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Theory and Decision

23.01.2024

Decomposition of interaction indices: alternative interpretations of cardinal–probabilistic interaction indices

verfasst von: Sébastien Courtin, Rodrigue Tido Takeng , Frédéric Chantreuil

Erschienen in: Theory and Decision

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Abstract

In cooperative game theory, the concept of interaction index is an extension of the concept of one-point solution that takes into account interactions among players. In this paper, we focus on cardinal–probabilistic interaction indices that generalize the class of semivalues. We provide two types of decompositions. With the first one, a cardinal–probabilistic interaction index for a given coalition equals the difference between its external interaction index and a weighted sum of the individual impact of the remaining players on the interaction index of the considered coalition. The second decomposition, based on the notion of the "decomposer", splits an interaction index into a direct part, the decomposer, which measures the interaction in the coalition considered, and an indirect part, which indicates how all remaining players individually affect the interaction of the coalition considered. We propose alternative characterizations of the cardinal–probabilistic interaction indices.

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In the context of inequality game theory (cooperative game theory applied to inequality theory), Chantreuil et al. (2019) and Tido Takeng et al. (2022) used the same approach to motivate the existence of interactions among the income inequalities.

Cardinal–probabilistic interaction indices are derived from the class of probabilistic interaction indices by additionally imposing the symmetry axiom to this class.

To avoid heavy notation, we adopt the following conventions: we will omit braces for singletons, e.g., by writing v(i), \(S\backslash i\) instead of \(v(\{i\})\), \(S\backslash \{i\}\). Similarly, for pairs, we will write ij instead of \(\{i,j\}\).

Note that \(\Delta _{S}v(\emptyset )=\lambda _v(S)\) and \(\Delta _{S}v(N\backslash S)=\displaystyle \sum _{T\supseteq S }\lambda _v(T)= \displaystyle \sum _{L\subseteq S}(-1)^{l}v(N\backslash L)\) (see Fujimoto et al. (2006)).

One can notice that when \(S=i\), according to Proposition 1, every semivalue is AID, especially the Shapley solution (Chantreuil et al., 2019).

Positivity: A solution \(\Psi\) satisfies the Positivity property if for all monotonic game \(v\in TU(N)\) and \(i\in N\), it holds that \(\Psi (v,i)\ge 0.\)

When \(S=i\), we obtain Theorem 3 of Casajus and Huettner (2018), page 39.

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Titel: Decomposition of interaction indices: alternative interpretations of cardinal–probabilistic interaction indices
verfasst von: Sébastien Courtin
Rodrigue Tido Takeng
Frédéric Chantreuil
Publikationsdatum: 23.01.2024
Verlag: Springer US
Erschienen in: Theory and Decision
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-023-09970-8

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