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Theory and Decision

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20.08.2022

Decompositions of inequality measures from the perspective of the Shapley–Owen value

verfasst von: Rodrigue Tido Takeng, Arnold Cedrick Soh Voutsa, Kévin Fourrey

Erschienen in: Theory and Decision | Ausgabe 2/2023

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Abstract

This article proposes three new decompositions of inequality measures, drawn from the framework of cooperative game theory. It allows the impact of players’ interactions, rather than players’ contributions to inequality, to be taken into consideration. These innovative approaches are especially suited for the study of income inequality when the income has a hierarchical structure: the income is composed of several primary sources, with the particularity that each of them is also composed of secondary sources. We revisit the Shapley–Owen value that quantifies the importance of each of these secondary sources in the overall income inequality. Our main contribution is to decompose this importance into two parts: the pure marginal contribution of the considered source and a weighted sum of pairwise interactions. We then provide an axiomatic characterization of each additive interaction decomposable (AID) coalitional value considered in this paper. We give an application of these decompositions in the context of inequality theory.

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Which is well known as a generalization of the Shapley Value.

Notice that, while \({\mathfrak {B}}\) is a coalitional structure over \({\mathfrak {U}}\), \(\pi {\mathfrak {B}}\) may not be a coalitional structure over \({\mathfrak {U}}\).

In that case, \({\mathfrak {B}}^N_{\pi }\) is the coalitional structure induced by \({\mathfrak {B}}_\pi\) over N, and \(\pi S_j\cap \pi S_{j}=\emptyset\) whenever \(j\ne k\).

For more detail, see Sect. 4.

M being the set of representatives of the structural coalitions in \({\mathfrak {B}}^N\).

In the literature, the notion of coalitional semivalue is also known as mixed modified semivalues (see Giménez & Puente, 2015). The semivalue induced by value \(\phi\) plays a role among the structural coalitions whereas the distribution among the players of \(S_{m_i}\) is obtained according to the semivalue induced by value \(\varphi\). Note that the two semivalues can be either equal (e.g., Shapley value for the Owen–Shapley value and Banzhaf value for the Owen-Banzhaf value) or different [e.g., Banzhaf value and Shapley value for the symmetric coalitional Banzhaf value (see Alonso-Meijide & Fiestras-Janeiro, 2002)].

A player i is veto in a game v if \(v(S)=0\) whenever \(i\notin S\).

A game v is monotonic if \(v(L)\le v(T)\) whenever \(L\subset T.\)

A game \(v\in \Gamma (N)\) is an additive game if, for every player \(i\in N\), there exists \(a_i\in {\mathbb {R}}\), such that \(v(T)=\sum _{i\in N}a_i\) for every \(T\subseteq N\).

Casajus and Huettner (2018) called the naive solution of player i.

Marichal et al. (2007) called the external generalized value of coalition T.

If the family of real numbers \(\{f^{r}(M,P)\}_{P\subseteq M\backslash \{\ell ,r\}}\) with \(f^{r}(M,P)=f(m,p)\) is a probability distribution over \(2^{M\backslash \{\ell ,r\}}\), then the second part of Eq. (3) can be interpreted as a weighted arithmetic mean of pairwise interactions of \(S_{\ell}\) with the other structural coalitions.

If we have \(\sum _{p=0}^{m-2}\sum _{k=0}^{s_{m_i}\!-1}\left( ^{s_{m_i}\!-1}_{~~k}\right) \times \left( ^{m-2}_{~p}\right) \times f(m,p)\cdot g(s_{m_i},k)=1\), \(g(s_{m_i},k)\ge 0\) with \({k=0,\dots ,s_{m_i}}\) and \(f(m,p)\ge 0\) with \({p=0,\dots ,m\!-\!2}\), then \(\Psi _i\left( v,{\mathfrak {B}}\right)\) can also be interpreted as the difference between a weighted sum of marginal contributions of \(i\) and a weighted arithmetic mean of pairwise interactions of \(i\) with the structural coalitions \(S_{\ell}\neq S_{m_i}\).

