The position value and the structures of graphs. (English) Zbl 1429.91027
Summary: The position value is an allocation rule based on the Shapley value of the link game from the original communication situation, in which cooperation is restricted by a graph. In the link games, feasible coalitions are connected but their structures are ignored. We introduce structure functions to describe the structures of connected sets, and generalize the link game and the position value to the setting with local structures. We modify an axiomatic characterization for the position value by M. Slikker [Int. J. Game Theory 33, No. 4, 505–514 (2005; Zbl 1091.91006); with A. van den Nouweland, Math. Soc. Sci. 64, No. 3, 266–271 (2012; Zbl 1260.90045)] to the generalized position value by component efficiency and balanced link contributions on local structures.
References:
| [1] | Algaba, E.; Bilbao, J. M.; Borm, P.; López, J. J., The position value for union stable systems, Math. Methods Oper. Res., 52, 221-236 (2000) · Zbl 1103.91310 |
| [2] | Borm, P.; Owen, G.; Tijs, S., On the position value for communication situations, SIAM J. Discret. Math., 5, 305-320 (1992) · Zbl 0788.90087 |
| [3] | Casajus, A., The position value is myerson value in a sense, Int. J. Game Theory, 36, 47-55 (2007) · Zbl 1210.91013 |
| [4] | Cesari, G.; Ferrari, M. M., On the position value for special classes of networks, (Petrosyan, L. A.; Mazalov, V. V., Recent Advances in Game Theory and Applications, Static & Dynamic Game Theory: Foundations & Applications (2016), Springer International Publishing: Springer International Publishing Switzerland), 29-47 · Zbl 1397.91030 |
| [5] | Gómez, D.; González-Arangüena, E.; Manuel, C.; Owen, G.; Pozo, M. D., A unified approach to the Myerson and the position value, Theory Decis., 56, 63-76 (2004) · Zbl 1107.91031 |
| [6] | Jackson, M.; Wolinsky, A., A strategic model of social and economic networks, J. Econ. Theory, 71, 44-74 (1996) · Zbl 0871.90144 |
| [7] | Kamijo, Y.; Kongo, T., Axiomatization of the Shapley value using the balanced cycle contributions property, Int. J. Game Theory, 39, 563-571 (2010) · Zbl 1211.91041 |
| [8] | Kongo, T., Difference between the position value and the Myerson value is due to exitence of coalition structures, Int. J. Game Theory, 39, 669-675 (2010) · Zbl 1211.91042 |
| [9] | Meessen, R., Communication Games (1988), Department of Mathematics, University of Nijmegen: Department of Mathematics, University of Nijmegen Netherlands, Master thesis |
| [10] | Myerson, R. B., Graphs and cooperation in games, Math. Oper. Res., 2, 225-229 (1977) · Zbl 0402.90106 |
| [11] | Myerson, R. B., Conference structures and fair allocation rules, Int. J. Game Theory, 9, 169-182 (1980) · Zbl 0441.90117 |
| [12] | Shapley, L. S., A value for \(n\)-person games, (Tucker, A. W.; Kuhn, H. W., Contributions to the Theory of Games II (1953), Princeton University Press: Princeton University Press Princeton, NJ), 307-317 · Zbl 0050.14404 |
| [13] | Slikker, M., A characterization of the position value, Int. J. Game Theory, 33, 505-514 (2005) · Zbl 1091.91006 |
| [14] | van den Nouweland, A.; Slikker, M., An axiomatic characterization of the position value for network situations, Math. Soc. Sci., 64, 266-271 (2012) · Zbl 1260.90045 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
