# American Institute of Mathematical Sciences

April  2018, 5(2): 95-107. doi: 10.3934/jdg.2018007

## Games with nested constraints given by a level structure

 Jalisco S/N, Valenciana, CP: 36240, CIMAT, A.C., Guanajuato, Gto, México

* Corresponding author: mvargas@cimat.mx

Received  December 2016 Revised  November 2017 Published  February 2018

In this paper we propose new games that satisfy nested constraints given by a level structure of cooperation. This structure is defined by a family of partitions on the set of players. It is ordered in such a way that each partition is a refinement of the next one. We propose a value for these games by adapting the Shapley value. The value is characterized axiomatically. For this purpose, we introduce a new property called class balance contributions by generalizing other properties in the literature. Finally, we introduce a multilinear extension of our games and use it to obtain an expression for calculating the adapted Shapley value.

Citation: Francisco Sánchez-Sánchez, Miguel Vargas-Valencia. Games with nested constraints given by a level structure. Journal of Dynamics & Games, 2018, 5 (2) : 95-107. doi: 10.3934/jdg.2018007
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##### References:
Characteristic function for maintenance cost of a highway system.
 S {1} {2} {3} {4} {2, 3} {1, 2, 3} {1, 2, 3, 4} v(S) 1 1 0 1 2 4 6
 S {1} {2} {3} {4} {2, 3} {1, 2, 3} {1, 2, 3, 4} v(S) 1 1 0 1 2 4 6
Games of classes for counties as players.
 R {1} {2} {1, 2} R {3} vC12(R) 1 2 4 vC22(R) 1
 R {1} {2} {1, 2} R {3} vC12(R) 1 2 4 vC22(R) 1
Game of classes for states as players.
 R {1} {2} {1, 2} vC13(R) 4 1 6
 R {1} {2} {1, 2} vC13(R) 4 1 6
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