The book introduces the reader not only to classical cooperative game theory (main game theoretical tools, game solutions and axiomatic value theory) but it also shows a wide range of applications of this theory for developing essential topics like cooperative potential, directed communication networks and cooperative theories of hierarchical organizations. It is worthwhile to emphasize that this book is very clearly and rigorously written. Besides, each of its chapters is accompanied by an appendix section containing proofs of the main theorems, and by a problem section, allowing this book to be used as a textbook for an advanced graduate course on game theory. On the other hand, the applications discussed in it are mainly based on some recent developments in the literature on the cooperative game theoretic analysis of social networks and hierarchical authority structures. The author focuses this volume mainly on the contributions made by himself and many of his co-authors. Thereby, this book can also be seen as a very valuable source for game theorists.
The book consists of six chapters:
Chapter 1 – Cooperative Game Theory, introduces and discusses the basic notions describing cooperative games in the characteristic function form and its generalization in the form of (Owen’s) multilinear extension.
Chapter 2 – The Core of a Cooperative Game, gives a wide discussion about the core as one of the fundamental solutions of a cooperative game. Here, the reader can also get acquainted with both the basic properties of the \(\Omega\)-core (the core based on a collection of coalitions) and with essential tools related with the core like balanced collections, strongly balanced collections, permission structures and lattices. The chapter ends with a discussion about two essential supersets of the core like the Weber set and the selectope.
Chapter 3 – Axiomatic Value Theory, discusses several formulae on the Shapley value of a cooperative game and its properties, and gives three mostly relevant axiomatization systems uniquely determining it; that is the axiomatizations of Shapley, Young and van der Brink. Besides, the Myerson value as a generalization of the Shapley value for game situations with constraints on coalition formation is discussed here.
Chapter 4 – The Cooperative Potential, introduces the reader to the potential theory of Hart and Mass-Colell (HM), a powerful tool for the investigation of properties of allocation rules (values of cooperative games). In particular, a big usefulness of this theory is shown and widely discussed in the context of the Shapley value. Among other things, the reductionist approach to \(HM\)-reduced games developed by those two authors is presented to show the \(HM\)-consistency property of that value. The second part of this chapter presents the approach of van der Brink and van der Laan to a special class of allocation rules of a cooperative game, called BL-share functions, and subsequently discusses the relationship between the rules from this class and its one element, the so-called Shapley share function. This is done with the help of the notion of share potentials.
Chapter 5 – Directed Communication Networks, considers some applications of the cooperative game theoretic notions that were covered in the previous chapters. It consists of two main parts. The first part discusses directed networks \(D\) on a finite set \(N\) of players, and three types of different measures on \(D^N\) (degree measure, \(\beta\)-measure and a class of iterated power measures), being (by definition) functions of the form \(m: D^N \rightarrow R^N\) with the interpretation that they measure the overall dominance of each player in the network \(D\) (here \(D^N\) is the collection of all directed networks on \(N\)). The second part of this chapter introduces three other notions defined on directed networks, cooperative network situations, network Myerson values and \(\alpha\)-hierarchical values, and discusses their properties.
Chapter 6 – Cooperative Theories of Hierarchical Organizations: Here the goal is to analyse the consequences of the implementation of different hierarchical authority structures on the set of the players in the context of a cooperative game with transferable utilities. This is based on games with permission structure where coalition formation is considered in two ways, by the conjunctive approach or disjunctive approach. The second part of this section discusses the Shapley value for these two permission structures and the reader can acquaint with several axiomatizations of the Shapley value on them (called the conjunctive and disjunctive permission value then).