On the Coalitional Rationality of the Banzhaf Value and Other Non-Efficient Semivalues

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.


Introduction
In an earlier work of the author [1], the Inverse Problem for the Shapley Value has been introduced and solved: for whatever n-vector, find out the set of all TU games for which the Shapley Value equals that a priori given vector.In a more recent work [2], the Inverse Problem was solved for Semivalues, including the Banzhaf Value.Like in the first case, the solution was called the Inverse Set, relative to the Semivalue, and was given by an explicit formula expressing all games associated with that Semivalue, defined by a fixed weight vector.In a recent work [3], it was also reminded that the Shapley Value, may not be coalitional rational, that is, in general, it does not belong to the Core of the game.In [3], another problem was introduced and solved: given a TU game, for which we know the Shapley Value, or some other efficient value, find out a game with the same Shapley Value, or the alternative efficient value, but in which the considered value is coalitional rational.For the Shapley Value, in technical words, in the Inverse Set relative to the given Shapley Value, find out a TU game for which this Shapley Value is coalitional rational.
In the present paper, the similar problem is now considered for the Banzhaf Value and the Semivalues, (values introduced in [4] [5], respectively).These values are usually non-efficient, so that they will not be in the Core; therefore we have to define now what it is a coalitional rational Banzhaf Value, and more general a Semi-value.The main idea will be that of going on the path met in a previous joint work [6].
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 Inverse Set for Semivalues
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 ( ) G N , with a dimension of 2 1 n − .In this space of games, the Semivalue of a game ( ) , SE N v , will be defined by a weight vector ( ) ∈ , satisfying some condition of normalization, precisely The Semivalue may be extended to the sets of games in Among the Semivalues, we have the popular particular cases of the Banzhaf Value, obtained for the weight vectors ( ) , where we have the components 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.
By computing the Banzhaf Value via formula (3), with the weight vector

and the Shapley
Value with the weight vector so that we may check that both are efficient and coalitional rational.It happened that the two values are equal, a fact discussed in [7].
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 ( ) , and this is not efficient, because the sum makes 59 14 4 ≠ , while the coalitional rationality 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 3  1 1 1 , , , which satisfies the condition (1), then from formula (3), we obtain ( ) Now, neither the efficiency, nor the coalitional rationality conditions are holding, because the sum of payoffs makes 77 14 6 ≠ , and does not hold; in other words, the last situation is the general case.Note that in (6) the Shapley Value is which is efficient and coalitional rational.Note the fourth situation, the well known fact that efficient Semivalues equal the Shapley Value, but this may be, or not, coalitional rational; the first case was met above.
Return to the set of games N G , the union of the vector spaces ( ), .

G T T N ∀ ⊆
Consider a game ( ) , N v , and compute the Semivalue ( ) , SE N v , by formula (3), for a weight vector n p , satisfying the normalization condition (1).Let us have ( ) where n L R ∈ .The Inverse Problem for Semivalues, discussed in [2], can be stated as follows: find out all TU games in the space

( )
G N for which the Semivalue associated with the weight vector n p , equals L. This problem has al- ready been solved in the above mentioned work of the author, and we shall sketch the solution, before going to the related problem of coalitional rationality.In the space defined by means of the formulas and ( ) 0 T w S = otherwise.Note, that the denominators t l t l p + + are obtained from the weight vector n p via the inverse Pascal triangle relationships (2).It is easy to see that these are linearly independent vectors and the set W has the cardinality 2 1 n − , hence this is a basis of the space.Thus, any TU game ( ) 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 [2], it has been proved that this is the general solution of the Inverse Problem for the Semivalue, that is it is offering an explicit expression for the games in the Inverse Set.Note also from (10) that the parenthesis in (11) has a null Semivalue, hence a basis of the null subspace is also shown in (11), namely with dimension 2 1.
n n − − Now, a new problem to be discussed in this paper will be introduced, starting with the concept of Power Game, to be used in connection with the coalitional rationality.

