# American Institute of Mathematical Sciences

October  2013, 9(4): 885-892. doi: 10.3934/jimo.2013.9.885

## Reduction and dynamic approach for the multi-choice Shapley value

 1 Department of Applied Mathematics, National Dong Hwa University, Hualien 974, Taiwan 2 Department of Applied Mathematics, National Pingtung University of Education, Pingtung 900, Taiwan

Received  March 2012 Revised  February 2013 Published  August 2013

In the framework of multi-choice games, we propose a specific reduction to construct a dynamic process for the multi-choice Shapley value introduced by Nouweland et al. [8].
Citation: Yan-An Hwang, Yu-Hsien Liao. Reduction and dynamic approach for the multi-choice Shapley value. Journal of Industrial and Management Optimization, 2013, 9 (4) : 885-892. doi: 10.3934/jimo.2013.9.885
##### References:
 [1] R. J. Aumann and L. S. Shapley, "Values of Non-Atomic Games," Princeton University Press, Princeton, N.J., 1974. [2] E. Calvo and J. C. Santos, A value for multichoice games, Mathematical Social Sciences, 40 (2000), 341-354. doi: 10.1016/S0165-4896(99)00054-2. [3] S. Hart and A. Mas-Colell, Potential, value and consistency, Econometrica, 57 (1989), 589-614. doi: 10.2307/1911054. [4] Y. A. Hwang and Y. H. Liao, Potentializability and consistency for multi-choice solutions, Spanish Economic Review, 10 (2008), 289-301. [5] M. Maschler and G. Owen, The consistent Shapley value for hyperplane games, International Journal of Game Theory, 18 (1989), 389-407. doi: 10.1007/BF01358800. [6] H. Moulin, On additive methods to share joint costs, The Japanese Economic Review, 46 (1995), 303-332. [7] R. Myerson, Conference structures and fair allocation rules, International Journal of Game Theory, 9 (1980), 169-182. doi: 10.1007/BF01781371. [8] A. van den Nouweland, J. Potters, S. Tijs and J. M. Zarzuelo, Core and related solution concepts for multi-choice games, ZOR-Mathematical Methods of Operations Research, 41 (1995), 289-311. doi: 10.1007/BF01432361. [9] L. S. Shapley, A value for $n$-person game, in "Contributions to the Theory of Games II" (eds. H. W. Kuhn and A. W. Tucker), Annals of Mathematics Studies, 28, Princeton University Press, Princeton, (1953), 307-317.

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##### References:
 [1] R. J. Aumann and L. S. Shapley, "Values of Non-Atomic Games," Princeton University Press, Princeton, N.J., 1974. [2] E. Calvo and J. C. Santos, A value for multichoice games, Mathematical Social Sciences, 40 (2000), 341-354. doi: 10.1016/S0165-4896(99)00054-2. [3] S. Hart and A. Mas-Colell, Potential, value and consistency, Econometrica, 57 (1989), 589-614. doi: 10.2307/1911054. [4] Y. A. Hwang and Y. H. Liao, Potentializability and consistency for multi-choice solutions, Spanish Economic Review, 10 (2008), 289-301. [5] M. Maschler and G. Owen, The consistent Shapley value for hyperplane games, International Journal of Game Theory, 18 (1989), 389-407. doi: 10.1007/BF01358800. [6] H. Moulin, On additive methods to share joint costs, The Japanese Economic Review, 46 (1995), 303-332. [7] R. Myerson, Conference structures and fair allocation rules, International Journal of Game Theory, 9 (1980), 169-182. doi: 10.1007/BF01781371. [8] A. van den Nouweland, J. Potters, S. Tijs and J. M. Zarzuelo, Core and related solution concepts for multi-choice games, ZOR-Mathematical Methods of Operations Research, 41 (1995), 289-311. doi: 10.1007/BF01432361. [9] L. S. Shapley, A value for $n$-person game, in "Contributions to the Theory of Games II" (eds. H. W. Kuhn and A. W. Tucker), Annals of Mathematics Studies, 28, Princeton University Press, Princeton, (1953), 307-317.
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