters in a society have more structured preferences. As voters’ preferences become more unstructured as measured by their distance from single-peakedness, it becomes much more likely that different voting rules will select different winners.A preliminary version of this paper was presented at the European Public Choice Society Conference in Berlin, Germany, April 15–18, 2004.     

Unequivocal majority and Maskin-monotonicity

The unequivocal majority of a social choice rule is a number of agents such that whenever at least this many agents agree on the top alternative, then this alternative (and only this) is chosen. The smaller the unequivocal majority is, the closer it is to the standard (and accepted) majority concept. The question is how small can the unequivocal majority be and still permit the Nash-implementability of the social choice rule; i.e., its Maskin-monotonicity. We show that the smallest unequivocal majority compatible with Maskin-monotonicity is n- ë \fracn-1m û{n-\left\lfloor \frac{n-1}{m} \right\rfloor} , where n ≥ 3 is the number of agents and m ≥ 3 is the number of alternatives. This value is equal to the minimal number required for a majority to ensure the non-existence of cycles in pairwise comparisons. Our result has a twofold implication: (1) there is no Condorcet consistent social choice rule satisfying Maskin-monotonicity and (2) a social choice rule satisfies k-Condorcet consistency and Maskin-monotonicity if and only if k 3 n- ë \fracn-1m û{k\geq n-\left\lfloor \frac{n-1}{m}\right\rfloor}.     

Soil-atmosphere exchange of carbon dioxide,methane and nitrous oxide in urban garden systems: impact of irrigation,fertiliser and mulch

Urban green spaces provide important ecosystem services, such as amenity, biodiversity, productivity, climate amelioration, hydrological and biogeochemical cycling. Intensively managed urban gardens can sequester carbon through vegetation growth and soil C increase, but may experience nitrous oxide (N₂O) emissions and reduced soil methane (CH₄) uptake from irrigation and fertiliser use. Soil atmosphere exchange of N₂O, CH₄ and carbon dioxide (CO₂) was measured in lawn and wood chip mulched garden areas in Melbourne, Australia in winter, spring and summer under various water and fertiliser regimes. Gas exchange before and after lawn fertiliser application was measured continuously for three weeks using an automated chamber system. Applying fertiliser led to a peak N₂O emission of >60 μg N m⁻² h⁻¹, but overall only weekly irrigation (10 mm) significantly increased mean soil N₂O emissions above that in other treatments. Under mulch, mean soil N₂O emissions (14.0 μg N m⁻² h⁻¹) were significantly smaller than from irrigated lawn (27.9 μg N m⁻² h⁻¹), whereas mean soil CH₄ uptake under mulch (−30.7 μg C m⁻² h⁻¹) was significantly greater (p < 0.01) than in any lawn treatment. Lawns were either a weak CH₄ sink or source. Soil C density (0–25 cm) under mulch (12.5 kg C m⁻²) was greater that under lawn (8.0 kg C m⁻²). On a carbon dioxide equivalent (CO₂-e) basis, soil N₂O emissions offset the benefits of soil CH₄ uptake. Mulched garden areas provide greatest C sequestration potential in soil and vegetation and the smallest non-CO₂ emissions, as soil CH₄ uptake offsets a large fraction of soil N₂O emissions. Results of this study suggest that reducing the irrigation and fertiliser application to lawns can help mitigate GHG emissions from urban garden systems, and increasing the area of mulched perennial garden beds can also provide net GHG benefits; however, this needs to be tested in other garden systems with different soil types and environmental conditions.     

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     

Ranking profiles of capability sets

In this paper, we present an innovative approach for ranking profiles of capability sets on the basis of equity. An interesting way of capturing the notion of equity is to take into account the extent to which each of the different functioning vectors is shared by the population under consideration (of size n). This is done by defining the ‘common capability sets’ enjoyed by k individuals as the set of functioning vectors simultaneously available to at least k individuals (1 ≤ k ≤ n). These sets are closely related to the original capability sets and have some interesting properties that are examined throughout the paper. We define and axiomatically characterize a capability profile ranking that lexicographically compares the different common capability sets. We would like to thank for helpful discussion and comments to Jorge Alcalde-Unzu, Ritxar Arlegi, Miguel A. Ballester, Jose Enrique Galdón, Antonio Nicolò, Jorge Nieto, Prasanta Pattanaik, John Weymark and the anonymous referees of this paper. The project is supported by Spanish’ Comision Interministerial de Ciencia y Technologia (SEC2003-08105) and by the European Commission (MRTN-CT-2003-504796).     

