TU Game Theory
Advances in Game Theory
1.1.3
Basic Properties
A game v is said to be
• super additive if
v(S ) + v(T ) ≤ v(S ∪ T ), ∀S , T ⊆ N with S ∩ T = ∅,
• inessential if it is additive so that
∑
v(S ) =
v({i}), ∀S ⊆ N
i∈S
1
A super additive game v is inessential iff v(N) =
A game is essential if it is not inessential.
∑
i∈N
v({i}).
• monotonic if v(S ) ≥ v(T ), ∀T ⊆ S ⊆ N,
• constant-sum or zero-sum if
v(S ) + v(−S ) = v(N), ∀S ⊆ N,
where −S denotes the complement of S relative to N,
• symmetric if v(S ) = v(T ), ∀S , T ⊆ N with |S | = |T |.
• convex if
v(S ) + v(T ) ≤ v(S ∪ T ) + v(S ∩ T ), ∀S , T ⊆ N,
or equivalently, if for all S , T , and i such that S ⊆ T ⊆ N \ {i},
v(S ∪ {i}) − v(S ) ≤ v(T ∪ {i}) − v(T ).
2
Theorem 1.1.1. A game v is convex if and only if for all S , T , and i such
that S ⊆ T ⊆ N \ {i},
v(S ∪ {i}) − v(S ) ≤ v(T ∪ {i}) − v(T ).
Proof. The direction =⇒ is straightforward, because by considering S ⊆ T
⊆ N \ {i}, we have
v(S ∪ {i}) + v(T ) ≤ v(S ∪ {i} ∪ T ) + v((S ∪ {i}) ∩ T )
= v(T ∪ {i}) + v(S ).
To see the converse, let S , R ⊆ N, and let
S \ R = {i, j, k, ..., l − 1, l}
3
Then,
v(S ) − v(S ∩ R) = v((S ∩ R) ∪ {i}) − v(S ∩ R)
+ v((S ∩ R) ∪ {i} ∪ { j}) − v((S ∩ R) ∪ {i})
+ v((S ∩ R) ∪ {i} ∪ { j} ∪ {k}) − v((S ∩ R) ∪ {i} ∪ { j})
+ ···
+ v(S ) − v((S ∩ R) ∪ {i} ∪ { j} ∪ {k} · · · ∪ {l − 1})
≤ v(R ∪ {i}) − v(R)
+ v(R ∪ {i} ∪ { j}) − v(R ∪ {i})
+ v(R ∪ {i} ∪ { j} ∪ {k}) − v(R ∪ {i} ∪ { j})
+ ···
+ v(R ∪ S ) − v(R ∪ {i} ∪ { j} ∪ {k} · · · ∪ {l − 1})
= v(R ∪ S ) − v(R).
4
Two games v and v0 with an identical player set N are said to be strategically
equivalent each other if for some a > 0 and (b1, . . . , bn),
∑
0
v (S ) = av(S ) +
bi, ∀S ⊆ N.
i∈S
(Show that any constant-sum game is strategically equivalent to the game
with the constant being replaced by zero. )
Given any game v, we may obtain strategically equivalent normalizations of
v as follows:
∑
0
0-normalization v (S ) = v(S ) − i∈S v({i}), ∀S ⊆ N
∑
v(S
)
−
i∈S v({i})
00
∑
0,1-normalization v (S ) =
v(N) − i∈N v({i})
5
1.1.4
Imputations
Given a coalitional game (N, v), the vector x = (x1, . . . , xn) ∈ <n will be
called an imputation if it satisfies
∑
total rationality
i∈N xi = v(N)
individual rationality xi ≥ v({i}), ∀i ∈ N
If x is an imputation of v, then it should be obvious that corresponding imputations x0 and x00 for 0-normalization v0 and 0,1-normalization v00, respectively, are given by
• xi0 = xi − v({i}), ∀i ∈ N
xi − v({i})
00
∑
• xi =
, ∀i ∈ N
v(N) − i∈N v({i})
6
1.2
Stable Sets
Let (N, v) be a coalitional game, and let A be the set of imputations. For
any x, y ∈ A, we define consecutively as follows:
∑
• x dom y ⇐⇒ ∃S ⊆ N s.t. i∈S xi ≤ v(S ), and xi > yi ∀i ∈ S
• Dom x = {y|y ∈ A, x dom y}
• For K ⊆ A,
Dom K = ∪ x∈K Dom x
Definition 1.2.1. K ⊆ A is called a stable set if
K = A \ Dom K.
