also wish to capture the notion of fairness in this outcome. While the notion is “fairness” is not captured in any one single definition, it can be represented reasonably by a few different properties.

In cooperative games, the ***shapley value*** derives a payoff vector $<\phi_1,\phi_2,\dots,\phi_n>$ based on each player’s marginal contribution to all possible sub-coalitions in the following equation, given that $v(.)$ is the characteristic function: $$\phi_{i}=\sum _{S\subseteq N\setminus \{i\}}{\frac {|S|!\;(N-|S|-1)!}{N!}}(v(S\cup \{i\})-v(S))$$ This payoff vector satisfies the following properties which capture the notion of “fairness” intuitively:

1.  **Symmetry**; if $v(C \cup i)-v(C) = v(C \cup j)-v(C)$ for all $C \subseteq N$, then $\phi_i = \phi_j$. That is, if player $i$ and player $j$ has the same marginal contribution to every possible coalition of players, then $\phi_i=\phi_j$

2.  **Null player**; if $v(C \cup i) = v(C)$ for all $C \subseteq N$, then $\phi_i=0$. That is, if player $i$ has zero marginal contribution to any subcoalition, then $\phi_i=0$

3.  **Deservedness** if $v(C \cup i) \leq v(C \cup j)$ for all $C \subseteq N$, then $\phi_i \leq \phi_j$. That is, if player $j$ has a larger marginal contribution than player $i$ for all possible subcoalitions, then $\phi_i \leq \phi_j$.

4.  **Efficiency**; $\sum^{n}_{i}\phi_i=v(N)$.

The 