The value of a player in n-person games

. The article decomposes the Shapley value into a value matrix which gives the value of every player to every other player in n-person games. Element Φ_ij(v) in the value matrix is positive, zero, or negative, dependent on whether row player i is beneficial, has no impact, or is not beneficial for column player j. The elements in each row and in each column of the value matrix sum up to the Shapley value of the respective player. The value matrix is illustrated by the voting procedure in the European Council of Ministers 1981–1995. Received: 9 September 1998/Accepted: 11 February 2000     

Multivariate Lorenz majorization

The Lorenz zonotope is a multivariate generalization of the Lorenz curve. It allows to define multivariate Lorenz majorization whose properties are studied.     

The measurement of opportunity inequality: a cardinality-based approach

We consider the problem of ranking distributions of opportunity sets on the basis of equality. First, conditional on a given ranking of individual opportunity sets, we define the notion of an equalizing transformation. Then, assuming that the opportunity sets are ranked according to the cardinality ordering, we formulate the analogues of the notions of the Lorenz partial ordering, equalizing (Dalton) transfers, and inequality averse social welfare functions – concepts which play a central role in the literature on income inequality. Our main result is a cardinality-based analogue of the fundamental theorem of inequality measurement: one distribution Lorenz dominates another if and only if the former can be obtained from the latter by a finite sequence of rank preserving equalizations, and if and only if the former is ranked higher than the latter by all inequality averse social welfare functions. In addition, we characterize the smallest monotonic and transitive extension of our cardinality-based Lorenz inequality ordering. Received: 2 May 1995 / Accepted: 11 October 1996     

Third-degree stochastic dominance and inequality measurement

We investigate the third-degree stochastic dominance order, which is receiving increasing attention in the field of inequality measurement. Observing that this partial order fails to satisfy the von Neumann–Morgenstern independence property in the space of random variables, we introduce the concepts of strong and local third-degree stochastic dominance, which do not suffer from this deficiency. We motivate these two new binary relations and characterize them in the spirit of the Lorenz characterization of the second-degree stochastic order, comparing our findings with the closest results in inequality literature. A preliminary version of this paper was presented at the second Canazei Winter School on Inequality and Collective Welfare Theory (IT2). We would like to thank all participants for their comments and suggestions. We are especially grateful to Rolf Aaberge, Peter Lambert, Maria G. Monti, Ernesto Savaglio, John Weymark, Claudio Zoli and two anonymous referees who provided very detailed and insightful comments.     

Income Share Elasticity and Stochastic Dominance

Income share elasticity is a function π which can describe the size distribution of income (Esteban in Intern Econ Rev 27:439–444, 1986). On the other hand, the conventional density representation of the latter gives parameters of first or second order stochastic dominance (SD), widely used to describe shifts in income distribution, to which inequality measures are attached. The paper draws a link between the two, by providing conditions such that a given shift to π is equivalent to a first or second order SD shift of the distribution of income. Some applications to Lorenz rankings are also provided.     

Symmetry extensions of “neutrality”

For n3 candidates, a system voting vector W ⁿ specifies the positional voting method assigned to each of the 2 ⁿ –(n+1) subsets of two or more candidates. While most system voting vectors need not admit any relationships among the election rankings; the ones that do are characterized here. The characterization is based on a particular geometric structure (an algebraic variety) that is described in detail and then used to define a partial ordering among system voting vectors. The impact of the partial ordering is that if W ⁿ ₁ W ⁿ ₂, then W ⁿ ₂ admits more kinds of (single profile) voting paradoxes than W ⁿ ₁. Therefore the partial ordering provides a powerful, computationally feasible way to compare system voting vectors. The basic ideas are illustrated with examples that completely describe the partial ordering for n=3 and n=4 candidates.This reearch was supported in part by NSF Grant IRI-9103180.     