Remark 1.2.1. K is a stable set ⇐⇒
x, y ∈ K ⇒ ¬[x dom y]
external stability z < K ⇒ ∃x ∈ K [x dom z]
internal stability
7
Definition 1.2.2. The set C = {x ∈ A| @z ∈ A[z dom x]} is called the core of
a game.
Problem 1.2.1. Show that for a nonempty core C,
1. C ⊆ K for all nonempty stable sets K,
2. C is a unique stable set if C itself is a stable set.
1.2.1
Stable Sets of Zero-Sum Three Person Games
v({1, 2, 3}) = v({1, 2}) = v({1, 3}) = v({2, 3}) = 1; v({i}) = 0, i ∈ {1, 2, 3}
objective solution K = {(1/2, 1/2, 0), (1/2, 0, 1/2), (0, 1/2, 1/2)}
discriminatory solution
Ki(c), for 0 ≤ c < 1/2 and i = 1, 2, 3, where
Ki(c) = {x ∈ A| xi = c, x j + xk = 1 − c}, and {i, j, k} = {1, 2, 3}
8
1.2.2
One Seller and Two Buyers
• player 1: the seller of a house with value a
• player 2: buyer with evaluation b
• player 3: buyer with evaluation c
• Assumption: a < b ≤ c
v({1}) = a; v({2}) = v({3}) = 0; v({1, 2}) = b; v({1, 3}) = c; v({2, 3}) = 0;
v({1, 2, 3}) = c.
K = {(p, 0, c − p)| b ≤ p ≤ c}
∪ {(p, d(p), c − p − d(p))| a ≤ p ≤ b, d(p) = 0 if p = b},
where the graph (p, d(p), c − p − d(p)) goes down within 30 degrees relative
to the vertical direction.
9
A
xB + xC
= v({B, C}) = 0
p=b
w0 = (p, 0, c − p)
where b ≤ p ≤ c
w0
z”
Y = (yA , yB, yC )
D
e
= (p, d(p),
c − p − d(p))
w
= (b − e, d(p),
c − b + [e − d(p)])
z
z’
Y
p=b−e
B
E
where
F
C
a≤ p=b−e≤b
xA + xC = v({A, C}) = c
xA + xB = v({A, B}) = b
The payoff d(p) is a compensation paid from player 3 with which player 2
ceases to be a competitor of player 3. The segment [(c, 0, 0), (b, 0, c − b)] is
the core and contained in every one of infinite number of stable sets.
10
1.3
The Core
In the previous section, the core was defined to be the set of all undominated imputations. When the game is super additive, the core can be obtained
as follows:
Theorem 1.3.1. If v is super additive, the core C is given by
∑
{
}
xi ≥ v(S ), ∀S ⊆ N .
C = x ∈ A
i∈S
Problem 1.3.1. Prove this theorem.
We may, however, define the core directly to be the set C given in the above
theorem. Hereafter, the core of a game v will be denoted by C(v).
11
Example 1.3.1 (The Core of 3-Person Games). Let (N, v) be a 0 - normalized 3-person game. Then:
C(v) , ∅ ⇐⇒
1. v({1, 2}) + v({1, 3}) + v({2, 3}) ≤ 2v({1, 2, 3})
2. v(S ) ≤ v(N) ∀S ⊆ N, |S | = 2
Proof. The necessity follows simply from adding both sides of the core
inequality. To prove the sufficiency, letting v({1, 2}) = a3, v({1, 3}) = a2,
v({2, 3}) = a1, we may take i ≥ 0 (i = 1, 2, 3) such that
a1 + a2 + a3 + 1 + 2 + 3 = 2v(N).
