The minimal covering set in large tournaments

We prove that in almost all large tournaments, the minimal covering set is the entire set of alternatives. That is, as the number of alternatives gets large, the probability that the minimal covering set of a uniformly chosen random tournament is the entire set of alternatives goes to one. In contrast, it follows from a result of (Fisher and Reeves, Linear Algebra Appl 217:83–85, 1995) that the bipartisan set contains about half of the alternatives in almost all large tournaments.     

Minimal rights based solidarity

In a model where individuals with different levels of skills exert different levels of effort, we propose to use individuals’ minimal rights to divide an extra amount of income generated by a change in the skill profile. Priority is given to individuals with a positive minimal right which ensures that the way redistribution is performed depends on the total sum of income available in society. We characterize two families of minimal rights based Egalitarian mechanisms. One family guarantees each individual her claim when claims are feasible. The other family guarantees a non-negative income after redistribution for all individuals.     

Minimal monotonic extensions of scoring rules

Noting the existence of social choice problems over which no scoring rule is Maskin monotonic, we characterize minimal monotonic extensions of scoring rules. We show that the minimal monotonic extension of any scoring rule has a lower and upper bound, which can be expressed in terms of alternatives with scores exceeding a certain critical score. In fact, the minimal monotonic extension of a scoring rule coincides with its lower bound if and only if the scoring rule satisfies a certain weak monotonicity condition (such as the Borda and antiplurality rule). On the other hand, the minimal monotonic extension of a scoring rule approaches its upper bound as its degree of violating weak monotonicity increases, an extreme case of which is the plurality rule with a minimal monotonic extension reaching its upper bound.

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Minimal manipulability: anonymity and unanimity

This paper is concerned with the minimal number of profiles at which a unanimous and anonymous social choice function for three alternatives is manipulable. The lower bound is derived and examples of social choice functions attaining the lower bound are given. It is conjectured that these social choice functions are in fact all minimally manipulable social choice functions. Since some of these social choice functions are Pareto optimal, it follows that the lower bound also holds for Pareto optimal and anonymous social choice functions. Some of the minimally manipulable Pareto optimal and anonymous social choice functions can be interpreted as status quo voting.     

Scales Measuring General Evaluation with Minimal Distortion

Recent factor-analytic studies demonstrating marked concept-rater-scaleinteraction using evaluative semantic differential scales posea problem for opinion researchers who lack the time, money,subjects, or trained personnel to perform a pretest factor analysisas the basis for selecting reliable scales. In this study, 225raters drawn from 10 sub-populations rated 9 concepts on 39evaluative scales and 2 marker scales. When these data weresubjected to 19 different principal factor analyses based onvarious concept-rater combinations, 6 scales loaded above .60on the dominant first factor of all analyses, 9 additional scalesloaded above .50 on the same factor for all analyses, and 7scales failed only once to load above .50. These scales arerecommended for use as a measure of general evaluation undercertain conditions     

Minimal winning coalitions in weighted-majority voting games

Riker's size principle for n-person zero-sum games predicts that winning coalitions that form will be minimal in that any player's defection will negate the coalition's winning status. Brams and Fishburn (1995) applied Riker's principle to weighted-majority voting games in which players have voting weights w ₁≥w ₂≥...≥w _n,and a coalition is winning if its members' weights sum to more than half the total weight. We showed that players' bargaining power tends to decrease as their weights decrease when minimal winning coalitions obtain, but that the opposite trend occurs when the minimal winning coalitions that form are “weight-minimal”, referred to as least winning coalitions. In such coalitions, large size may be more harmful than helpful. The present paper extends and refines our earlier analysis by providing mathematical foundations for minimal and least winning coalitions, developing new data to examine relationships between voting weight and voting power, and applying more sophisticated measures of power to these data. We identify all sets of minimal and least winning coalitions that arise from different voting weights for n≤6 and characterize all coalitions that are minimal winning and least winning for every n. While our new analysis supports our earlier findings, it also indicates there to be less negative correlation between voting weight and voting power when least winning coalitions form. In this context, players' powers are fairly insensitive to their voting weights, so being large or small is not particularly important for inclusion in a least winning coalition.     

Uncovered set choice rules

I study necessary and sufficient conditions for a choice function to be rationalized in the following sense: there exists a total asymmetric relation T (a tournament) such that, for each feasible (finite) set, the choice set coincides with the uncovered set of T restricted to that feasible set. This notion of ‘maximization’ offers testable restrictions on observable choice behavior.     

