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.
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