The intermediate set and limiting superdifferential for coalitional games: between the core and the Weber set

Abstract

We introduce the intermediate set as an interpolating solution concept between the core and the Weber set of a coalitional game. The new solution is defined as the limiting superdifferential of the Lovász extension and thus it completes the hierarchy of variational objects used to represent the core (Fréchet superdifferential) and the Weber set (Clarke superdifferential). It is shown that the intermediate set is a non-convex solution containing the Pareto optimal payoff vectors that depend on some chain of coalitions and marginal coalitional contributions with respect to the chain. A detailed comparison between the intermediate set and other set-valued solutions is provided. We compute the exact form of intermediate set for all games and provide its simplified characterization for the simple games and the glove game.

Conditions of the Green Transformation. The Case of the European Union

“…To study the relationships between analysed dimensions of green growth and contextual indicators of green transformation, the taxonomic measure of development applying the Weber median vector was used. In the literature (Młodak, 2014;Adam & Kroupa, 2017;Szopik-Depczyńska et al, 2018) one may indicate many examples of the application of this synthetic measure to calculate rankings comprising many various dimensions of the socio-economic development of the countries, regions or cities etc. The Weber median is presented in the literature as a multi-dimensional generalization of the classical notion of the median (Szopik-Depczyńska et al, 2018).…”