Variations on the Shapley value

Theorem 3. A solution is a quasivalue if and only if it is a random-order value.
Note that the Shapley value is a random-order value with b(τ) = 1/n! for every
τ ∈ OR. 2
It can be easily verified that for every b ∈ ∆(OR) there exist n random variables
(zi)i∈N that are jointly distributed in the cube [0, 1] N with a density function and
therefore with absolutely continuous marginal distribution functions (αi)i∈N, such
that for every τ ∈ OR
(4.2) b(τ) = prob(z τ −1 (i) < z τ −1 (j) for every 1 ≤ i < j ≤ n).
This enables us to interpret ϕ b in terms of random arrival times, as follows. We
think of the value of the random variable zi as the time of arrival of player i in the
“room”. The marginal contribution of i, when he arrives at time t, is his marginal
contribution to the set of other players who arrived up to time t. Define
(4.3) N(t) = {j ∈ N : zj ≤ t}.
Then N(t) is a random coalition in C and it can be easily verified that
(4.4) ϕ b v(i) =
1
0
E(v(N(t) ∪ i) − v(N(t) \ i)|zi = t)dαi(t),
where the integral in (4.4) is the Riemann-Stieltjes integral. Thus, we have shown
that a solution ψ is a random order value, if and only if there exist n random
variables (zi)i∈N, which are jointly distributed in the cube [0, 1] N , and have a
2 Observe that for sufficiently large n the affine map b → ϕ b is not one-to-one. Indeed, the
affine dimension of the convex set ∆(OR) is n! − 1, while the affine dimension of the set of
quasivalues is less than n(2 n−1 −1), because each quasivalue ψ is uniquely determined by a vector
(p 1 , p 2 , . . . , p n ) in ×i∈N ∆(C i ) which, in addition, satisfies the necessary linear conditions for the
efficiency of ψ. It is still to be checked whether the uniform distribution over OR is the only one
that gives rise to the Shapley value.
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