Aggregated statistical rankings are arbitrary

In many areas of mathematics, statistics, and the social sciences, the intriguing, and somewhat unsettling, paradox occurs where the “parts” may give rise to a common decision, but the aggregate of those parts, the “whole”, gives rise to a different decision. The Kruskal-Wallis nonparametric statistical test on n samples which can be used to rank-order a list of alternatives is subject to such a Simpson-like paradox of aggregation. That is, two or more data sets each may individually support a certain ordering of the samples under Kruskal-Wallis, yet their union, or aggregate, yields a different outcome. An analysis of this phenomenon yields a computable criterion which characterizes which matrices of ranked data, when aggregated, can give rise to such a paradox. Received: 6 November 2001/Accepted: 7 March 2002     

Robust stochastic dominance: A semi-parametric approach

Lorenz curves and second-order dominance criteria, the fundamental tools for stochastic dominance, are known to be sensitive to data contamination in the tails of the distribution. We propose two ways of dealing with the problem: (1) Estimate Lorenz curves using parametric models and (2) combine empirical estimation with a parametric (robust) estimation of the upper tail of the distribution using the Pareto model. Approach (2) is preferred because of its flexibility. Using simulations we show the dramatic effect of a few contaminated data on the Lorenz ranking and the performance of the robust semi-parametric approach (2). Since estimation is only a first step for statistical inference and since semi-parametric models are not straightforward to handle, we also derive asymptotic covariance matrices for our semi-parametric estimators.     

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     

Dynastic inequality,mobility and equality of opportunity

One often heard counter to the concern about rising income and wealth inequality is that it is wrong to focus on inequality of outcomes in a “snapshot”. Intergenerational mobility and “equality of opportunity”, so the argument goes, is what matters for normative evaluation. In response to this counter, we ask what pattern of intergenerational mobility leads to lower inequality not between individuals but between the dynasties to which they belong? And how does this pattern in turn relate to commonly held views on what constitutes equality of opportunity? We revive and revisit here our earlier contributions which were in the form of working papers (Kanbur and Stiglitz 1982, 1986) in order to engage with the current debate. Focusing on bistochastic transition matrices in order to hold constant the steady state snapshot income distribution, we develop an explicit partial ordering which ranks matrices on the criterion of inequality between infinitely lived dynasties. A general interpretation of our result is that when comparing two transition matrices, if one matrix is “further away” from the identity matrix then it will lead to lower dynastic inequality. More specifically, the result presents a computational procedure to check if one matrix dominates another on dynastic inequality. We can also assess “equality of opportunity”, defined as identical prospects irrespective of starting position. We find that this is not necessarily the mobility pattern which minimizes dynastic inequality.     

Lorenz curves for various processes: A pluralistic approach to equity

Theoretical Lorenz curves for a bureaucracy withn seniority levels and similar curves for a simple stochastic economy with capitalistic opportunity to move up or down are derived. In both cases it is argued that equality of distribution is not an appropriate ideal. But the main thrust is that income alone is not a sufficient criterion of equity. A more general Aristotelian viewpoint is argued for. The marked increase in the equity of education between 1940 and 1984 in the United States is taken as a good example. But even within education, it is argued, detailed policies will realistically satisfy no simple set of equity principles.     

Utility-gap Dominances and Inequality Orderings

The justification for using Lorenz dominance as an inequality ranking condition has been based on the aggregate social welfare comparison and the Pigou–Dalton principle of transfers. Since both the aggregating aspect of the social welfare function and certain implications of the principle of transfers are debatable, ordering conditions stronger than Lorenz dominance are worth exploring. A particularly interesting direction to pursue is to follow the frequently invoked notion that inequality is the “gap” between the rich and the poor. This paper follows this notion to formally propose a unified utility-gap concept and characterizes several utility-gap based conditions as general stronger-than-Lorenz-dominance ranking criteria. Specifically, we propose utility-gap dominance which requires all pair-wise utility-gaps in one distribution to be uniformly smaller than those of the other distribution. We then explore a conceptually weaker dominance concept – quasi dominance – which imposes conditions only on the gap between each person’s utility and some reference utility point of the distribution. I am grateful to two anonymous referees and Peter Lambert for their very constructive comments and suggestions on an earlier version of the paper. The usual caveat applies.     

