We establish an equivalence between three criteria for comparing distributions of an ordinal variable taking finitely many values. The first criterion is the possibility of going from one distribution to the other by a finite sequence of increments and/or Hammond transfers. The latter transfers are like the Pigou-Dalton ones, but without the requirement that the amount transferred be fixed. The second criterion is the unanimity of all comparisons of the distributions performed by a class of additively separable social evaluation functions. The third criterion is a new statistical test based on a weighted recursion of the cumulative distribution. We also identify an exact test for the possibility of going from one distribution to another by a finite sequence of Hammond transfers only. An illustration of the usefulness of our approach for evaluating distributions of self-reported happiness level is also provided.

One way to study connectivity in visual cortical areas is by examining spontaneous neural activity. In the absence of visual input, such activity remains shaped by the underlying neural architecture and, presumably, may still reflect visuotopic organization. Here, we applied population connective field (CF) modeling to estimate the spatial profile of functional connectivity in the early visual cortex during resting state functional magnetic resonance imaging (RS-fMRI). This model-based analysis estimates the spatial integration between blood-oxygen level dependent (BOLD) signals in distinct cortical visual field maps using fMRI. Just as population receptive field (pRF) mapping predicts the collective neural activity in a voxel as a function of response selectivity to stimulus position in visual space, CF modeling predicts the activity of voxels in one visual area as a function of the aggregate activity in voxels in another visual area. In combination with pRF mapping, CF locations on the cortical surface can be interpreted in visual space, thus enabling reconstruction of visuotopic maps from resting state data. We demonstrate that V1 ➤ V2 and V1 ➤ V3 CF maps estimated from resting state fMRI data show visuotopic organization. Therefore, we conclude that—despite some variability in CF estimates between RS scans—neural properties such as CF maps and CF size can be derived from resting state data.

We provide foundations for robust normative evaluation of distributions of two attributes, one of which is cardinally measurable and transferable between individuals and the other is ordinal and non-transferable. The result that we establish takes the form of an analogue to the standard Hardy-Littlewood-Pólya theorem for distributions of one cardinal attribute. More specifically, we identify the transformations of the distributions which guarantee that social welfare increases according to utilitarian unanimity provided that the utility function is concave in the cardinal attribute and that its marginal utility with respect to the same attribute is non-increasing in the ordinal attribute. We establish that this unanimity ranking of the distributions is equivalent to the ordered poverty gap quasi-ordering introduced by Bourguignon [12]. Finally, we show that, if one distribution dominates another according to the ordered poverty gap criterion, then the former can be derived from the latter by means of an appropriate and finite sequence of such transformations. Introductory remarksThe normative foundations of the comparison of distributions of a single attribute between a given number of individuals are by now well-established. They originate in the equivalence between three statements that are considered relevant answers to the question of when a distribution x can be considered normatively better than a distribution y. Given two distributions x and y with equal means, these statements, the equivalence of which was first established by Hardy et al. [30] and popularised later on among economists by Kolm [32], Atkinson [3], Dasgupta et al. [15], Sen [47], Fields and Fei [21] among others, are the following:(a) Distribution x can be obtained from distribution y by means of a finite sequence of progressive -or equivalently Pigou-Dalton -transfers. (b) All utilitarian ethical observers who assume that individuals convert the attribute into wellbeing by means of the same non-decreasing and concave utility function rank distribution x above distribution y. (c) The Lorenz curve of distribution x lies nowhere above and somewhere below that of y, or equivalently, for all poverty lines, the poverty gap is no greater in distribution x than in distribution y and it is smaller for at least one poverty line.This remarkable result, which can be generalised in a number of ways, points to three different aspects of the inequality measurement process. 1 The first statement aims at capturing the very notion of inequality reduction by associating it with elementary transformations of the distributions. The second statement is fundamentally normative and it assumes that society has an aversion to inequality which, in the utilitarian framework, is reflected by the concavity of the utility function. To some extent the first statement helps in clarifying the meaning of the restriction imposed on the utility function in the second statement. While these two conditions shed light on two different facets of the inequality concept, they do ...