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     

Notes on Craven's conjecture

Craven's conjecture says that 2 ^n–1 is the maximum number of linear orders on {1, 2,...n} that simultaneously satisfy certain restrictions on three-element subsets of {1, 2,...n}. This is true for n=3, but the maximum exceeds 2 ^n–1 for n4. There is a set of nine linear orders on {1, 2, 3, 4} that satisfy the restrictions. As n gets large, the ratio of the size of the maximum satisfying set to 2 ^n–1 approaches infinity.     

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     

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.     

Topological aggregation, the case of an infinite population

 The literature on infinite Chichilnisky rules considers two forms of anonymity: a weak and a strong. This note introduces a third form: bounded anonymity. It allows us to prove an infinite analogue of the “Chichilnisky– Heal-resolution” close to the original theorem: a compact parafinite CW-complex X admits a bounded anonymous infinite rule if and only if X is contractible. Furthermore, bounded anonymity is shown to be compatible with the finite and the [0, 1]-continuum version of anonymity and allows the construction of convex means in infinite populations. With X=[0, 1], the set of linear bounded anonymous rules coincides with the set of medial limits. Received: 30 October 1993/Accepted: 22 April 1996     

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.     

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     

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     

Serum oxidizability potential is associated with age and exercise test results

The purpose of this paper is to find out whether serum oxidizability potential measured before an exercise test (EXT) correlates with age and ischemic heart disease severity. Oxidizability potential was determined in 3 age groups, in gr. I patients < 45 years, in gr. II age range = 45 - 70years {\hbox{range}} = {45} - {7}0{\hbox{years}} , and in gr. III patients > 70 years. Included subjects had chronic ischemic heart disease (IHD) and underwent a symptom-limited EXT upon initiation of a cardiac rehabilitation program. The thermo-chemiluminescence (TCL) assay was used to assess serum oxidizability potential. This assay is based on heat-induced oxidation of serum, leading to the formation of electronically excited species in the form of unstable carbonyls, which further decompose into stable carbonyls and light energy (low chemiluminescence). Measured photons emission is represented by a kinetic curve which is described by its amplitude and slope (=ratio). We assessed the correlations of TCL ratio with age, exercise duration, metabolic equivalents (METS), maximal heart rate (mHR), maximal systolic BP, >1 mm S-T depression, diabetes, hypertension, smoking, left ventricular ejection fraction (LVEF)> or <40%, previous myocardial infarction, and aorto-coronary bypass surgery. A high TCL ratio (%) correlated well with METS (r = 0.82), mHR (r = 0.77) and with exercise-induced S-T segment shift (r = 0.86, p < 0.05). A lower serum oxidizability potential, expressed as a low TCL ratio, thus suggestive of a previous high oxidative stress, was found in the two older age groups compared to gr. I (<45 years), p = 0.041, and in particular, in gr. III patients with low LVEF%. The TCL ratio (%) in gr. III was 188.7 ± 14.5, 192 ± 17 in gr. II, and 214 ± 13 in gr. I (p < 0.05), and was 166 ± 13.1 in gr. III with LVEF < 45% as compared to 271 ± 15.7 in gr. I patients with LVEF > 45% (p < 0.01). A trend for lower TCL ratio (%) was found in diabetic, hypertensive, and post-coronary bypass surgery patients. A paradoxically low TCL ratio (low oxidizability potential) was observed in patients without S-T depression compared to patients with S-T depression (189 ± 22 vs. 201 ± 15, p = NS), due to the fact these patients had a much lower LVEF% and a lower exercise capacity. Serum oxidizability potential is associated with age, EXT parameters, results, and IHD severity. TCL ratio is an “easy-to-measure marker” that might be incorporated into risk assessment and prediction in aged IHD patients.     

