Abstract
In earlier works, we introduced the Inverse Problem, relative to the Shapley Value, as follows: for a given n-dimensional vector L, find out the transferable utilities’ games , such that The same problem has been discussed further for Semivalues. A connected problem has been considered more recently: find out TU-games for which the Shapley Value equals L, and this value is coalitional rational, that is belongs to the Core of the game . Then, the same problem was discussed for other two linear values: the Egalitarian Allocation and the Egalitarian Nonseparable Contribution, even though these are not Semivalues. To solve such problems, we tried to find a solution in the family of so called Almost Null Games of the Inverse Set, relative to the Shapley Value, by imposing to games in the family, the coalitional rationality conditions.
 In the present paper, we use the same idea, but a new tool, an Alternative Representation of Semivalues. To get such a representation, the definition of the Binomial Semivalues due to A. Puente was extended to all Semivalues. Then, we looked for a coalitional rational solution in the Family of Almost Null games of the Inverse Set, relative to the Shapley Value.
 In each case, such games depend on a unique parameter, so that the coalitional rationality will be expressed by a simple inequality, determined by a number, the coalitional rationality threshold. The relationships between the three numbers corresponding to the above three efficient values have been found. Some numerical examples of the method are given.
Highlights
In earlier works, we introduced the Inverse Problem, relative to the Shapley Value, as follows: for a given n-dimensional vector L, find out the transferable utilities’ games (N,v), such that SH (N,v) = L
We looked for a coalitional rational solution in the Family of Almost Null games of the Inverse Set, relative to the Shapley Value
In each case, such games depend on a unique parameter, so that the coalitional rationality will be expressed by a simple inequality, determined by a number, the coalitional rationality threshold
Summary
As we have seen above, we obtained the games (10) found in the Family of Almost Null Games, relative to the Shapley Value, expressed in terms of the weights for this value. Remember the connection between the initial weights for the games with n players and the subgames with n-1 players, that we called the inverse Pascal triangle; this will be first used below in deriving from (10) the new expressions for an Alternative Representation of the Family of the Almost Null Games in the Inverse Set. The main weight appearing in (10) is: pn−1(n −1) =. From (10), (11) and (12), we get the Alternative Representation for the Family of Almost Null Games in the Inverse Set, relative to the Shapley Value: w(N −{i}) = a − Li , i N 1... Will be a solution of the inverse problem with coalitional rationality, for a game in the Family of Almost Null Games of the Inverse Set, relative to the Shapley Value.
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