Define the payoff vector x by
xi = v(N) − (ai + i), i = 1, 2, 3.
Then,
x1 + x2 + x3 = 3v(N) − 2v(N) = v(N),
12
implying x to satisfy the total rationality. By condition (2), we may take
0 ≤ i ≤ v(N) − ai for i = 1, 2, 3 to obtain xi ≥ 0 for i = 1, 2, 3. Hence, x
is an imputation. That x belongs to the core follows from xi ≤ v(N) − ai for
i = 1, 2, 3.
Theorem 1.3.2 (the core of symmetric games). Let v be a symmetric game.
Then,
v(S ) v(N)
C(v) , ∅ ⇐⇒
≤
, ∀S ⊆ N.
|S |
n
Proof. The sufficiency part is obvious. To show the necessity, let s be any
integer with 1 ≤ s ≤ n and let S ⊆ N be any coalition with |S | = s. Then, for
some payoff vector x,
∑
v(S ) ≤ x(S ) :=
xi, ∀S ⊆ N.
i∈S
13
Adding both sides for all S ⊆ N with |S | = s, we have
(
)
( )
∑
n−1
n
x(S ) =
x(N).
v(S ) ≤
s
s−1
|S |=s
Hence,
n!
(n − 1)!
v(S ) ≤
v(N),
(n − s)!s!
(n − s)!(s − 1)!
which leads to the conclusion.
Theorem 1.3.3 (the core of convex games, Shapley (1972)). The core of a
convex game is nonempty.
14
Proof. (Shapley [40, 1972]). Let x be a payoff vector defined by
xi = v({1, .., i − 1, i}) − v({1, ..., i − 1}), ∀i ∈ N.
It is easy to see that x is an imputation.
Let S ( N, and let
N \ S = { j(1), ..., j(n − s)},
where j(1) ≤ · · · ≤ j(n − s), and s = |S |. For j(1), put
T = {1, ..., j(1) − 1, j(1)}.
Then,
S ∪ T = S ∪ { j(1)}, S ∩ T = {1, ..., j(1) − 1}.
Hence,
v(S ) + v({1, ..., j(1) − 1, j(1)} ≤ v(S ∪ { j(1)}) + v({1, ..., j(1) − 1}),
15
so that
x j(1) = v({1, ..., j(1) − 1, j(1)}) − v({1, ..., j(1) − 1})
≤ v(S ∪ { j(1)}) − v(S ).
Next, putting S 0 = S ∪ { j(1)} and considering T 0 = T ∪ { j(2)}, we obtain
x j(2) ≤ v(S ∪ { j(1)} ∪ { j(2)}) − v(S ∪ { j(1)}).
Continuing this process, we finally obtain
x j(n−s) ≤ v(N) − v(S ∪ { j(1)} ∪ · · · ∪ { j(n − s − 1)}).
Adding both sides of these n − s inequalities, we obtain
∑
x j(k) ≤ v(N) − v(S ).
Hence, v(S ) ≤
k=1,...,n−s
∑
i∈S
xi.
16
1.3.1
Balanced Games
Definition 1.3.1. A family B of nonempty, proper subsets of N is balanced iff
there exist positive weights (balancing weights) δS for S ∈ B such that
∑
δS = 1 f or all i ∈ N
S ∈B, S 3i
Example 1.3.2. N = {1, 2, 3}, B= {{1, 2}, {2, 3}, {1, 3}}, δS =
1
2
Example 1.3.3. Take a game (N, v) and any nonempty S ( N. Then, the
family
B = {R ( N | |R| = |S |}
is balanced with the balancing weights
1
δR = ( n−1 ), ∀R ∈ B.
|R|−1
17
Definition 1.3.2. A game (N, v) is balanced iff for every balanced family B,
∑
δS v(S ) ≤ v(N).