Essentially lexicographic aggregation

In the paper a new approach to lexicography is developed by which in the general framework of ordered blocks with a monotonic basis it is shown that a nontrivial ordering is translation-invariant if and only if it is essentially lexicographic of degree n. Here, the latter means that the ordering can be represented by an ordinary lexicographic ordering in n dimensions. As an application it is shown that a nontrivial social welfare ordering on Euclidean space possesses a useful invariance property (cardinality and non comparability) if and only if the ordering is essentially lexicographic of a strong kind in that it can be obtained from ordinary lexicography by permutation, cutting-off and order reversal with respect to components. This result generalizes the characterization of lexical individual dictatorship obtained by Gevers and d'Aspremont and it provides, within the social welfare approach, a strong version of Arrow's impossibility theorem by not invoking any Pareto principle at all.The author thanks K. Arrow, W. Gaertner, L. Gevers, and two anonymous referees for helpful hints and suggestions.     

Meditation Lowers Stress and Supports Forgiveness Among College Students: A Randomized Controlled Trial

Objective and Participants: The authors evaluated the effects on stress, rumination, forgiveness, and hope of two 8-week, 90-min/wk training programs for college undergraduates in meditation-based stress-management tools. Methods: After a pretest, the authors randomly allocated college undergraduates to training in mindfulness-based stress reduction (MBSR; n = 15), Easwaran's Eight-Point Program (EPP; n = 14), or wait-list control (n = 15). The authors gathered pretest, posttest, and 8-week follow-up data on self-report outcome measures. Results: The authors observed no post-treatment differences between MBSR and EPP or between posttest and 8-week follow-up (p > .10). Compared with controls, treated participants (n = 29) demonstrated significant benefits for stress (p < .05, Cohen's d = -.45) and forgiveness (p < .05, d = .34) and marginal benefits for rumination (p < .10, d = -.34). Conclusions: Evidence suggests that meditation-based stress-management practices reduce stress and enhance forgiveness among college undergraduates. Such programs merit further study as potential health-promotion tools for college populations.     

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     

An informationally parsimonious impartial observer theorem

In Harsanyi's impartial observer theorem, an impartial observer determines a social ordering of the lotteries on the set of social alternatives based on a sympathetic but impartial concern for all individuals in society. This ordering is derived from a more primitive ordering on the set of all extended lotteries. An extended lottery is a lottery which determines both the observer's personal identity and the social alternative. We establish a version of Harsanyi's theorem in which the observer is only required to have preferences on the extended lotteries in which there is an equal chance of being any person in society. Received: 19 June 1996 / Accepted: 30 December 1996     

Richness orderings

An index of richness in a society is a measure of the extent of its affluence. This paper presents an analytical discussion on several indices of richness and their properties. It also develops criteria for ordering alternative distributions of income in terms of their richness. Given a line of richness, an income level above which a person is regarded as rich, and depending on the redistributive principle, it is shown that the ranking relation can be implemented by seeking dominance with respect to the generalized Lorenz curve of the rich or the affluence profile of the society. When the line of richness is assumed to be variable, we need to employ the stochastic dominance conditions for ordering the income distributions.     

On the power of poverty orderings

This paper investigates the possibility of increasing the ordering power of additively separable poverty measures beyond the condition of second degree stochastic dominance by considering third degree stochastic dominance. For a fixed poverty line, the ordering power can be significantly enhanced by using the third degree criterion. For a range of poverty lines, the marginal power of the third degree criterion over the second degree depends critically upon the lower bound of the range; if the lower bound poverty line is arbitrarily close to zero, the two criteria coincide. The implications of a strong version of the transfer sensitivity axiom are also considered. Received: 20 November 1995/Accepted: 16 February 1998     

Lorenz non-consistent welfare and inequality measurement

Typical welfare and inequality measures are required to be Lorenz consistent which guarantees that inequality decreases and welfare increases as a result of a progressive transfer. We explore the implications for welfare and inequality measurement of substituting the weaker absolute differentials and deprivation quasi-orderings for the Lorenz quasi-ordering. Restricting attention to distributions of equal means, we show that the utilitarian model - the so-called expected utility model in the theory of risk - does not permit one to make a distinction between the views embedded in the differentials, deprivation and Lorenz quasi-orderings. In contrast it is possible within the dual model of M. Yaari (Econometrica 55 (1987), 99–115) to derive the restrictions to be placed on the weighting function which guarantee that the corresponding welfare orderings are consistent with the differentials and deprivation quasi-orderings respectively. Finally we drop the equal mean condition and indicate the implications of our approach for the absolute ethical inequality indices.