# 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 and Games, 2018, 5 (2) : 95-107. doi: 10.3934/jdg.2018007
##### References:
 [1] M. Álvarez-Mozos and O. Tejada, Parallel characterizations of a generalized shapley value and a generalized banzhaf value for cooperative games with level structure of cooperation, Decision Support Systems, 52 (2011), 21-27. [2] R. J. Aumann and J. H. Dreze, Cooperative games with coalition structures, International Journal of Game Theory, 3 (1974), 217-237.  doi: 10.1007/BF01766876. [3] J. M. Bilbao and P. H. Edelman, The shapley value on convex geometries, Discrete Applied Mathematics, 103 (2000), 33-40.  doi: 10.1016/S0166-218X(99)00218-8. [4] E. Calvo, J. J. Lasaga and E. Winter, The principle of balanced contributions and hierarchies of cooperation, Mathematical Social Sciences, 31 (1996), 171-182.  doi: 10.1016/0165-4896(95)00806-3. [5] U. Faigle and W. Kern, The shapley value for cooperative games under precedence constraints, International Journal of Game Theory, 21 (1992), 249-266.  doi: 10.1007/BF01258278. [6] M. Gómez-Rúa and J. Vidal-Puga, Balanced per capita contributions and level structure of cooperation, Top, 19 (2011), 167-176.  doi: 10.1007/s11750-009-0122-3. [7] J. C. Harsanyi, A simplified bargaining model for the n-person cooperative game, Part of the Theory and Decision Library book series, 28 (1960), 44-70.  doi: 10.1007/978-94-017-2527-9_3. [8] S. Hart and A. Mas-Colell, Potential, value, and consistency, Econometrica: Journal of the Econometric Society, 57 (1989), 589-614.  doi: 10.2307/1911054. [9] E. Kalai and D. Samet, On weighted shapley values, International Journal of Game Theory, 16 (1987), 205-222.  doi: 10.1007/BF01756292. [10] Y. Kamijo, The collective value: A new solution for games with coalition structures, Top, 21 (2013), 572-589.  doi: 10.1007/s11750-011-0191-y. [11] G. Koshevoy and D. Talman, Solution concepts for games with general coalitional structure, Mathematical Social Sciences, 68 (2014), 19-30.  doi: 10.1016/j.mathsocsci.2013.12.004. [12] R. B. Myerson, Graphs and cooperation in games, Mathematics of Operations Research, 2 (1977), 225-229.  doi: 10.1287/moor.2.3.225. [13] R. B. Myerson, Conference structures and fair allocation rules, International Journal of Game Theory, 9 (1980), 169-182.  doi: 10.1007/BF01781371. [14] G. Owen, Values of graph-restricted games, SIAM Journal on Algebraic Discrete Methods, 7 (1986), 210-220.  doi: 10.1137/0607025. [15] G. Owen, Multilinear extensions of games, in The Shapley value: essays in honor of Lloyd S. Shapley (ed. A. E. Roth), Cambridge University Press, 1988, chapter 10, 139-151. [16] G. Owen and E. Winter, The multilinear extension and the coalition structure value, Games and Economic Behavior, 4 (1992), 582-587.  doi: 10.1016/0899-8256(92)90038-T. [17] G. Owen, Values of games with a priori unions, in Mathematical Economics and Game Theory, Springer, 141 (1977), 76-88. [18] R. Stanley, Enumerative Combinatorics, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1986. [19] R. van den Brink, A. Khmelnitskaya and G. van der Laan, An owen-type value for games with two-level communication structure, Annals of Operations Research, 243 (2016), 179-198.  doi: 10.1007/s10479-015-1808-6. [20] E. Winter, A value for cooperative games with levels structure of cooperation, International Journal of Game Theory, 18 (1989), 227-240.  doi: 10.1007/BF01268161.

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##### References:
 [1] M. Álvarez-Mozos and O. Tejada, Parallel characterizations of a generalized shapley value and a generalized banzhaf value for cooperative games with level structure of cooperation, Decision Support Systems, 52 (2011), 21-27. [2] R. J. Aumann and J. H. Dreze, Cooperative games with coalition structures, International Journal of Game Theory, 3 (1974), 217-237.  doi: 10.1007/BF01766876. [3] J. M. Bilbao and P. H. Edelman, The shapley value on convex geometries, Discrete Applied Mathematics, 103 (2000), 33-40.  doi: 10.1016/S0166-218X(99)00218-8. [4] E. Calvo, J. J. Lasaga and E. Winter, The principle of balanced contributions and hierarchies of cooperation, Mathematical Social Sciences, 31 (1996), 171-182.  doi: 10.1016/0165-4896(95)00806-3. [5] U. Faigle and W. Kern, The shapley value for cooperative games under precedence constraints, International Journal of Game Theory, 21 (1992), 249-266.  doi: 10.1007/BF01258278. [6] M. Gómez-Rúa and J. Vidal-Puga, Balanced per capita contributions and level structure of cooperation, Top, 19 (2011), 167-176.  doi: 10.1007/s11750-009-0122-3. [7] J. C. Harsanyi, A simplified bargaining model for the n-person cooperative game, Part of the Theory and Decision Library book series, 28 (1960), 44-70.  doi: 10.1007/978-94-017-2527-9_3. [8] S. Hart and A. Mas-Colell, Potential, value, and consistency, Econometrica: Journal of the Econometric Society, 57 (1989), 589-614.  doi: 10.2307/1911054. [9] E. Kalai and D. Samet, On weighted shapley values, International Journal of Game Theory, 16 (1987), 205-222.  doi: 10.1007/BF01756292. [10] Y. Kamijo, The collective value: A new solution for games with coalition structures, Top, 21 (2013), 572-589.  doi: 10.1007/s11750-011-0191-y. [11] G. Koshevoy and D. Talman, Solution concepts for games with general coalitional structure, Mathematical Social Sciences, 68 (2014), 19-30.  doi: 10.1016/j.mathsocsci.2013.12.004. [12] R. B. Myerson, Graphs and cooperation in games, Mathematics of Operations Research, 2 (1977), 225-229.  doi: 10.1287/moor.2.3.225. [13] R. B. Myerson, Conference structures and fair allocation rules, International Journal of Game Theory, 9 (1980), 169-182.  doi: 10.1007/BF01781371. [14] G. Owen, Values of graph-restricted games, SIAM Journal on Algebraic Discrete Methods, 7 (1986), 210-220.  doi: 10.1137/0607025. [15] G. Owen, Multilinear extensions of games, in The Shapley value: essays in honor of Lloyd S. Shapley (ed. A. E. Roth), Cambridge University Press, 1988, chapter 10, 139-151. [16] G. Owen and E. Winter, The multilinear extension and the coalition structure value, Games and Economic Behavior, 4 (1992), 582-587.  doi: 10.1016/0899-8256(92)90038-T. [17] G. Owen, Values of games with a priori unions, in Mathematical Economics and Game Theory, Springer, 141 (1977), 76-88. [18] R. Stanley, Enumerative Combinatorics, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1986. [19] R. van den Brink, A. Khmelnitskaya and G. van der Laan, An owen-type value for games with two-level communication structure, Annals of Operations Research, 243 (2016), 179-198.  doi: 10.1007/s10479-015-1808-6. [20] E. Winter, A value for cooperative games with levels structure of cooperation, International Journal of Game Theory, 18 (1989), 227-240.  doi: 10.1007/BF01268161.
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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