Shapley value

1.1. A VALUE FOR N-PERSON GAMES(L.S.SHAPLEY 1952) 2
• AXIOM1 For each π ∈ Π(U), φ πi [πv] = φ i [v].
• AXIOM2 For each carrier N of v, ∑ N φ i[v] = v(N).
• AXIOM3 For any two games v and w, φ[v + w] = φ[v] + φ[w].
Lemma 1.5. If N is a finite carrier of v then for i /∈ N φ i [v] = 0.
Proof. Take i /∈ N. Both N and N ∪ {i} are carriers of v and
v(N) = v(N ∪ {i}). Hence φ i [v] = 0 by Axiom 2 ( ∑ ∑
N φ i[v] =
N∪{i} φ i[v])
Definition 1.6. For given a set R ⊂ N define a function v R :
v R = { 1 if S ⊇ R
0 if S R.
• cv R is game where c > 0, and then R is a carrier.
• r, s, n... = ♯(R), ♯(S), ♯(N)...
Lemma 1.7. For c > 0, 0 < r < ∞, we have
φ i [cv R ] = {
c/r if i ∈ R
0 if i /∈ R.
Proof. Take i, j ∈ R and choose π ∈ Π(U) s.t. π(R) = R and
π(i) = j. Then πv R = v R by Axiom 1 φ j [cv R ] = φ i [cv R ] (φ j [cv R ] =
φ πi [πcv R ]) by Axiom 2. c = cv R (R) = ∑ j∈R φ j[cv R ] = rφ i [cv R ]
Lemma 1.8. Any game with finite carrier is a linear combination
of symmetric games v R :
v =
∑
c R (v)v R ,
R⊆N,R≠∅
where N is any finite carrier of v the coefficients are independent of
N, and are given by
c R (v) = ∑ T ⊆R(−1) r−t v(T ) (0 < r < ∞).
Proof. If S ⊆ N then v(S) = ∑ R⊆N c R(v)v R = ∑ R⊆S c R(v) =
∑
R⊆S
∑T ⊆R (−1)r−t v(T ) = ∑ ( )
T ⊆S [∑ s s − t
r=t
]v(T ) = v(S).
r − t
In general we have v(S) = v(S ∩ N) = ∑ R⊆N c R(v)v R (S ⋂ ∑
N) =
R⊆N c R(v)v R (S).