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Economics - Game Theory / Mathematical Methods | The Pre-Kernel as a Tractable Solution for Cooperative Games

The Pre-Kernel as a Tractable Solution for Cooperative Games

An Exercise in Algorithmic Game Theory

Meinhardt, Holger Ingmar

2014, XXXIII, 242 p. 8 illus., 3 illus. in color.

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  • Characterizes a fair division rule of game theory by convex analysis
  • Proposes tractable formula to solve fair division problems in real life situations
  • Provides algorithms to implement vectorized and parallel computer programs designed to solve fair division problems​

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.

Content Level » Research

Keywords » Convex Analysis - Cooperative Game Theory - Fair Division - Fairness - Fenchel-Moreau Conjugation - Pre-Kernel Solutions

Related subjects » Applications - Economic Theory - Game Theory / Mathematical Methods - Theoretical Computer Science

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