On the Coalitional Rationality of the Shapley Value and Other Efficient Values of Cooperative TU Games

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.

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.

KEYWORDS

Efficiency, Shapley Value, Coalitional Rationality, Least Square Values, Inverse Problem, Inverse Set

Efficiency, Shapley Value, Coalitional Rationality, Least Square Values, Inverse Problem, Inverse Set

Cite this paper

Dragan, I. (2014) On the Coalitional Rationality of the Shapley Value and Other Efficient Values of Cooperative TU Games.*American Journal of Operations Research*, **4**, 228-234. doi: 10.4236/ajor.2014.44022.

Dragan, I. (2014) On the Coalitional Rationality of the Shapley Value and Other Efficient Values of Cooperative TU Games.

References

[1] Owen, G. (1995) Game Theory. 2nd Edition, Academic Press, New York.

[2] Shapley, L.S. (1953) A Value for n-Person Games. Annals of Mathematics Studies, No. 28, 307-317.

[3] Dragan, I. (1991) The Potential Basis and the Weighted Shapley Value. Libertas Mathematica, 12, 139-146.

[4] Ruiz, L., Valenciano, F. and Zarzuelo, J. (1998) The Least Square Prenucleolus and the Least Square Nucleolus, Two Values for TU Games Based upon the Excess Vector. IJGT, 25, 113-134.

[5] Keane, M. (1969) Some Topics in n-Person Game Theory. Ph.D. Thesis, Northwestern University, Evanston.

[6] Dragan, I. (2006) The Least Square Value and the Shapley Value for Cooperative TU Games. Top, 14, 61-73.

http://dx.doi.org/10.1007/BF02579002

[1] Owen, G. (1995) Game Theory. 2nd Edition, Academic Press, New York.

[2] Shapley, L.S. (1953) A Value for n-Person Games. Annals of Mathematics Studies, No. 28, 307-317.

[3] Dragan, I. (1991) The Potential Basis and the Weighted Shapley Value. Libertas Mathematica, 12, 139-146.

[4] Ruiz, L., Valenciano, F. and Zarzuelo, J. (1998) The Least Square Prenucleolus and the Least Square Nucleolus, Two Values for TU Games Based upon the Excess Vector. IJGT, 25, 113-134.

[5] Keane, M. (1969) Some Topics in n-Person Game Theory. Ph.D. Thesis, Northwestern University, Evanston.

[6] Dragan, I. (2006) The Least Square Value and the Shapley Value for Cooperative TU Games. Top, 14, 61-73.

http://dx.doi.org/10.1007/BF02579002