*2.3. Shapley Value*

The Shapley Value [28] is a solution concept for coalitional games along with the core, nucleolus and Pareto optimal, among others. Given a coalitional game (N , *<sup>v</sup>*), there is a unique feasible payoff division *x*(*v*) = *ϕ* (N , *v*) that divides the full payoff of the grand coalition. The Shapley Value can be defined as [29],

$$p\_i(\mathcal{N}, v) = \frac{1}{N!} \sum\_{R=1}^{N!} \left[ v(P\_i(R) \cup i) - v(P\_i(R)) \right] \tag{1}$$

where *R* is the set of all *N*! orderings of N , *Pi*(*R*) is the set of players preceding *i* in the ordering *R* and *<sup>ϕ</sup>i*(N , *v*) is the expected marginal contribution over all orders of player *i* to the set of players who are preceding it [26].

The Shapley Value also satisfies the following axioms [30]:


• Efficiency: The efficiency axioms states that the entire payoff is divided among the players, so <sup>∑</sup>N1 *ϕi* = *v*(N ), where *ϕi* is the Shapley Value of player *i*.
