Shapley Value

Definition

The Shapley value (Shapley 1953) probably is the most eminent (single-valued) solution concept for cooperative games with transferable utility (TU games)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 φ_i (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...