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     

Affect-Regulation Expectancies among Gamblers

Factor scores on a gambling expectancy questionnaire (GEQ) were used to subtype 132 university students who gamble regularly (37.9% male; M age = 22.6 years, SD = 6.04) as: Reward Expectancy Gamblers (Reward EGs)—have strong expectations that gambling augments positive mood, Relief Expectancy Gamblers (Relief EGs)—have strong expectations that gambling relieves negative affect, and Non-Expectancy Gamblers (Non-EGs)—have neither strong expectation. Gambling on a high-low card game was compared across subtypes following priming for either “relief” or “reward” affect-regulation expectancies with the Scrambled Sentence Test (SST). The hypothesized Prime type × GEQ subtype interaction was not significant. When a more stringent set of criteria for GEQ subtyping was imposed, the “purified” sub-sample (n = 54) resulted in the hypothesized statistically significant Prime type × GEQ subtype interaction. Relief EGs gambled more after being primed with the construct “relief of negative emotions” compared to after being primed with the construct “augmentation of positive emotion.” Planned orthogonal contrasts showed a significant linear increase in number of bets made across GEQ subtypes when prime type corresponded to GEQ subtype. The results suggest a need for components in gambling treatment programs that address clients’ expectancies that gambling can provide a specific desirable emotional outcome.     

On the multi-preference approach to evaluating opportunities

The purpose of the paper is to provide a general framework for analyzing “preference for opportunities.” Based on two simple axioms a fundamental result due to Kreps is used in order to represent rankings of opportunity sets in terms of multiple preferences. The paper provides several refinements of the basic representation theorem. In particular, a condition of “closedness under compromise” is suggested in order to distinguish the flexibility interpretation of the model from normative interpretations which play a crucial role in justifying the intrinsic value of opportunities. Moreover, the paper clarifies the link between the multiple preference approach and the “choice function” approach to evaluating opportunities. In particular, it is shown how the well-known Aizerman/Malishevski result on rationalizability of choice functions can be obtained as a corollary from the more general multiple preference representation of a ranking of opportunity sets. Received: 3 September 1996 / Accepted: 18 August 1997     

The majoritarian compromise is majoritarian-optimal and subgame-perfect implementable

It is shown that the Majoritarian Compromise of Sertel (1986) is subgame-perfect implementable on the domain of strict preference profiles, although it fails to be Maskin-monotonic and is hence not implementable in Nash equilibrium. The Majoritarian Compromise is Pareto-optimal and obeys SNIP (strong no imposition power), i.e. never chooses a strict majority's worst candidate. In fact, it is “majoritarian approving” i.e. it always picks “what's good for a majority” (alternatives which some majority regards as among the better “effective” half of the available alternatives). Thus, being Pareto-optimal and majoritarian approving, it is majoritarian-optimal. Finally, the Majoritarian Compromise is measured against various criteria, such as consistency and Condorcet-consistency. Received: 31 January 1995/Accepted: 22 July 1998     

Smallest tournaments not realizable by $${\frac{2}{3}}$$-majority voting

Define the predictability number α(T) of a tournament T to be the largest supermajority threshold for which T could represent the pairwise voting outcomes from some population of voter preference orders. We establish that the predictability number always exists and is rational. Only acyclic tournaments have predictability 1; the Condorcet voting paradox tournament has predictability ; Gilboa has found a tournament on 54 alternatives (i.e. vertices) that has predictability less than , and has asked whether a smaller such tournament exists. We exhibit an 8-vertex tournament that has predictability , and prove that it is the smallest tournament with predictability <  . Our methodology is to formulate the problem as a finite set of two-person zero-sum games, employ the minimax duality and linear programming basic solution theorems, and solve using rational arithmetic. D. Shepardson was supported by a NSF Graduate Research Fellowship during the course of this work.     

On the equilibrium of voting games with abstention and several levels of approval

We study the core of “(j, k) simple games”, where voters choose one level of approval from among j possible levels, partitioning the society into j coalitions, and each possible partition facing k levels of approval in the output (Freixas and Zwicker in Soc Choice Welf 21:399–431, 2003). We consider the case of (j, 2) simple games, including voting games in which each voter may cast a “yes” or “no” vote, or abstain (j = 3). A necessary and sufficient condition for the non-emptiness of the core of such games is provided, with an important application to weighted symmetric (j, 2) simple games. These results generalize the literature, and provide a characterization of constitutions under which a society would allow a given number of candidates to compete for leadership without running the risk of political instability. We apply these results to well-known voting systems and social choice institutions including the relative majority rule, the two-thirds relative majority rule, the United States Senate, and the United Nations Security Council.     

