Definition
The Shapley value (Shapley 1953) probably is the most eminent (single-valued) solution concept for cooperative games with transferable utility (TU games)Footnote 1. A (TU) game is a pair (N, v) consisting of a nonempty and finite set of players N and a coalition function\( v\in \mathbb{V}(N):= \left\{f:2N\to \mathrm{\mathbb{R}}|f\left(\O \right)=0\right\} \). Subsets of the player set are called coalitions. The coalition function assigns a worthv(S) to any coalition S, which indicates what the players in S can achieve by cooperation. A (single-valued) solution conceptφ on N assigns to any game (N, v) a payoff vector \( \varphi \left(N,v\right)\in {\mathrm{\mathbb{R}}}^N \), where φi (N, v) denotes the payoff of player i in the game (N, v).
The Shapley value on the player set N is defined as follows: A rank order on N is a bijection \( \rho :N\to \left\{1,\dots, \left|N\right|\right\} \) with the interpretation that i is the ρ (i)th player in ρ; R (N) denotes the set of all...
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Notes
- 1.
Peleg and Sudhölter (2007) provide a good introduction to cooperative game theory.
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Casajus, A., Labrenz, H. (2019). Shapley Value. In: Marciano, A., Ramello, G.B. (eds) Encyclopedia of Law and Economics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7753-2_437
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