_{1}

In the Inverse Set relative to a Semivalue, we are looking for a new game for which the Semivalue of the original game is coalitional rational. The problem is solved by means of the Power Game of the given game. The procedures of building the new game, as well as the case of the Banzhaf Value are illustrated by means of some examples.

In an earlier work of the author [

In the present paper, the similar problem is now considered for the Banzhaf Value and the Semivalues, (values introduced in [

Precisely, for each game, we associate a Power Game, relative to the Semivalue, and the value of the original game, efficient in the Power Game, is coalitional rational, if it belongs to the Core of the Power Game, called the Power Core. If the Semivalue happens also to be an efficient one, for the original game, then the Power Game is the game itself, so that the new definition of the coalitional rationality should be extending the definition given above for the efficient values. Note that, on the other hand, the Semivalue of the original game is always efficient in the Power Game, hence in the general case we should discuss only the coalitional rationality. An example with some details will be given for a Banzhaf Value, which is efficient for a particular game.

In the following, the concept of Semivalue as well as the solution of the Inverse Problem for Semivalues, will be sketched in the second section. The concept of Power Game and the solution of the above new problem connected to coalitional rationality will be shown in the third section, where several three person games will also illustrate the results, for some non-efficient Semivalues, defined by a priori given weight vectors.

More remarks, about the efficient values, and the computational means for finding solutions of the problem, will be shown in the last section.

The set of all cooperative TU games, on a finite set of player N, with two operations, addition and scalar multiplication, is a linear vector space, to be denoted by

The Semivalue may be extended to the sets of games in

for

Among the Semivalues, we have the popular particular cases of the Banzhaf Value, obtained for the weight vectors

together with the weight vectors derived by means of (2).

The first is usually non-efficient, while the second is always efficient. We mean that the Banzhaf Value may be efficient for some games and non-efficient for others, while the Shapley Value is efficient for all games. An example showing the first situation follows.

Example 1: a) Consider the game

By computing the Banzhaf Value via formula (3), with the weight vector

It happened that the two values are equal, a fact discussed in [

For those Semivalues which are efficient, only in the case of some games, there is no need to define again the coalitional rationality.

b) The fact that this may not happen is shown by the game

for the Banzhaf Value, which is a particular Semivalue. With the same weight vector, we find

nality conditions are satisfied. Clearly, we have to discuss also the coalitional rationality for non-efficient Semivalues.

c) The most general case is the one when for some game and some value, both the efficiency and the coalitional rationality conditions do not hold. For example, if we take for the same game (6) the Semivalue defined

by the weight vector

Now, neither the efficiency, nor the coalitional rationality conditions are holding, because the sum of payoffs

makes

does not hold; in other words, the last situation is the general case.

Note that in (6) the Shapley Value is

Return to the set of games

The Inverse Problem for Semivalues, discussed in [

defined by means of the formulas

and

for some values of the constant coefficients. In the earlier work, it has been shown the following auxiliary results:

These equalities, as well as the linearity of the Semivalues, shown by formula (3), give

In [

with dimension

For any game

Formula (13) shows a nice interpretation of the Power Game: for each coalition in the original game, the worth of the coalition in the Power Game equals the total win of its members, when they use the value

However, it will be easier to use a computational formula for the Power Core proved also in the same paper [

Now, an arbitrary game in the almost null subfamily of the Inverse Set, relative to the Semivalue

and depends on the parameter

Similarly we obtain

As noticed above, this game belongs to the almost null subfamily of Inverse Set, for any value of the parameter, and we may check that the Semivalue of any game in this family equals the Semivalue of the original game. From (15), based upon the definition (13), we can compute the Power Game of

Instead, the Power Game of

For

and the efficiency is holding, because we have also

In this way, we almost proved the following main result:

Theorem: Suppose that the Semivalue, associated with a given weight vector

the Semivalue is in the Power Core, if and only if the parameter

where the minimum is taken over the index

Proof: Taking into account the above formulas (17), (18), for the Power Game, the Power Core is given by

from which the result (20) follows, as the efficiency is obvious. ■

Note that the inequality (20) does not depend on the weight vector and it is the same as in [

Example 2: Consider the same three person game of Example 1a, shown in (5). Compute the Semivalue asso-

ciated with the weight vector

Obviously, the conditions

makes 1000; hence, it does not belong to the Core, even though the coaltional rationality conditions hold. Compute the characteristic function for all the coalitions of size two in an arbitrary game belonging to the almost null subfamily of the Inverse Set, relative to the Semivalue and to the vector shown in formula (22), by using the formulas (15). We get

We may check that the Semivalue of this game is the same as the one in the original game, for whatever value of the parameter

where the Semivalue denoted by L, is provided by (22). We have

By taking for example

where the null worth of the singletons have been omitted. We may check that the Semivalue equals L, and it belongs to the Core of the Power Game for

It follows that the Semivalue of the original game and of the game in the almost null family (26) are coalitional rational as it is in the Core of (27), Hence, the game (26) is a solution for our problem, that is the Semivalue of the original game is coalitional rational, in the above introduced sense. In the next section we shall consider a Semivalue, corresponding to another weight vector, for which neither the efficiency, nor the coalitional rationality will hold, in the original game.

The following remarks are appropriate:

a) Note that the Semivalue of the Power Game (27) is

not the same as the Semivalue (22), as the Power Game is not in the almost null subfamily of the Inverse Set. Moreover, it does not belong to the Power Core, as it is not efficient, to be easily checked.

b) It is well known that the Semivalues are efficient if and only if they are Shapley Values. Hence, the above theory applies also to the Shapley Value. However, due to the efficiency the Power Game is the game itself, so that after getting the game belonging to the almost null family, we should get the coalitional rationality condi-

tion from this game. Indeed, looking at the games (15), and using the fact that

(17), and from here we obtain (20), which gives the same coalitional rationality condition as in the study of coalitional rationality for efficient values (see [

c) The above results may be applied to the Banzhaf Value, which is a Semivalue defined by the weights

Returning to the game in our example 1, where the Banzhaf Value is

that happens to be efficient, and if the parameter is chosen to satisfy (20) with equal sign,

The Power Game for (31) is

derived from (15), the Banzhaf Value of original game and the value of the parameter

Example 3. Consider the game (6) of example 1c, and the Semivalue with the weight vector

result of the above Theorem. We may check that this Semivalue is neither efficient, nor coalitional rational.

From formula (20) we see that the Semivalue will be coalitional rational if we have

satisfy the above inequality. Now, find out a game in the almost null subset of the Inverse Set, from formula (15),

where we take

This game is

From formulas (17), (18), we compute the Power Game and we obtain the game

We see that the Semivalue is efficient in the Power Game (34) of the Game (33) and we check easily that it is also coalitional rational; hence (33) is the solution of our problem.

d) An interesting case is the case of the Binomial Semivalues, a class of Semivalues introduced in [

The present work is a continuation of our earlier work [

IrinelDragan, (2015) On the Coalitional Rationality of the Banzhaf Value and Other Non-Efficient Semivalues. Applied Mathematics,06,2069-2076. doi: 10.4236/am.2015.612182