S ∈B
Theorem 1.3.4. (Bondareva [8, 1963], Shapley [37, 1967]) The core of a TU
coalitional game is nonempty if and only if it is balanced.
1.3.2
Proof by LP Duality Theorem
∑
∑
• Primal. min nj=1 c j x j s.t. nj=1 ai j x j ≥ bi, i = 1, . . . , m.
matrix form: min cx s.t. Ax ≥ b
• Dual. max
∑m
∑m
s.t. i=1 yiai j = c j, j = 1, . . . , n,
yi ≥ 0, i = 1, . . . , n.
matrix form: max by s.t. yA = c, y ≥ 0.
i=1 bi yi
18
Theorem 1.3.5 (Duality Theorem). If either the Primal or the Dual has a
finite optimal solution, then the other also has a finite optimal solution, and
the optimal values are the same.
In order to make use of this theorem, consider the following problems.
• Primal. min
• Dual. max
s.t.
∑
∑
∑n
j=1
S (N
S (N
S3j
x j s.t.
∑
j∈S
x j ≥ v(S ) ∀S ( N.
δS v(S )
δS = 1, j = 1, . . . , n; δS ≥ 0 ∀S ( N.
19
To put in the matrix form, let us define:
• A = (aSj ) is the (2n − 2) × n matrix with rows in a given order.
• row vector aS is the characteristic vector of S ⊆ N,
i.e., aS = (aS1 , . . . , aSn ), aSj = 1 if j ∈ S and aSj = 0 if j < S .
• 1 = (1, . . . , 1) ∈ <n+.
N
• v ∈ <2 \({N}∪{∅}) with elements v(S ) arranged in the given order.
N
• δ ∈ <2 \({N}∪{∅}) with elements δS arranged in the given order.
Then:
• Primal: min 1x s.t. Ax ≥ v
• Dual: max δv s.t. δA = 1, δ ≥ 0
20
The game (N, v) is thus balanced if and only if the problem Dual has an
optimal solution {δ∗S }S (N satisfying
∑
δ∗S v(S ) ≤ v(N).
S (N
Proof. (Proof of Bondareva=Shapley Theorem). If v is balanced, then
there exists an optimal solution {δ∗S }. Hence by the duality theorem the Primal
also has an optimal solution x∗ = (x1∗ , . . . , xn∗ ) satisfying
n
∑
j=1
x∗j
=
∑
δ∗S v(S ) ≤ v(N),
S (N
which shows that the core is nonempty.
21
Conversely, if the core of v is nonempty, then the Primal has an optimal
∑n ∗
solution such that j=1 x j ≤ v(N). Thus, by the duality theorem, the Dual has
an optimal solution {δ∗S }, which implies that the game v is balanced, because
∑
δS v(S ) ≤
S (N
∑
δ∗S v(S ) =
S (N
n
∑
x∗j ≤ v(N)
j=1
for all solutions {δS } of Dual.
Problem 1.3.2. Prove the necessary and sufficient condition for a symmetric
game to have a nonempty core by showing the balancedness of the game.
1.4
TU Market Games
Assumption 1.4.1. The preference relation for each i ∈ N over X = <m
+ ×
< is represented by a continuous function ui : <m
+ → < such that for each
i ∈ N,
(xi, ξ) i (xi0, ξ0) iff ui(xi) + ξ ≥ ui(xi0) + ξ0
22
Remark 1.4.1. The value ui(xi) is called a transferable utility (TU) for xi,
and ξ is called money.
Proposition 1.4.1. Let U : <m
+ × < → < be the utility function given by
U(x, m) = u(x) + m, where u is a transferable utility. Then,
U is quasiconcave ⇔ u is concave.
