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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