Variations on the Shapley value

density function, such that ψv(i) is given by the right-hand side of (4.4), where
(αi)i∈N are the marginal distribution functions.
Owen (1972) defined the multilinear extension of a game v. Each coalition S
can be identified with an extreme point, eS, of the cube [0, 1] N , where eS is the
characteristic function of S. Therefore, each game v can be considered as a function
on the extreme points of the cube and hence it can be uniquely extended to a
multilinear function on the cube, which is denoted by ¯v. Owen showed that if the
random variables that define b via (4.2) are independent, then for every i:
(4.5) ϕ b v(i) =
1
0
∂¯v
(α(t))dαi(t).
∂xi
In particular, the Shapley value is determined by n i.i.d. random variables that are
uniformly distributed in [0, 1]. Therefore,
where e = eN.
ϕv(i) =
1
0
∂¯v
(te)dt,
∂xi
The presentation of the solution ϕ b , in (4.5), gives rise to another natural solu-
tion. A solution ψ is a path value if there exists an absolutely continuous nonde-
creasing path α : [0, 1] → [0, 1] N with α(0) = 0 and α(1) = e such that for every
game v and every player i, ψv(i) is defined by the right-hand side of (4.5). Alter-
natively, ψ is a path value if there exists b ∈ ∆(OR) such that ψ = ϕ b , and there
exist n independent random variables that are distributed in [0, 1] with absolutely
continuous distribution functions such that (4.2) holds. It can be verified that not
every quasivalue is a path value.
Remarks.
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