_{1}

^{*}

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.

Consider the cooperative TU game:

This is a constant sum game, hence the Core of the game is empty (see [

where N is the set of players, and

and the efficiency is obvious, as the sum of components makes

In an earlier work of the author (see [

where

and

The vector space of TU games may be identified with

where there are

Theorem (Dragan, 1991): The set of vectors (5) form a basis of the space of TU games with the set of players

where

Proof: The vectors (5) form a basis, because their number is the dimension of the space and they are linearly independent. The Shapley Values of the basic vectors (5) are:

(see [

If the Shapley Value is L, then the coefficients

Example 1. Consider a general three person TU game and let us use the above theorem, in order to derive the coalitional rationality conditions for the three person case, that is the inequalities defining the Core of any game in the Inverse Set. From (5) and (8), we find the expressions for the characteristic function of any three person TU game:

The Core of this family of games, when

as the efficiency is already holding. If

In our game (1) above, this condition is

so that, in the Inverse Set we have games with the Shapley Value coalitional rational and also games with the Shapley Value not coalitional rational. Our game (1) is obtained from (12) for

Now, we can verify that the Shapley Value is the same as before, i.e. we have

and the Shapley Value is kept the same, and again it is coalitional rational.

The discussion connected to the three person TU games suggests how can we behave in the case of games with any number of players, which will be considered in the next section.

Consider now the general case of an

Theorem 1. Let

called the almost null family. Then, a game in this family is coalitional rational if and only if

where the minimum is taken over the index

Proof. Return to (8), the general expansion of games in the Inverse Set, relative to the Shapley Value, when the value equals L. Consider, like in the Example 1, games with all parameters

where the null values of the characteristic function have been omitted. Like in the three person case, if

as the efficiency condition is already holding. Obviously, if

Obviously, (19) reduces to the above condition (14), that we got for

Example 2. Consider a new three person game, namely

with the Shapley Value

which is not coalitional rational; the inequality (19) is

and for the parameter

We may verify that the Shapley Value is the same, which now is efficient and coalitional rational. If we take,

for example

tional rational. Notice that (20) allows us to find the characteristic function of the coalitions with size

An extension of the Shapley Value is the family of Least Square Values introduced by L. Ruiz, F. Valenciano and J. Zarzuelo (see [

Let

He proved that the problem has a unique optimal solution, called in [

where

It is obvious that this is a family of efficient values, depending on the chosen function; he proved also that in this family is also included the Shapley Value, obtained for

with

It is easily seen, that the weights

but now, this basis consists of the games defined as follows

and

where there are

Theorem (Dragan, 2006): The set of vectors (33) form a basis of the space of cooperative TU games with the set of players N, and if the Least Square Value of a game

where

Proof. The vectors (33) form a basis because they are linearly independent and their number equals the di- mension of the space. The LS-values of the basic vectors (33) are:

(see [

Thus, like in Section 2, the expansion (34) becomes (35). □

We have shown that the TU games with the LS-value L, and the weights defined by the function m, or

Theorem 2. Let

Then, there are games in this family with L as a coalitional rational LS-value, if and only if

where the minimum is taken over the index

Proof. Return to (35), the general expression of the games in the Inverse Set, relative to the LS-values. As in the case of the Shapley Value, for the family of games obtained when their coordinates are

If

Notice that (40) is a formula similar to (19), which is obtained in the case of the Shapley Value. Also, in terms of the other weights appearing in the quadratic programming problem (27), we obtain from (31) the inequality

We proved a result similar to Theorem 1 given in the previous section.

Example 3. Consider the same game as in the previous section, and let us take as weights in the quadratic programming problem the numbers

We have

The solution of the quadratic programming problem, the LS-value, is the vector

ly, this is an efficient LS-value for the game; but we would like to be also coalitional rational in a TU game from the set obtained by intersecting

After taking the value which satisfies the inequality with an equal sign, and computing the values of the cha- racteristic function for coalitions of size

We may easily check that the LS-value is in the Core of this game. Of course, we can take other values

In the present paper, it has been shown how we can get, for an a priori given Nonnegative