The Core and Shapley Function for Games on Augmenting Systems with a Coalition Structure
In this paper, we first introduce the model of games on augmenting systems with a coalition structure, which can be seen as an extension of games on augmenting systems. The core of games on augmenting systems with a coalition structure is defined, and an equivalent form is discussed. Meantime, the Shapley function for this type of games is given, and two axiomatic systems of the given Shapley function are researched. When the given games are quasi convex, the relationship between the core and the Shapley function is discussed, which does coincide as in classical case. Finally, a numerical example is given.
[1] R. Aumann and J. Dreze, Cooperative games with coalition structures,
International Journal of Game Theory 3 (1974) 217-237.
[2] G. Owen, Values of games with a priori unions, In: Henn, R., Moeschlin,
O. (Eds.), Lecture Notes in Economics and Mathematical Systems,
Essays in Honor of Oskar Morgenstern, Springer Verlag, Nueva York,
1977.
[3] G. Owen, Characterization of the Banzhaf-Coleman index, SIAM Journal
of Applied Mathematics 35 (1978) 315-327.
[4] R. Amer, F. Carreras and J. M. Gimenez, The modified Banzhaf
value for games with coalition structure: an axiomatic characterization,
Mathematical Social Sciences 43 (2002) 45-54.
[5] M. J. Albizuri, Axiomatizations of the Owen value without efficiency,
Mathematical Social Sciences 55 (2008) 78-89.
[6] G. Hamiache, A new axiomatization of the Owen value for games with
coalition structures, Mathematical Social Sciences 37 (1999) 281-305.
[7] A. B. Khmelnitskaya and E. B. Yanovskaya, Owen coalitional value
without additivity axiom, Mathematical Methods of Operations Research
66 (2007) 255-261.
[8] R. B. Myerson, Graphs and cooperation in games, Mathematics of
Operations Research 2 (1997) 25-229.
[9] U. Faigle and W. Kern, The Shapley value for cooperative games under
precedence constraints, International Journal of Game Theory 21 (1992)
249-266.
[10] J. M. Bilbao, Axioms for the Shapley value on convex geometries,
European Journal of Operational Research 110 (1998) 368-376.
[11] J. M. Bilbao, T. S. H. Driessen, A. J. Losada and E. Lebron, The Shapley
value for games on matroids: The static model, Mathematical Methods
of Operations Research 53 (2001) 333-348.
[12] J. M. Bilbao, T. S. H. Driessen, A. J. Losada and E. Lebron, The Shapley
value for games on matroids: The dynamic model, Mathematical
Methods of Operations Research 56 (2002) 287-301.
[13] J. M. Bilbao, Cooperative games under augmenting systems. SIAM
Journal of Discrete Mathematics 17 (2003) 122-133.
[14] E. Algaba, J. M. Bilbao, R.Van den Brink and A. J. Losada, Axiomatizations
of the Shapley value for cooperative games on antimatroids,
Mathematical Methods of Operations Research 57 (2003) 49-65.
[15] J. M. Bilbao and M. Ordonez, Axiomatizations of the Shapley value
for games on augmenting systems, European Journal of Operational
Research 196 (2009) 1008-1014.
[16] Y. Sprumont, Population monotonic allocation schemes for cooperative
games with transferable utility, Games and Economic Behavior 23
(1990) 378-394.
[1] R. Aumann and J. Dreze, Cooperative games with coalition structures,
International Journal of Game Theory 3 (1974) 217-237.
[2] G. Owen, Values of games with a priori unions, In: Henn, R., Moeschlin,
O. (Eds.), Lecture Notes in Economics and Mathematical Systems,
Essays in Honor of Oskar Morgenstern, Springer Verlag, Nueva York,
1977.
[3] G. Owen, Characterization of the Banzhaf-Coleman index, SIAM Journal
of Applied Mathematics 35 (1978) 315-327.
[4] R. Amer, F. Carreras and J. M. Gimenez, The modified Banzhaf
value for games with coalition structure: an axiomatic characterization,
Mathematical Social Sciences 43 (2002) 45-54.
[5] M. J. Albizuri, Axiomatizations of the Owen value without efficiency,
Mathematical Social Sciences 55 (2008) 78-89.
[6] G. Hamiache, A new axiomatization of the Owen value for games with
coalition structures, Mathematical Social Sciences 37 (1999) 281-305.
[7] A. B. Khmelnitskaya and E. B. Yanovskaya, Owen coalitional value
without additivity axiom, Mathematical Methods of Operations Research
66 (2007) 255-261.
[8] R. B. Myerson, Graphs and cooperation in games, Mathematics of
Operations Research 2 (1997) 25-229.
[9] U. Faigle and W. Kern, The Shapley value for cooperative games under
precedence constraints, International Journal of Game Theory 21 (1992)
249-266.
[10] J. M. Bilbao, Axioms for the Shapley value on convex geometries,
European Journal of Operational Research 110 (1998) 368-376.
[11] J. M. Bilbao, T. S. H. Driessen, A. J. Losada and E. Lebron, The Shapley
value for games on matroids: The static model, Mathematical Methods
of Operations Research 53 (2001) 333-348.
[12] J. M. Bilbao, T. S. H. Driessen, A. J. Losada and E. Lebron, The Shapley
value for games on matroids: The dynamic model, Mathematical
Methods of Operations Research 56 (2002) 287-301.
[13] J. M. Bilbao, Cooperative games under augmenting systems. SIAM
Journal of Discrete Mathematics 17 (2003) 122-133.
[14] E. Algaba, J. M. Bilbao, R.Van den Brink and A. J. Losada, Axiomatizations
of the Shapley value for cooperative games on antimatroids,
Mathematical Methods of Operations Research 57 (2003) 49-65.
[15] J. M. Bilbao and M. Ordonez, Axiomatizations of the Shapley value
for games on augmenting systems, European Journal of Operational
Research 196 (2009) 1008-1014.
[16] Y. Sprumont, Population monotonic allocation schemes for cooperative
games with transferable utility, Games and Economic Behavior 23
(1990) 378-394.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:52547", author = "Fan-Yong Meng", title = "The Core and Shapley Function for Games on Augmenting Systems with a Coalition Structure", abstract = "In this paper, we first introduce the model of games on augmenting systems with a coalition structure, which can be seen as an extension of games on augmenting systems. The core of games on augmenting systems with a coalition structure is defined, and an equivalent form is discussed. Meantime, the Shapley function for this type of games is given, and two axiomatic systems of the given Shapley function are researched. When the given games are quasi convex, the relationship between the core and the Shapley function is discussed, which does coincide as in classical case. Finally, a numerical example is given.
", keywords = "Cooperative game, augmenting system, Shapley function, core.", volume = "6", number = "8", pages = "881-6", }
