The Shapley value is a rule for distributing the total value produced by a finite set 
of 
players. Let 
denote the value produced by each subset 
, with 
. The Shapley value assigned to player 
is
![phi_i(v)=sum_(S subset= N\{i})(|S|!(n-|S|-1)!)/(n!)[v(S union {i})-v(S)].](/images/equations/ShapleyValue/NumberedEquation1.svg) |
The difference 
is the marginal contribution of player 
to 
.
The factorial coefficient is the probability that exactly the players in 
precede 
in a uniformly random ordering of all players. Thus, 
is the expected marginal contribution
of player 
over all possible player orderings.
-Person Games." In