If \(\Psi ^{{\mathrm{CS}}}=\Psi ^{{\mathrm{Sh}}}\) or \(\Psi ^{{\mathrm{Sh}}}=\Psi ^{{\mathrm{Bz}}},\) then \(\sum _{p=0}^{m-2}\sum _{k=0}^{s_{m_i}\!-1}\left( ^{s_{m_i}\!-1}_{~~k}\right) \times \left( ^{m-2}_{~p}\right) \times 2f(m,p)\cdot g(s_{m_i},k)=1\). \(\Psi _i\left( v,{\mathfrak {B}}\right)\) can also be interpreted as the difference between a weighted sum of marginal contributions of i and the half of a weighted arithmetic mean of pairwise interactions of \(i\) with the structural coalitions \(S_{\ell}\neq S_{m_i}\).

Albizuri and Zarzuelo (2004) (Theorem 5) showed that the Shapley–Owen value is the only coalitional value on \((v,{\mathfrak {B}}) \in G(N)\) that satisfies efficiency, rearrangement, and additivity.

The second part of the decomposition can also be interpreted as the half of a weighted arithmetic mean of pairwise interactions of \(i\) with the other players \(j\neq i\).

Other axioms of the inequality measure could be integrated here, such as the strict Schur-convexity, for instance. However, we do not impose more restrictions on the inequality function as our decompositions can be applied with any inequality measure.

See Chantreuil and Trannoy (2011) for the presentation of other games.

Note that all considered sources have nonnegative values. If this is not the case then, one must apply transformations to the distribution to have nonnegative values or take an inequality measure suitable for such a situation, such as the Gini coefficient developed by Raffinetti et al. (2015).

These observations are also verified with a null game, where \(\lambda =0\).

We consider the Quotient game where structural coalitions are viewed as players (Alonso-Meijide et al., 2007).

The Gini index, noted G, for n individuals ordered from the lowest income to the highest income is denoted by: \(G = \frac{2}{n} \sum _{i=1}^{n} (p_i - L_i)\), where \(p_i\) and \(L_i\) are the respective cumulative share of population and the cumulative share of income.

Albizuri, M. J., & Zarzuelo, J. M. (2004). On coalitional semivalues. Games and Economic Behavior, 49, 221–243.CrossRef

Alonso-Meijide, J. M., Carreras, F., Fiestras-Janeiro, M. G., & Owen, G. (2007). A comparative axiomatic characterization of the Banzhaf–Owen coalitional value. Decision Support Systems, 43, 701–712.CrossRef

Alonso-Meijide, J. M., & Fiestras-Janeiro, G. M. (2002). Modification of the Banzhaf value for games with a coalition structure. Annals of Operation Research, 109, 213–227.CrossRef

Auvray, C., & Trannoy, A. (1992). Décomposition par source de l’inégalité des revenus à l’aide de la valeur de Shapley. Sfax.

Casajus, A., & Huettner, F. (2018). Decomposition of solutions and the Shapley value. Games Economic Behavior, 108, 37–48.CrossRef

Chameni, N. (2006a). Linking Gini to Entropy: Measuring inequality by an interpersonal class of indices. Economics Bulletin, 4(5), 1–9.

Chameni, N. (2006b). A note on the decomposition of the coefficient of variation squared: Comparing entropy and Dagum’s methods. Economics Bulletin, 4(8), 1–8.

Chantreuil, F., Courtin, S., Fourrey, K., & Lebon, I. (2019). A note on the decomposability of inequality measures. Social Choice and Welfare, 54, 1–16.

Chantreuil, F., & Trannoy, A. (2011). Inequality decomposition values. Annals of Economics and Statistics, 101(102), 6–29.

Chantreuil, F., & Trannoy, A. (2013). Inequality decomposition values: The trade-off between marginality and efficiency. The Journal of Economic Inequality, 11(1), 83–98.CrossRef

Courtin, S., Tido Takeng, R., & Chantreuil, F. (2020). Decomposition of interaction indices: Alternative interpretations of cardinal-probabilistic interaction indices. https://hal.archives-ouvertes.fr/hal-02952516. Accessed 29 Sep 2020

Dagum, C. (1997). A new approach to the decomposition of the Gini income inequality ratio. Empirical Economics, 22, 515–531.CrossRef

Dubey, P., Neyman, A., & Weber, R. J. (1981). Value theory without efficiency. Mathematics of Operations Research, 6, 122–128.CrossRef