Problem 1.4.1. Prove this proposition.
Definition 1.4.1. A TU market game is a coalitional game (N, v) defined as
follows: For each S ⊆ N,
∏
∑
∑ }
{∑
m
<+ ,
xi =
wi ,
v(S ) = max
ui(xi) x = (x1, . . . , xn) ∈
i∈N
i∈S
i∈S
i∈S
where ui(·) is a concave transferable utility function of player i.
The vector wi ∈ <m
+ is the initial endowments of player i. The vector x
such that xi ∈ <m
+ for all i ∈ N is called an allocation. The allocation x that
23
satisfies the resource constraint
∑
xi =
i∈S
∑
wi
i∈S
in coalition S will be called an S − allocation.
Theorem 1.4.1. (Shapley and Shubik [41]) Every TU market game is balanced.
Proof. Let B be any balanced collection, and for each S ⊆ N, let f S = ( fiS )i∈S
be the S -allocation that gives v(S ). Define the allocation f by
∑
fi =
δS fiS ,
S ∈B, S 3i
which is a convex combination of fiS , S ∈ B, S 3 i.
24
The allocation f is an N−allocation, since
∑
∑ ∑
∑ ∑
fi =
δS fiS =
δS
fiS
i∈N
i∈N S ∈B, S 3i
S ∈B
i∈S


∑  ∑
∑
∑ ∑




δS
wi =
δS  wi =
wi.
=
S ∈B
i∈S
i∈N
S ∈B, S 3i
i∈N
By the concavity of ui, we have
∑
∑ ∑
∑ ∑
S
δS v(S ) =
δS
ui ( f i ) =
δS ui( fiS )
S ∈B
S ∈B

 S ∈B, S 3i
∑  ∑
 ∑
S
ui( fi) ≤ v(N),
≤
ui 
δS fi  =
i∈S
i∈N
S ∈B, S 3i
i∈N
i∈N
which shows that the game is balanced.
Definition 1.4.2. A game (N, v) is totally balanced iff every subgame of (N, v)
is balanced.
25
Crollary 1.4.1. Every market game is totally balanced.
This is so because every subgame of a market game is a market game; and
then, it is balanced by the above theorem.
1.4.1
The Direct Market
Definition 1.4.3. A direct market (DM, for short) is a market (N, e, u∗) where
e = (ei)i∈N , ei ∈ <+N is the characteristic vector of i ∈ N, and u∗ : <+N → <
is the common utility function of the players which is homogeneous of degree
1, concave and continuous.
Remark 1.4.2. The common utility function u∗ is superadditive, i.e.,
u∗(x + y) ≥ u∗(x) + u∗(y), ∀x, y ∈ <+N
Problem 1.4.2. Prove this.
26
Remark 1.4.3. If (N, v) is the market game generated by the DM, then v(S ) =
u∗(eS ). To see this, the superadditivity of u∗ implies,
∑
(∑ )
∗
∗
u (xi) ≤ u
xi = u∗(eS ), ∀S ⊆ N
v(S ) = max
i∈S
i∈S
∑
∑
where max runs over all the S-allocations x (i.e., i∈S xi = i∈S ei := eS ).
On the other hand, since u∗(eS ) = |S |u∗(eS /|S |) and xi = eS /|S | for all
i ∈ S is an S -allocation, we have v(S ) ≥ u∗(eS ). Hence, v(S ) = u∗(eS ).
We now associate with every game (N, v) a direct market (N, e, u∗) such
that
∑
∑
∗
u (x) = max
δS v(S ) s.t.
δS eS = x, and δS ≥ 0 ∀S ⊆ N.
Note that
∑
S ⊆N
S ⊆N
δS eS = eR =⇒
∑
S ⊆N
S ⊆R δS eS
= eR
Problem 1.4.3. Prove that u∗ is concave. Homogeneity and continuity are
obvious.
