Download THE NUCLEOLUS AND THE SHAPLEY VALUE

THE NUCLEOLUS AND THE SHAPLEY VALUE
Given a game v and a payoff vector x, the excess of coalition S against x
is defined to be
e(S , x) = v(S ) − x(S ),
∑
where x(S ) = i∈S xi . Let e(x) be the 2n -dimensional vector of excesses,
the components of which are arranged in the non-increasing order, that is,
el (x) ≥ el+1 (x), l = 1, . . . , 2n−1 .
We say e(x) is lexicographically less than e(y) if el (x) < el (y) for the
smallest l for which el (x) , el (y).
Definition 0.0.1 (Schmeidler, 1969). The set of imputations x for which the
vector e(x) is lexicographically minimal is said to be the nucleolus.
Theorem 0.0.1. N , ∅.
Proof. We show first that for each k ≥ 1, the function ek is continuous. To
see this, first note for each k = 1, 2, ..., 2n that
{
}
ek (x) = max min{e(x, S ) | S ∈ F} | F ⊆ 2N , |F| = k
Then, ek (·) is continuous on A, the set of imputations, since it is defined by
min and max of a finite number of continuous functions.
Now, define
A1 = {x ∈ A| e1 (x) ≤ e1 ( x̄), ∀ x̄ ∈ A}
Ak = {x ∈ Ak−1 | ek (x) ≤ ek ( x̄), ∀ x̄ ∈ Ak−1 }, k = 2, ..., 2n .
It is enough to show that A2n is nonempty. First, since e1 (·) is continuous
on A and A is compact, A1 is compact and nonempty. Similarly, since e2 (·)
is continuous on A1 and A1 is compact, A2 is also compact and nonempty.
Continuing this in finitely many times, we will arrive at the conclusion that
A2n is nonempty.
Theorem 0.0.2. The nucleolus N of a game v is a singleton.
Theorem 0.0.3. The nucleolus is contained in the nonempty core.
Problem 0.0.1. Show that the nucleolus of a symmetric game (N, v) is the
equal distribution ν = (v(N)/n, . . . , v(N)/n).
Department of Economics, Keio University, 2-15-45 Mita, Tokyo 108-8345.
e-mail: [email protected] Do not quote without a permission of the author.
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0.0.1. Special Cases.
Proposition 0.0.1. For each i ∈ N and S ⊆ N, let
• di = v(N)
∑ − v(N \ {i})
• d(S ) = i∈S di
and assume that
v(N) − d(N)
(1) di +
≥ v({i}), ∀i ∈ N
n
(
v(N) − d(N) ) v(N) − d(N)
(2) v(S ) − d(S ) + |S |
≤
, ∀S ( N
n
n
Then, the nucleolus is given by the payoff vector x∗ where
xi∗ = di +
v(N) − d(N)
, ∀i ∈ N.
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The amount di is player i’s marginal contribution to N. On the other hand,
the amount (v(N) − d(N))/n is independent of i and may be called a dividend. Condition 1 says that the payoff vector x∗ satisfies the individual
rationality. Condition 2 implies that the excess of any coalition against this
payoff vector does not exceed the dividend.
Problem 0.0.2. Find the nucleolus of the game satisfying
(|S | − 1)
v(N)
≥ v(S ) ≥ 0, ∀S ( N.
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0.0.2. The Bankruptcy Problem. The bankruptcy game due to Aumann and
Maschler [2, 1985] is the TU coalitional game (N, v) defined as follows:


∑ 

v(S ) = max 0, E −
di  ∀S ⊆ N,
i∈N\S
where
N = {1, . . . , n}: the set of creditors,
E ≥ 0: the estate
∑
di : the debt to creditor i ∈ N satisfying i∈N di ≥ E, di ≥ 0, ∀i ∈ N.
E = 100
E = 200
E = 300
d1 = 100 d2 = 200 d3 = 300
100/3
100/3
100/3
50
75
75
50
100
150
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0.1. The Shapley Value. Let (N, v) be a coalitional game. The well known
one-point solution concept, the Shapley value, is given by the formula:
1 ∑
(1)
(φv)i =
(|S | − 1)!(n − |S |)![v(S ) − v(S \ {i})]
n! S ⊆N, S 3i
1 ∑
(2)
=
[v(S i ∪ {i}) − v(S i )],
n! R
where R runs through all permutations of N, and S i is the set of players
preceeding to i in the permutation R.
There are many ways to obtain the Shapley value. We shall show a standard axiomatic approach due to Aumann [1].
Definition 0.1.1. Let i, j ∈ N. Then, i and j are substitutes each other if
for all S ⊆ N with i < S and j < S , we have
v(S ∪ {i}) = v(S ∪ { j}).
Definition 0.1.2. Player i is a null player if for all S ⊆ N with i < S ,
v(S ∪ {i}) = v(S ).
Definition 0.1.3. Let G N ⊆ <2 −1 be the set of all games v. Then the function φ : G N → <n satisfying the conditions below is called the Shapley
value (or the value ) on G N .
symmetry: For all substitutes i and j in v, (φv)i = (φv) j .
null player:∑For all null players in v, (φv)i = 0.
efficiency: i∈N (φv)i = v(N).
additivity: (φ(v + w))i = (φv)i + (φw)i .
n
Theorem 0.1.1 (Shapley 1953). For all v ∈ G N , the value exists uniquely.
Proof. (uniqueness). We first show the uniqueness of φ. Let us define, for
all nonempty T ⊆ N, the following game vT , called the T-unanimity game.



1 if T ⊆ S ,
vT (S ) = 
0 if T * S .

Then, for any real number α, it is easy to see that in game αvT ,
(a): every member in N \ T is a null player in αvT ,
(b): every member in T is a substitute with every member in T .
Hence, by (a) and the null player condition, φ(αvT )i = 0 for all i < T . Also,
by (b) and the symmetry condition,∑φ(αvT )i = φ(αvT ) j for all i, j ∈ T .
Further, by the efficiency condition, i∈N φ(αvT )i = α. Hence, for all i ∈ T ,
∑
α=
φ(αvT ) j = |T |φ(αvT )i ,
j∈T
or

α


 |T | if i ∈ T,
φ(αvT )i = 

0 if i < T.
Now, we note the following items consecutively.
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i: G N ⊆ <2 −1
ii: There are 2n − 1 T -unanimity games vT .
iii: If {vT : T ⊆ N, T , ∅} is linearly independent, then {vT } forms a
basis of G N .
iv: Any game v ∈ G N can be uniquely represented by a linear combination of {vT }.
∑
v: By the additivity, for any linear combination ki=1 αi vTi , we have
k
k
∑
∑
φ(αi vTi ).
φ( αi vTi ) =
n
i=1
i=1
Thus, for the uniqueness of φ, it will be enough to prove that {vT } is
linearly independent.
Suppose to the contrary that it is not linearly independent. Then, for some
nonempty T, T i ⊆ N, (i = 1, . . . , j), we have
∑
vT =
βi vTi .
In this expression, we may take T smallest so that |T | ≤ |T i | for all i. Since
all these j + 1 coalitions are different each other, we have
j
j
∑
∑
1 = vT (T ) =
βi vTi (T ) =
βi · 0 = 0.
i=1
i=1
This is a contradiction, which completes the proof of the uniqueness of φ.
(existence). Existence will be proved simply by checking that the formula (1) in fact satisfies the four conditions.
References
[1] R.J.Aumann, Lectures on Game Theory, Westview Press, 1989.
[2] R.J.Aumann and M.Maschler, ”Game Theoretic Analysis of a Bankruptcy Problem
From the Talmud,” Journal of Economic Theory 36 (1985), 195–213.
[3] D.Schmeidler, ”The Nucleolus of a Characteristic Function Game,” SIAM Journal of
Applied Mathematics 17 (1969), 1163-1170.
[4] L.S.Shapley, ”A Value for n-Person Games,” in Contribution to the Theory of Games
II Annals of Mathematical Studies no. 28, ed. by H.W.Kuhn and A.W.Tucker, Princeton University Press, 307–317, 1953.
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