The core of games on ordered structures and graphs (Q5902298)

A cooperative game in coalitional form is an ordered pair \(\langle N,v\rangle\), where \(N=\left\{ 1,2,\ldots,n\right\}\) (the set of players) and \(v:2^{N}\rightarrow \mathbb{R}\) is a map, assigning to each coalition \(S\in 2^{N}\) a real number, such that \(v(\emptyset )=0\) [\textit{T. Driessen}, Cooperative games, solutions and applications. Dordrecht etc.: Kluwer Academic Publishers (1988; Zbl 0686.90043)]. From the authors abstract: ``In cooperative games, the core [\textit{D. B. Gillies}, Ann. Math. Stud. 40, 47--85 (1959; Zbl 0085.13106)] is the most popular solution concept, and its properties are well known. In the classical setting of cooperative games, it is generally assumed that all coalitions can form, i.e., they are all feasible. In many situations, this assumption is too strong and one has to deal with some unfeasible coalitions.'' Based on these results, the author defines a game on a subcollection of the power set of the set of players. This line of vision ``has many implications on the mathematical structure of the core, depending on the precise structure of the subcollection of feasible coalitions. Many authors have contributed to this topic, and we give a unified view of these different results.'' Reviewer's remark: In my opinion this paper is enjoyable to read and truly an enrichment in cooperative game theory. The paper's contents is widely innovative and pioneering. Many ideas are quite smart. Further, the author gives a quite good literature review. The paper is organized in an attentive way. The purpose of the paper is given perfectly. All theoretical results and examples are explained evidently. Finally, the paper is very well motivated, structured, written; theory, methods and applications vary in an appropriate way. I fully recommend the paper for researchers and graduate students in the area.