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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 |
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.
Keywords:
Nested constraints,
level structure,
Shapley value,
class balanced contributions,
multilinear extension.
Mathematics Subject Classification:
Primary: 91A12, 91A44; Secondary: 91A46.
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
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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. |
show all references
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. |
| 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 |
Table 2.
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 |
Table 3.
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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