ABSTRACT
In the theory of cooperative transferable
utilities games, (TU games), the Efficient Values, that is those which show how
the win of the grand coalition is shared by the players, may not be a good
solution to give a fair outcome to each player. In an earlier work of the
author, the Inverse Problem has been stated and explicitely solved for the Shapley
Value and for the Least Square Values. In the present paper, for a given vector,
which is the Shapley Value of a game, but it is not coalitional rational,
that is it does not belong to the Core of the game, we would like to find out a
new game with the Shapley Value equal to the a priori given vector and for
which this vector is also in the Core of the game. In other words, in the
Inverse Set relative to the Shapley Value, we want to find out a new game, for
which the Shapley Value is coalitional rational. The results show how such a game
may be obtained, and some examples are illustrating the technique. Moreover, it
is shown that beside the original game, there are always other games for which
the given vector is not in the Core. The similar problem is solved for the
Least Square Values.