Ebert, U. (2010). The decomposition of inequality reconsidered: Weakly decomposable measures. Mathematical Social Sciences, 60(2), 94–103.CrossRef

Firpo, S., Fortin, N., & Lemieux, T. (2018). Decomposing wage distributions using recentered influence function regressions. Econometrics, 6, 28. https://doi.org/10.3390/econometrics6020028.CrossRef

Fujimoto, K., Kojadinovic, I., & Marichal, J.-L. (2006). Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices. Games and Economic Behavior, 55, 72–99.CrossRef

Giménez, J. M., & Puente, M. A. (2015). A method to calculate generalized mixed modified semivalues: Application to the Catalan parliament (legislature 2012–2016). TOP, 23, 669–684.CrossRef

Grabisch, M., & Roubens, M. (1999). An axiomatic approach to the concept of interaction among players in cooperative games. International Journal of Game Theory, 28, 547–565.CrossRef

Harsanyi, J. C. (1959). A bargaining model for cooperative n-person games. In R. D. Luce & A. W. Tucker (Eds.), Contribution to the theory of games IV (Vol. 2, pp. 325–355). Princeton University Press.

Jenkins, S. P., & Van Kerm, P. (2006). Trends in income inequality, pro-poor income growth, and income mobility. Oxford Economic Papers, 58(3), 531–548.CrossRef

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

Lerman, R. I., & Yitzhaki, S. (1985). Income inequality effects by income source: A new approach and applications to the United-States. The Review of Economics and Statistics, 67(1), 151.CrossRef

Marichal, J. L., Kojadinovic, I., & Fujimoto, K. (2007). Axiomatic characterizations of generalized values. Discrete Applied Mathematics, 155, 26–43.CrossRef

Mornet, P., Zoli, C., Mussard, S., Sadefo-Kamdem, J., Seyte, F., & Terraza, M. (2013). The (\(\alpha,\beta\))-multi-level \(\alpha\)-Gini decomposition with an illustration to income inequality in France in 2005. Economic Modelling, 35, 944–963.CrossRef

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

Mussard, S., Pi Alperin, M. N., Seyte, F., & Terraza, M. (2006). Extensions of Dagum’s Gini decomposition. Statistica & Applicazioni, 4(2), 5–30.

Mussard, S., & Savard, L. (2012). The Gini multi-decomposition and the role of Gini’s transvariation: Application to partial trade liberalization in the Philippines. Applied Economics, 44(10), 1235–1249.CrossRef

Mussini, M. (2013). On decomposing inequality and poverty changes over time: A multi-dimensional decomposition. Economic Modelling, 33, 8–18.CrossRef

Owen, G. (1975). Multilinear extensions and the Banzhaf value. Naval Research Logistics Quarterly, 22, 741–750.CrossRef

Owen, G. (1977). Values of games with a priori unions. In R. Hein & O. Moeschlin (Eds.), Essays in mathematical economics and game theory. Springer.

Owen, G. (1981). Modification of the Banzhaf–Coleman index for games with a priori unions. In M. J. Holler (Ed.), Power, voting, and voting power. Physica-Verlag.

Raffinetti, E., Siletti, E., & Vernizzi, A. (2015). On the Gini coefficient normalization when incomes with negative values are considered. Statistical Methods & Applications, 24(3), 507–521.CrossRef

Shapley, L. S. (1953). A value for n-person games. Annals of mathematics studies 28. In H. W. Kuhn & A. W. Tucker (Eds.), Contribution to the theory of games (Vol. II, pp. 307–317). Princeton University Press.

Shorrocks, A. F. (1982). Inequality decomposition by factor components. Econometrica, 50(1), 193.CrossRef

Shorrocks, A. F. (2013). Decomposition procedures for distributional analysis: A unified framework based on the Shapley value. The Journal of Economic Inequality, 11(1), 99–126.CrossRef

Theil, H. (1967). Economics and information theory. North-Holland.

Titel: Decompositions of inequality measures from the perspective of the Shapley–Owen value
verfasst von: Rodrigue Tido Takeng
Arnold Cedrick Soh Voutsa
Kévin Fourrey
Publikationsdatum: 20.08.2022
Verlag: Springer US
Erschienen in: Theory and Decision / Ausgabe 2/2023
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-022-09893-w

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