Abstract
This paper discusses the so-called multigraph, in which multiple connections (arcs) between two given nodes and loops are possible. The authors present the concept of using Shapley values to analyse both elements (nodes and arcs) of a multigraph. The results obtained allow to evaluate the importance of a given element (node or arc) as an element of a whole structure of the graph. The authors proposed a new cooperative game solution and Shapley value which may determine the evaluation of multigraph elements (which may, among others, determine the cost of its use or the volume of flows).
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Notes
- 1.
There are several claims regarding modified assumptions. E.g., Young (1985) demonstrated that Shapley value is the only value that satisfies effectiveness, symmetry, and strong monotonicity, van den Brink (2001), by modifying contributions to the coalition, Myerson (1977), with balanced contributions to the coalition.
- 2.
In a directed graph such node is source or target node.
- 3.
\( card\left\{ B \right\}\, or\, card\left( B \right) \) represents cardinality of a set B.
- 4.
Note that the graph structure is not explicitly used here, but it can be assumed that this structure affects the probabilities of a given coalition of players (vertices).
References
Aumann, R.J.: Recent developments in the theory of the Shapley value. In: Proceedings of the International Congress of Mathematicians, Helsinki, pp. 995–1003 (1978)
Bertini, C.: Shapley value. In: Dowding, K. (ed.) Encyclopedia of Power, pp. 600–603. SAGE Publications, Los Angeles (2011)
Bertini, C., Mercik, J., Stach, I.: Indirect control and power. Oper. Res. Decis. 26, 7–30 (2016)
van den Brink, R.: An axiomatization of the Shapley value using a fairness property. Int. J. Game Theory 30, 309–319 (2001)
Hart, S.: Shapley value. In: Eatwell, J., Milgate, M., Newman, P. (eds.) Game Theory, pp. 210–216. Palgrave Macmillan, UK (1989)
Mercik, J., Ramsey, D.M.: The effect of Brexit on the balance of power in the European union council: an approach based on pre-coalitions. In: Mercik, J. (ed.) Transactions on Computational Collective Intelligence XXVII. LNCS, vol. 10480, pp. 87–107. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70647-4_7
Myerson, R.: Graphs and cooperation in games. Math. Oper. Res. 2, 225–229 (1977)
Owen, G.: Values of games with a priori unions. In: Henn, R., Moeschlin, O. (eds.) Mathematical Economics and Game Theory. LNE, vol. 141, pp. 76–88. Springer, Heidelberg (1977). https://doi.org/10.1007/978-3-642-45494-3_7
Shapley, L.S.: A value for n-person games. In: Kuhn, H., Tucker, A.W. (eds.) Contributions to the Theory of Games II. Annals of Mathematics Studies, vol. 28, pp. 307–317. Princeton University Press (1953)
Stach, I.: Sub-coalitional approach to values. In: Mercik, J. (ed.) Transactions on Computational Collective Intelligence XXVII. LNCS, vol. 10480, pp. 74–86. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70647-4_6
Young, H.: Monotonic solutions of cooperative games. Int. J. Game Theory 14, 65–72 (1985)
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Forlicz, S., Mercik, J., Stach, I., Ramsey, D. (2018). The Shapley Value for Multigraphs. In: Nguyen, N., Pimenidis, E., Khan, Z., Trawiński, B. (eds) Computational Collective Intelligence. ICCCI 2018. Lecture Notes in Computer Science(), vol 11056. Springer, Cham. https://doi.org/10.1007/978-3-319-98446-9_20
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