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<p>One large class of relations used in the measurement of social welfare and risk consists of relations induced by finitely generated cones. Within this class, we develop a general approach to investigate the ordering of distributions. We provide an equivalence between the statement that distributions <i>x</i> and <i>y</i> are ordered, and (1) the possibility of expressing <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(x-y\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>-</mo> <mi>y</mi> </mrow> </math></EquationSource> </InlineEquation> as a positive combination of a subset of linearly independent vectors from the generators of the cone, (2) the existence of a relation defined on a simplicial cone such that <i>x</i> and <i>y</i> are ordered by this latter relation, (3) the existence of a generalized inverse <i>G</i> of the matrix whose columns generate the cone, such that the product of <i>G</i> and the vector <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(x-y\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>-</mo> <mi>y</mi> </mrow> </math></EquationSource> </InlineEquation> results in a non-negative vector. We illustrate the results in the context of a discrete version of the cone of inframodular transfers.</p>

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Ordering distributions on a finitely generated cone

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  • 发表日期：

    2025年7月23日

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  • Ramses H. Abul Naga

摘要

One large class of relations used in the measurement of social welfare and risk consists of relations induced by finitely generated cones. Within this class, we develop a general approach to investigate the ordering of distributions. We provide an equivalence between the statement that distributions x and y are ordered, and (1) the possibility of expressing \(x-y\) x - y as a positive combination of a subset of linearly independent vectors from the generators of the cone, (2) the existence of a relation defined on a simplicial cone such that x and y are ordered by this latter relation, (3) the existence of a generalized inverse G of the matrix whose columns generate the cone, such that the product of G and the vector \(x-y\) x - y results in a non-negative vector. We illustrate the results in the context of a discrete version of the cone of inframodular transfers.