The Power Game of the Inverse Set and the Coalitional Rationality
For any game ( ) , N v , and any value Φ defined on N G , the Power Game of ( ) derived from the given game by ( ) ( ) , , .
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 Φ .Thus, for a Semivalue, that is if ∀ ⊆ then, by (13), the Semivalue of the original game ( ) , N v is efficient in any subgame of the Power Game.The Core of the Power Game is called the Power Core.Now, the natural approach in discussing how to share fairly the total win of the grand coalition in the given game is that of following the ideas introduced in the earlier paper [6]: we shall consider the Semivalue of the given game as efficient and coalitional rational, if it belongs to the Power Core.Hence, to check whether, or not, this happens, we should compute the Power Core.
However, it will be easier to use a computational formula for the Power Core proved also in the same paper [2]; this will be done next, because we shall compute only the worth of coalitions of size 1 n − .The problem similar to the one discussed above may be stated as: for a given game ( ) , but in which the Semivalue of the given game is also efficient and coalitional rational.Now, there are two cases: either the Semivalue is in the Core of the Power Game for the given game, or not.In the first case, the Semivalue is considered coalitional rational; in the other case, from ( ) , N v we build a new game ( ) , N w in the Inverse Set, where this happens.In the following, we shall confine ourselves to a solution ( ) , N w in the subfamily of the Inverse Set, to be called the null subfamily.In other words, if the Semivalue is not in the Core of the Power Game of the initial game, then for this game ( ) , N v , we want the Semivalue of this given game to belong to the Core of the new game ( ) , N w , from the almost null subfamily.
Now, an arbitrary game in the almost null subfamily of the Inverse Set, relative to the Semivalue 0 L ≥ , is given by and depends on the parameter N c .From ( 14) and ( 8), as all the characteristic functions of the basic games are null when 2 S n ≤ − , we get ( ) 0, ,

{ }
S N i = − , from ( 14) and ( 8),we get 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 ( ) , N w , denoted ( ) , N π , and its Core.
Instead, the Power Game of ( ) , N w may be easier computed by using the formula from the previous work mentioned above, precisely , .

{ }
T N i = − ,from ( 8) and ( 16) we get 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 ( ) , is 0 L ≥ .Then, in a game belonging to the subfamily of the Inverse Set relative to the Semivalue, defined by the Semivalue is in the Power Core, if and only if the parameter N c satisfies the inequality { } ( ) where the minimum is taken over the index i N ∈ .Proof: Taking into account the above formulas (17), (18), for the Power Game, the Power Core is given by 0, 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 [2], the case of the Shapley Value and other efficient values.This is not surprising, because this theorem is also applicable to the Shapley Value, as the Shapley Value is a particular Semivalue.Other remarks will be discussed later, for the moment let us illustrate the results contained in the theorem.
Example 2: Consider the same three person game of Example 1a, shown in (5).Compute the Semivalue associated with the weight vector 3 .
Returning to the game in our example From formula (20) we see that the Semivalue will be coalitional rational if we have 25 3 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 [8], and discussed in [9].

Conclusion
The present work is a continuation of our earlier work [2], on the Inverse Problem of Semivalues.By discussing the new problem of the game with the same value, but coalitional rational, we believe that we give also a motivation for the concept of Inverse Set.We introduced this new problem and we offered an explicit numerical procedure for solving the problem, as seen in the examples that are following the theory given above.A nice introduction is the case of efficient values considered in [3], as well as the application to the case of Binomial Semivalues shown in [9].
Value where the components of the weight vectors are that the Semivalue is not coalitional rational, as it does not satisfy the above inequality.Now, find out a game in the almost null subset of the Inverse Set, from formula (15), formulas (17), (18), we compute the Power Game and we obtain the game 1, where the Banzhaf Value is , the Banzhaf Value of original game and the value of the parameter N c , chosen to satisfy the coalitional rationality.We may check that the Banzhaf Value of the original game belongs to the Core of (32), hence it is coalitional rational.
. As already noticed, this is neither efficient, nor coalitional rational.Let us apply the result of the above Theorem.We may check that this Semivalue is neither efficient, nor coalitional rational.