Pre-kernel as a tractable solution for cooperative games : an exercise in algorithmic game theory / Holger Ingmar Meinhardt.

Includes bibliographical references and index.

1. Introduction--
2. Some Solution Schemes and Game Properties --
3. The Shapley Value and (Pre-Kernel) as a Fairness Concept --
4. Fair Division in Cournot Markets --
5. Some Preliminary Results --
6. A Pre-Kernel Characterization and Orthogonal Projection --
7. Characterization of the Pre-Kernel by Solution Sets --
8. Algorithms for Computing the Pre-Kernel --
9. An Upper Dimension Bound of the Pre-Kernel --
10. Concluding Remarks--

Bibliography--
Author Index--
Subject Index.

This present book provides an alternative approach to study the pre-kernel solution of transferable utility games based on a generalized conjugation theory from convex analysis. Although the pre-kernel solution possesses an appealing axiomatic foundation that lets one consider this solution concept as a standard of fairness, the pre-kernel and its related solutions are regarded as obscure and too technically complex to be treated as a real alternative to the Shapley value. Comprehensible and efficient computability is widely regarded as a desirable feature to qualify a solution concept apart from its axiomatic foundation as a standard of fairness. We review and then improve an approach to compute the pre-kernel of a cooperative game by the indirect function. The indirect function is known as the Fenchel-Moreau conjugation of the characteristic function. Extending the approach with the indirect function, we are able to characterize the pre-kernel of the grand coalition simply by the solution sets of a family of quadratic objective functions.