Minimal conversely consistent extension of the men-optimal solution

This study pertains to two-sided, one-to-one matching problems and considers the best-known solution concept: the men-optimal solution. The men-optimal solution fails to satisfy consistency as well as converse consistency. Furthermore, the minimal consistent extension of the men-optimal solution equals the core. In this article, we compute the minimal conversely consistent extension of the men-optimal solution as a correspondence which associates with each problem the set consisting of the men-optimal matching, and all stable and men-barterproof matchings for this problem.     

Responsible Gambling: General Principles and Minimal Requirements

Many international jurisdictions have introduced responsible gambling programs. These programs intend to minimize negative consequences of excessive gambling, but vary considerably in their aims, focus, and content. Many responsible gambling programs lack a conceptual framework and, in the absence of empirical data, their components are based only on general considerations and impressions. This paper outlines the consensus viewpoint of an international group of researchers suggesting fundamental responsible gambling principles, roles of key stakeholders, and minimal requirements that stakeholders can use to frame and inform responsible gambling programs across jurisdictions. Such a framework does not purport to offer value statements regarding the legal status of gambling or its expansion. Rather, it proposes gambling-related initiatives aimed at government, industry, and individuals to promote responsible gambling and consumer protection. This paper argues that there is a set of basic principles and minimal requirements that should form the basis for every responsible gambling program.     

A smallest tournament for which the Banks set and the Copeland set are disjoint

Given a tournament T, a Banks winner of T is the first vertex of any maximal (with respect to inclusion) transitive subtournament of T; a Copeland winner of T is a vertex with a maximum out-degree. In this paper, we show that 13 is the minimum number of vertices that a tournament must have so that none of its Copeland winners is a Banks winner: for any tournament with less than 13 vertices, there is always at least one vertex which is a Copeland winner and a Banks winner simultaneously. Received: 2 May 1997 / Accepted: 30 September 1997     

On the tournament equilibrium set

An example is provided showing that Schwartz's tournament equilibrium set is not identical to the minimal covering set of Dutta.     

Minimal treatments and problem gamblers: A preliminary investigation

In view of the increasing popularity of minimal intervention treatments for problem drinking, a self-help manual for people who wish to reduce or stop gambling was prepared. Twenty-nine (ACT residents) who responded to advertisements for help with problem gambling were allocated to either of two minimal treatments, Manual (only) and Manual & Interview. On average, clients from both groups reduced the frequency of their gambling sessions, frequency of overspending, and amount spent per week in the first three months and next three months after first contact, but expenditure per session increased from three to six months, after an initial improvement. There was no evidence that a single in-depth interview added to the effectiveness of the manual.This project was funded by a grant from the Australian National University Faculties Research Fund.     

At Home in Japan: Beyond the Minimal House

London, Geffrye Museum, March 22–;August 29, 2011     

Minimal manipulability and interjacency for two-person social choice functions

This paper investigates the number of profiles at which a social choice function is manipulable. It is found that there is a lower bound on this number when the social choice function is assumed non-dictatorial. Also, any number between this lower bound and the maximal number of profiles may be obtained as the number of manipulable profiles of some social choice function. Received: 30 November 1993 / Accepted: 14 April 1997     

An axiomatic characterization of the lexicographic maximin extension of an ordering over a set to the power set

The lexicographic maximin extension of an ordering is an important and widely used tool in social choice theory. We provide an axiomatization of it by means of five axioms. When the basic ordering is linear the following four (independent) axioms are sufficient: (1) Gärdenfors principle; (2) Neutrality; (3) Strong Fishburn monotonicity; and (4) Extension. Our result may also have applications in the theory of individual choice under uncertainty.     

Truncated Egalitarian solutions

A family of solutions to the bargaining problem with a variable population, generalizing the Egalitarian solution, is introduced under the name of Truncated Egalitarian solutions. A Truncated Egalitarian solution solves small problems at thier Egalitarian outcome and larger problems at some fixed and therefore non-optimal point, violations of optimality being more severe for large than for small groups. Any solution satisfying Symmetry, Independence of Irrelevant Alternatives, Continuity and Monotonicity (when the claims of more agents have to be accomodated, all agents should sacrifice) coincides with a Truncated Egalitarian solution except perhaps when only two agents are involved, in which case utility substitutions are possible in some limited range.     

Bargaining over a finite set of alternatives

We analyze bilateral bargaining over a finite set of alternatives. We look for "good" ordinal solutions to such problems and show that Unanimity Compromise and Rational Compromise are the only bargaining rules that satisfy a basic set of properties. We then extend our analysis to admit problems with countably infinite alternatives. We show that, on this class, no bargaining rule choosing finite subsets of alternatives can be neutral. When rephrased in the utility framework of Nash (1950), this implies that there is no ordinal bargaining rule that is finite-valued. Professor Sertel passed away on January 25, 2003.