Abstract
The Shapley value of a cooperative transferable utility game distributes the dividend of each coalition in the game equally among its members. Given exogenous weights for all players, the corresponding weighted Shapley value distributes the dividends proportionally to their weights. A proper Shapley value, introduced in Vorob’ev and Liapounov (Game Theory and Applications, vol IV. Nova Science, New York, pp 155–159, 1998), assigns weights to players such that the corresponding weighted Shapley value of each player is equal to her weight. In this contribution we investigate these proper Shapley values in the context of monotone games. We prove their existence for all monotone transferable utility games and discuss other properties of this solution.
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Notes
Another type of weighted Shapley value is defined in Kalai and Samet (1987) where, besides having weights, the players are partitioned and there is an ordering of the elements of the partition such that only the players in the ‘highest level’ of a coalition share in the corresponding dividend.
Note that this is not a weighted Shapley value because the weights, in general, will be different for different games.
Queueing games being 2-games follows directly from (3) in the proof of Lemma 1 in Maniquet (2003).
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Miroslav Zelený was supported by the research project MSM 0021620839 financed by MSMT.
The preliminary draft of this paper was presented at the 19th Stony Brook Game Theory Festival of the Game Theory Society. We appreciate valuable comments from the workshops participants, in particular from Sergiu Hart. We also thank Jean Derks for valuable comments on a previous draft.
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van den Brink, R., Levínský, R. & Zelený, M. On proper Shapley values for monotone TU-games. Int J Game Theory 44, 449–471 (2015). https://doi.org/10.1007/s00182-014-0439-5
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DOI: https://doi.org/10.1007/s00182-014-0439-5