Abstract
We characterize the minimal Hilbert basis of the Hammond order cone, and present several novel applications of the resulting basis. From the basis, we extract an invertible matrix, that provides a numerical representation of the Hammond order relation. The basis also enables the construction of a space—that we call the Hammond order lattice—where order-extensions of the Hammond order (i.e. more complete relations) may be derived. Finally, we introduce a class of maximal linearly independent Hilbert bases, in which the specific results derived in relation to the Hammond order cone, are shown to hold more generally.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Apouey, B., Silber, J., Xu, Y.: On inequality-sensitive and additive achievement measures based on ordinal data. Rev. Income Wealth 66, 267–286 (2020)
Chateauneuf, A., Moyes, P.: A non-welfarist approach to inequality measurement. In: McGillivray, M. (ed.) Inequality, Poverty and Well-Being, pp. 22–65. Palgrave Macmillan, Basingstoke (2006)
Giles, F., Pulleyblank, W.: Total dual integrality and integer polyhedra. Linear Algebra Appl. 25, 191–196 (1979)
Gravel, N., Magdalou, B., Moyes, P.: Ranking distributions of an ordinal variable. Econ. Theory 71, 33–80 (2021)
Gruber, P.: Convex and Discrete Geometry. Springer, Heidelberg (2007)
Hammond, P.: Equity, arrow’s conditions, and Rawls’ difference principle. Econometrica 44, 793–804 (1976)
Magdalou, B.: A model of social welfare improving transfers. J. Econ. Theory 196, 105318 (2021)
Marshall, A., Walkup, D., Wets, R.: Order-preserving functions: applications to majorization and order statistics. Pac. J. Math. 23, 569–584 (1967)
Seth, S., Yalonetzky, G.: Assessing deprivation with an ordinal variable: theory and application to sanitation deprivation in Bangladesh. World Bank Econ. Rev. 35, 793–811 (2020)
Funding
Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Funding for open access charge: Universidad de Málaga / CBUA.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
I am indebted to Brice Magdalou as well as two anonymous reviewers for very detailed and constructive comments. I also wish to thank Mauro Papi and Gaston Yalonetzky for discussions. Any errors are mine.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Abul Naga, R.H. The minimal Hilbert basis of the Hammond order cone. Econ Theory Bull 10, 191–215 (2022). https://doi.org/10.1007/s40505-022-00226-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40505-022-00226-2