Abstract
In Chap. 17 we have seen that the Shapley value of a game does not have to be in the core of the game, nor even an imputation (Problem 17.5). In this chapter we introduce a set-valued extension of the Shapley value, the Weber set, and show that it always contains the core (Sect. 18.1). Next, we study so-called convex games and show that these are exactly those games for which the core and the Weber set coincide. Hence, for such games the Shapley value is an attractive core selection (Sect. 18.2). Finally, we study random order values (Sect. 18.3), which fill out the Weber set, and the subset of weighted Shapley values, which still cover the core (Sect. 18.4).
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References
Derks, J. (1992). A short proof of the inclusion of the core in the Weber set. International Journal of Game Theory, 21, 149–150.
Derks, J., Haller, H., & Peters, H. (2000). The Selectope for cooperative games. International Journal of Game Theory, 29, 23–38.
Ichiishi, T. (1981). Super-modularity: Applications to convex games and to the greedy algorithm for LP. Journal of Economic Theory, 25, 283–286.
Kalai, E., & Samet, D. (1987). On weighted Shapley values. International Journal of Game Theory, 16, 205–222.
Littlechild, S. C., & Owen, G. (1974). A simple expression for the Shapley Value in a special case. Management Science, 20, 370–372.
Monderer, D., Samet, D., & Shapley, L. S. (1992). Weighted values and the core. International Journal of Game Theory, 21, 27–39.
Owen, G. (1972). Multilinear extensions of games. Management Science, 18, 64–79.
Shapley, L. S. (1971). Cores of convex games. International Journal of Game Theory, 1, 11–26.
Weber, R. J. (1988). Probabilistic values for games. In Roth, A. E. (Ed.), The Shapley value. Essays in honor of Lloyd S. Shapley (pp. 101–119). Cambridge: Cambridge University Press.
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Peters, H. (2015). Core, Shapley Value, and Weber Set. In: Game Theory. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46950-7_18
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DOI: https://doi.org/10.1007/978-3-662-46950-7_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-46949-1
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