Identifying habitat sinks: a case study of Cooper’s hawks in an urban environment

We studied a population of Cooper’s hawks (Accipiter cooperii) in Tucson, Arizona from 1994 to 2005. High rates of mortality of nestlings from an urban-related disease prompted speculation that the area represented an ecological trap and habitat sink for Cooper’s hawks. In this paper, we used estimates of survival and productivity from 11years of monitoring to develop an estimate of the rate of population change, λ, for Cooper’s hawks in the area. We used a Cormack–Jolly–Seber approach to estimate survival of breeding hawks, and a stochastic, stage-based matrix to estimate λ. Despite the urban-related disease, the estimate of λ indicated that the area does not function as a habitat sink for Cooper’s hawks (= 1.11 ± 0.047; P = 0.0073 for the null of λ ≤ 1). Because data required to reliably identify habitat sinks are extensive and difficult to acquire, we suggest that the concept of habitat sinks be applied cautiously until substantiated with reliable empirical evidence.     

Assessing Overweight,Obesity, Diet,and Physical Activity in College Students

Abstract

The authors surveyed 738 college students aged 18 to 27 years to assess overweight, obesity, dietary habits, and physical activity. They used BMI (body mass index) > 25 kg/m² or BMI > 85th percentile and BMI > 30 kg/m¹ or BMI > 95th percentile to estimate overweight and obesity in those aged < 19 years. To define overweight and obesity in those > 20 years, they used BMI > 25 kg/m² and > 30 kg/m². They found overweight rates of 21.6% using BMI directly and 16.2% using BMI percentile and obesity rates of 4.9% using BMI directly and 4.2% using BMI percentile. More than 69% of the participants reported < 5 servings of fruits and vegetables per day and more than 67% reported < 20 g of fiber per day; participants reported physical activity on fewer than 3 d/wk. Most college students are not meeting dietary and physical activity guidelines, suggesting the need for prevention interventions and increased understanding of overweight in college students.     

The differences of sarcopenia-related phenotypes: effects of gender and population

Sarcopenia is a serious condition especially in the elderly population mainly characterized by the loss of skeletal muscle mass and strength with aging. Extremity skeletal muscle mass index (EMMI) (sum of skeletal muscle mass in arms and legs/height²) is gaining popularity in sarcopenia definition (less than two standard deviations below the mean of a young adult reference group), but little is known about the gender- and population-specific differences of EMMI. This study aimed at investigating the differences of EMMI, arm muscle mass index (AMMI), and leg muscle mass index (LMMI) between gender groups and populations (Chinese vs. Caucasians). The participants included 1,809 Chinese and 362 Caucasians with normal weight aged from 19 to 45 years old. Extremity muscle mass, arm muscle mass, and leg muscle mass were measured by using dual energy x-ray absorptiometry. Independent sample t tests were used to analyze the differences in muscle mass indexes between the studied groups. All the study parameters including EMMIs, AMMIs, and LMMIs were significantly higher (P ≤ 0.0003) in the Caucasian group than in the Chinese group and also higher in the male group than in the female group, and these significant differences (P ≤ 0.0005) remained after adjusting for age by simple regressions. The detected differences of muscle mass indexes between different gender and ethnic groups may provide important implications in their different risk of future sarcopenia.     

The inverse Banzhaf problem

Let ${\mathcal{F}}Let F{\mathcal{F}} be a family of subsets of the ground set [n] = {1, 2, . . . , n}. For each i ? [n]{i \in [n]} we let p(F,i){p(\mathcal{F},i)} be the number of pairs of subsets that differ in the element i and exactly one of them is in F{\mathcal{F}}. We interpret p(F,i){p(\mathcal{F},i)} as the influence of that element. The normalized Banzhaf vector of F{\mathcal{F}}, denoted B(F){B(\mathcal{F})}, is the vector (B(F,1),...,B(F,n)){(B(\mathcal{F},1),\dots,B(\mathcal{F},n))}, where B(F,i)=\fracp(F,i)p(F){B(\mathcal{F},i)=\frac{p(\mathcal{F},i)}{p(\mathcal{F})}} and p(F){p(\mathcal{F})} is the sum of all p(F,i){p(\mathcal{F},i)}. The Banzhaf vector has been studied in the context of measuring voting power in voting games as well as in Boolean circuit theory. In this paper we investigate which non-negative vectors of sum 1 can be closely approximated by Banzhaf vectors of simple voting games. In particular, we show that if a vector has most of its weight concentrated in k < n coordinates, then it must be essentially the Banzhaf vector of some simple voting game with n − k dummy voters.     

Single-peaked preferences with several commodities

We consider the problem of allocating m commodities among n agents with single-peaked preferences. When m≥2 and n=2 any strategy-proof and efficient solution is dictatorial. We propose an extension of the Uniform Rule that (in the two-agents case) is the only one that satisfies strategy-proofness, envy-freeness, and a weak requirement related to efficiency. Alternatively, the envy-freeness property may be replaced by weak-anonymity. Received: 7 November 1997/Accepted: 1 August 2000