Shapley value

1.1. A VALUE FOR N-PERSON GAMES(L.S.SHAPLEY 1952) 4
Proof. by Axiom3.
Definition 1.14. v and w are strategically equivalent iff w = cv + a,
where c is a positive constant and a is an additive set-function on U
with finite carrier.
Proposition 1.15. v and w are strategically equivalent then
φ i (w) = cφ i (v) + a(i) (all i ∈ U)
Proof. by Axiom3 and propsition1.11 applied to the inessential game
a, φ i is linear and homogeneous in v.
Definition 1.16. v is constant-sum iff v(S)+v(U−S) = v(U)
U).
(all S ⊆
Proposition 1.17. If v is constant-sum then its value is given by
the formula:
∑
φ i [v] = 2[ r n (s)v(S)] − v(N) (all i ∈ N),
S⊆N,i∈S
where N is any finite carrier of v.
Proof.
∑
φ i [v] = r n (s)v(S) −
∑
r n (t + 1)v(T )
=
= 2[
S⊆N,i∈S
∑
S⊆N,i∈S
∑
r n (s)v(S) −
r n (s)v(S)] −
T ⊆N,i/∈T
∑
S⊆N,i∈S
∑
S⊆N,i∈S
S⊆N,i∈S
r n (n − s + 1)[v(N) − v(S)]
v(N).
by propsition1.11 proof.
Definition 1.18. the centroid of imputation set by θ:
θ i = v(i) + 1 n [v(N) − ∑ v(j)].
j∈N
Example1
For two-person games, three-person constant-sum games, and inessential
games, we have φ = θ.
1.two-person games
by theorem φ 1 = ∑ 1 inS⊆1,2 r 2(s)[v(S) − v(S − 1)] = r 2 (1)(v(1) −
v(∅)) + r 2 (1)(v(1, 2) − v(2)) = 1 2 v(1) + 1 2 v(1, 2) − 1 2v(2) = v(1) +