The cost of getting local monotonicity

Titelangaben

Freixas, Josep ; Kurz, Sascha:
The cost of getting local monotonicity.
In: European Journal of Operational Research. Bd. 251 (2016) Heft 2 . - S. 600-612.
ISSN 0377-2217
DOI: https://doi.org/10.1016/j.ejor.2015.11.030

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Abstract

In 1982 Manfred Holler introduced the Public Good index. In its unnormalized version, i.e., the raw measure, it counts the number of times that a player belongs to a minimal winning coalition. Unlike the Banzhaf index, it does not count the remaining winning coalitions in which the player is crucial. Holler noticed that his index does not satisfy local monotonicity, a fact that can be seen either as a major drawback or as an advantage.
In this paper we consider a convex combination of the two indices and require the validity of local monotonicity. We prove that the cost of obtaining it is high, i.e., the achievable new indices satisfying local monotonicity are closer to the Banzhaf index than to the Public Good index. All these achievable new indices are more solidary than the Banzhaf index, which makes them as very suitable candidates to divide a public good.
As a generalization we consider convex combinations of either: the Shift index, the Public Good index, and the Banzhaf index, or alternatively: the Shift Deegan-Packel, Deegan-Packel, and Johnston indices.