Izabella Stach

In our sub-coalition approach, players and their blocs termed sub-coalitions form a grand coalition following the queue bargaining model. So, sub-coalitional versions of marginal contribution values need to consider the marginal contributions of blocs of players. This article is a novel contribution in two aspects. The first aspect consists of introducing a new sub-coalitional value. This value is obtained by a modification of the Solidarity value, introduced by Nowak and Radzik in 1994, and an egalitarian way to divide the blocs' contributions. The second aspect consists of applying the queue bargaining model to provide new formulations for the Sub-coalitional Egalitarian Shapley value introduced in 2017 and the new sub-coalitional value. Additionally, using a combinatorial approach, we prove that the Solidarity value can be calculated using the queue model where players share the marginal contributions with their predecessors, which is the same idea as in procedure 4 proposed by Malawski in 2013.

The purpose of this paper is to introduce new methods to measure the indirect control power of firms in complex corporate shareholding structures using the concept of power indices from cooperative game theory. The proposed measures vary in desirable properties satisfied, as well as in the bargaining models of power indices used to construct them. Hence, they can be used to produce different pictures of the coalitional strength of firms in control of other firms in mutual shareholding networks with the presence of cycles. Precisely, in the framework of Karos and Peters from 2015, ten power indices substitute the original Shapley and Shubik power index in a modular fashion. In this way, we obtain a set of new measures called aggregated indices. The float shareholders typically hold less than 5 percent of the outstanding shares, which is an uncertain element of indirect control in complex shareholding structures. The fuzzy number seems appropriate to model these shareholders’ behavior. The novelty is that we model the behavior of float using Z-fuzzy numbers. The new methods are tested in an example.

The aim of the article is to propose a new method of valuation of a company, considering its ownership relations with other companies. For this purpose, the concept of the Shapley value from cooperative game theory is used as the basis for assessing such dependent companies. The paper presents proposals for Shapley value calculation algorithms for our model. We expand our model by discussing personal relations in addition to ownership relations and point out how intuitionistic fuzzy sets may be helpful in this context. As a result, we propose two new expanded models. In the first probabilistic model, we apply Pearson’s correlation coefficient, in the second, we use a correlation coefficient between intuitionistic fuzzy sets to determine the personal relationships. Finally, we present and interpret results for a real-world economic network with 17 companies.

This paper proposes a new representation for the Public Help Index θ (briefly, PHI θ). Based on winning coalitions, the PHI θ index was introduced by Bertini et al. in (2008). The goal of this article is to reformulate the PHI θ index using null player free winning coalitions. The set of these coalitions unequivocally defines a simple game. Expressing the PHI θ index by the winning coalitions that do not contain null players allows us in a transparent way to show the parts of the power assigned to null and non-null players in a simple game. Moreover, this new representation may imply a reduction of computational cost (in the sense of space complexity) in algorithms to compute the PHI θ index if at least one of the players is a null player. We also discuss some relationships among the Holler index, the PHI θ index, and the g^np index (based on null player free winning coalitions) proposed by Álvarez-Mozos et al. in (2015).