The pre-kernel as a tractable solution for cooperative games. An exercise in algorithmic game theory (Q362353)

In this monograph the author ``provides an alternative approach to study the pre-kernel solution of transferable utility games based on a generalized conjugation theory from convex analysis''. The book consists of ten chapters and an appendix. Chapter 1 serves as an introduction in which the state of the art with related works and a discussion of the presented results are provided. In Chapter 2, the basic definitions, notations, some solution schemes and game properties, needed for the remaining part of the monograph, are introduced. In Chapter 3 first the author presents the Shapley value as a standard of fairness, and then the pre-kernel -- as an alternative fair solution to be used to divide the proceeds of mutual cooperation. Chapter 4 concerns fair division in Cournot markets. After introducing a cooperative oligopoly game without transferable technologies, its properties and computing aspects, axiomatic treatments of the Shapley value, of the (pre)kernel and the nucleolus are provided. The author terminates this chapter with a general discussion of the fair division rules. In Chapter 5, some preliminary results from convex analysis and cooperative game theory, with a special focus on the Moore-Penrose matrix, are reconsidered. Chapter 6 concerns a pre-kernel characterization and orthogonal projection. In Chapter 7, a characterization of the pre-kernel by solution sets is discussed. In Chapter 8 the author presents several algorithms for computing the pre-kernel, a numerical example as well as a general discussion of the algorithms. An upper dimension bound of the pre-kernel is presented and discussed in Chapter 9. Chapter 10 provides some concluding remarks. In the appendix the author discusses some empirical results concerning the proposed algorithms.