Top

Social Choice and Welfare

Published in:

20-03-2019 | Original Paper

A note on the decomposability of inequality measures

Authors: Frédéric Chantreuil, Sébastien Courtin, Kevin Fourrey, Isabelle Lebon

Published in: Social Choice and Welfare | Issue 2/2019

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

We propose a decomposition of inequality measures. By taking the example of the decomposition of income inequality by components, we show that this decomposition fits the definition of two elements: the sum of pure marginal contributions of income components and the sum of the pairwise interactions of all income components. This decomposition relies on the Shapley function and remains valid for a decomposition by subgroups and by components.

previous article Conformity and truthful voting under different voting rules

next article What proportion of sincere voters guarantees efficiency?

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

See also Murofushi and Soneda (1993), Grabish (1996, 1997), Grabish et al. (2000), and Kojadinovic (2002, 2004, 2005).

For instance, if there are two components (\(m=2\)), the coalition considered to evaluate the interaction between i and j only includes these two components (\(t=0\)). Thus, in this particular case, the weight of this interaction is \(\frac{(2-0-2)!(0+1)!}{2!} = \frac{0!1!}{2!} = \frac{1}{2}\). Both components will have an importance equals to their PMC, plus half of value of their interaction.

Bourguignon F (1979) Decomposable income inequality measures. Econometrica 47:901–920CrossRef

Chantreuil F, Lebon I (2015) Gender contribution to income inequality. Econ Lett 133:27–32CrossRef

Chantreuil F, Trannoy A (2011) Inequality decomposition values. Ann Econ Stat 101(102):6–29

Chantreuil F, Trannoy A (2013) Inequality decomposition values: the trade-off between marginality and efficiency. J Econ Inequal 11:83–98CrossRef

Cowell F (1980) Decomposable income inequality measures. Rev Econ Stud 47:520–531CrossRef

Cowell F, Fiori CV (2011) Inequality decompositions: a reconciliation. J Econ Inequal 9:509–528CrossRef

Grabish M (1996) The representation of importance and interaction of features by fuzzy measures. Pattern Recogn 17:567–575CrossRef

Grabish M (1997) Alternative representations of discrete fuzzy measures for decision making. Int J Uncertain Fuzziness Knowl-Based Syst 5:587–607CrossRef

Grabish M, Marichal JL, Roubens M (2000) Equivalent representations of set functions. Math Oper Res 25:157–178CrossRef

Kojadinovic I (2002) Modeling interaction phenomena using fuzzy measures: On the notions of interaction and independence. Fuzzy Sets Syst 135:317–340CrossRef

Kojadinovic I (2004) Estimation of the weights of interacting criteria from the set of profiles by means of information-theoretic functionals. Eur J Oper Res 155:740–751CrossRef

Kojadinovic I (2005) An axiomatic approach to the measurement of the amount of interaction among criteria or players. Fuzzy Sets Syst 152(3):417–435CrossRef

Murofushi T, Soneda S (1993)Techniques for reading fuzzy measures (iii): interaction index. In: Proceedings of the 9th, Fuzzy System Symposium, Saporo, Japan, pp 693–696

Mussard S, Savard L (2012) The gini multi-decomposition and the role of Gini’s transvariation: application to partial trade liberalization in the philippines. Appl Econ 44(10):1235–1249CrossRef

Owen G (1972) Multilinear extension of games. Manag Sci 18:64–79CrossRef

Shapley LS (1953) A value for n-person games. In: Contrib. Theory of Games, II. Ann Math Stud 28:307–317

Shorrocks A (1980) The class of additively decomposable inequality measures. Econometrica 48:613–625CrossRef

Shorrocks A (1982) Inequality decomposition by factor component. Econometrica 50:193–211CrossRef

Shorrocks A (1984) Inequality decomposition by population subgroups. Econometrica 51:1369–1385CrossRef

Shorrocks A (1988) Aggregation issues in inequality measurement. In: Eichhorn W (ed) Measurement in economics. Physica-Verlag, New York

Shorrocks A (2013) Decomposition procedures for distributional analysis: a unified framework based on the shapley value. J Econ Inequal 11:99–126CrossRef

Title: A note on the decomposability of inequality measures
Authors: Frédéric Chantreuil
Sébastien Courtin
Kevin Fourrey
Isabelle Lebon
Publication date: 20-03-2019
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 2/2019
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-019-01183-9

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 2/2019

Professor Bezalel Peleg (1936–2019)

Random matching under priorities: stability and no envy concepts

Power index rankings in bicameral legislatures and the US legislative system

What proportion of sincere voters guarantees efficiency?

Conformity and truthful voting under different voting rules

Truth-tracking judgment aggregation over interconnected issues