Correlates of sport participation among community-dwelling elderly people in Germany: a cross-sectional study

The aims of this study were (1) to analyze the sport participation in a cohort of community-dwelling elderly people in Germany and (2) to evaluate associations between sport participation, sociodemographic factors, cardiovascular risk factors, and health status. In a monitored prospective cohort study (getABI), 6,880 unselected patients ≥65 years have been followed up by 344 general practitioners beginning in 2001. As part of the 5-year follow-up telephone interview, a sample of 1,376 participants was interviewed on sport participation. The association between participation in at least one sporting activity (“sporty” yes/no) during the past week (cycling) or month (other sports) and the following parameters was analyzed by logistic regression: age, sex, immigration background, education, waist circumference, smoking, self-reported health, history of vascular events, diabetes mellitus, lipometabolic disorder, and arterial hypertension. Analysis of activities (n = 1,304; median age 76 (70–94) years; 55.1% women) showed that 27.6% of participants rode a bicycle during the previous week. During the previous month, 24.9% of participants did gymnastics or strength training, and 16.5% swam. Of all participants, 53.8% were sporty. Multivariate analysis revealed several independent factors to be associated with being sporty (p < 0.05): younger age, male sex, higher education, nonsmoking, better self-reported health, and not being diagnosed with diabetes. Immigration background, waist circumference, history of vascular events, lipometabolic disorder, and hypertension did not show a statistically significant association (p ≥ 0.05) with sport participation. Summing up, the most frequently performed sporting activities were cycling, gymnastics or strength training, and swimming. Sport participation was associated with, for example, age and sex.     

Strategy-proof allocation of multiple public goods

This paper characterizes strategy-proof social choice functions (SCFs), the outcome of which are multiple public goods. Feasible alternatives belong to subsets of a product set . The SCFs are not necessarily “onto”, but the weaker requirement, that every element in each category of public goods A _k is attained at some preference profile, is imposed instead. Admissible preferences are arbitrary rankings of the goods in the various categories, while a separability restriction concerning preferences among the various categories is assumed. It is found that the range of the SCF is uniquely decomposed into a product set in general coarser than the original product set, and that the SCF must be dictatorial on each component B _l . If the range cannot be decomposed, a form of the Gibbard–Satterthwaite theorem with a restricted preference domain is obtained.     

Strategy-proofness versus efficiency on restricted domains of exchange economies

 Strategy-proofness has been shown to be a strong property, particularly on large domains of preferences. We therefore examine the existence of strategy-proof and efficient solutions on restricted, 2-person domains of exchange economies. On the class of 2-person exchange economies in which agents have homothetic, strictly convex preferences we show, as Zhou (1991) did for a larger domain, that such a solution is necessarily dictatorial. As this proof requires preferences exhibiting high degrees of complementarity, our search continues to a class of linear preferences. Even on this “small” domain, the same negative result holds. These two results are extended to many superdomains, including Zhou’s. Received: 9 June 1995/Accepted: 8 January 1996     

Calendars and keys: The classification of “home” and “work”

This article presents a discussion of the relationship between classification systems and individuals' everyday activities. The concept of “boundary work” is defined as the practices that concretize and give meaning to mental frameworks by placing, maintaining, and challenging cultural categories. “Home” and “work” provide a case study for examining boundary work across a range of realm relationships, from those that are highly “integrating” to those that are highly “segmenting.” Boundary practices involving calendars and keys, clothes and appearances, eating and drinking, money, people and their representations (like photographs and gifts), talk styles and conversations, reading materials and habits, and work breaks (including lunches and vacations) are discussed. Mary Douglas's work on categorical purity helps illustrate the relationship between cognitive order and visible behavior seen in the boundary work of home and work. This article includes material that was published in a larger study,Home and Work: Negotiating Boundaries Through Everyday Life (1996 University of Chicago Press), after it was presented at the 1995 Eastern Sociological Society meetings in Philadelphia.