27
Definition 1.4.4. Given a game (N, v) and the associated DM (N, e, u∗), the
game (N, v̄) will be called the totally balanced cover of (N, v) iff for any
R ⊆ N,
∑
v̄(R) = max
δS v(S )
S ⊆R
∑
s.t.
δS eS = eR and δS ≥ 0 ∀S ⊆ R.
S ⊆R
Remark 1.4.4. v̄(R) ≥ v(R) ∀R ⊆ N.
Remark 1.4.5. The totally balanced cover (N, v̄) is a market game. This is so
by Remark 1.4.3 and the fact that the market game (N, v M ) defined on the DM
which is generated from a given game (N, v) satisfies that v M (S ) = u∗(eS ) =
v̄(S ) for all S ⊆ N.
28
Lemma 1.4.1. A game (N, v) is balanced if and only if v(N) = v̄(N).
Proof. By definition, the game (N, v) is balanced if and only if v(N) ≥ v̄(N),
which is equivalent to v(N) = v̄(N).
Crollary 1.4.2. A game (N, v) is totally balanced if and only if v = v̄.
Proof. For any nonempty R ⊆ N, the totally balanced cover of the subgame
(R, v) is (R, v̄). Hence by lemma 1.4.1 the result follows.
Theorem 1.4.2. (Shapley and Shubik [41, 1969]). A game is a market game
if and only if it is totally balanced.
Proof. Every market game is totally balanced (Corollary 1.4.1). A totally
balanced game coincides with its totally balanced cover (Corollary 1.4.2).
The totally balanced cover is a market game.
Crollary 1.4.3. A convex game is totally balanced.
29
Proof. A convex game has a nonempty core; and hence, it is balanced. Every
subgame of a convex game is convex, which implies that a convex game is
totally balanced.
Crollary 1.4.4. A convex game is a market game.
Remark 1.4.6. Let BAL, T BAL and CON be the collections of balanced
games, totally balanced games and convex games, respectively. Then,
BAL ) T BAL ) CON
The inclusions are proper. The games v1 and v2 given below satisfy that
v1 ∈ BAL \ T BAL and v2 ∈ T BAL \ CON.
• v1 is a symmetric game satisfying
v1(S ) v1(N)
1.
≤
for all S ⊆ N; so that v1 is balanced.
|S |
|N|
v1(S ) v1(T )
2.
>
for some S ( T ( N; so that v1 is not totally balanced.
|S |
|T |
30
• v2 is a symmetric game defined by




0
v2(S ) = 


a|S |
if |S | = 1
if |S | > 1
where a > 0. Then:
1. v2 is totally balanced, since for all |R| = 1, |S | > 1 and S ⊆ T ⊆ N,
v2(T )
v2(R) v2(S )
<
=a=
|R|
|S |
|T |
2. v2 is not convex, since v({1, 2}) − v({2}) = 2a > v({1, 2, 3}) − v({2, 3}) =
a.
0=
31
1.4.2
A Public Good Game
Definition 1.4.5. Let v be defined by
∑
v(S ) = max( ui(y) − c(y)) f or all S ⊆ N,
y≥0
i∈S
where y is a quantity of a public good, ui(·) is an increasing transferable
utility function, and c : <+ → R is an increasing linear cost function. Then,
(N, v) is said to be a T U public good game.
Theorem 1.4.3. A T U public good game (N, v) is convex.
Problem 1.4.4. Prove this theorem.
32
1.4.3
Competitive Outcomes and the Core of Market Games
Let xi ∈ <m
+ be an allocation to player i ∈ N, and let ui (xi ) + ξi be the
transferable utility of i for xi, with ξi being the net amount of money of i
after the trade.
∏
Lemma 1.4.2. Allocation (x∗, ξ) = ((x1∗ , ξ1), . . . , (xn∗ , ξn)) ∈ i∈N <+m+1 and
price vector (p∗, 1) = (p∗1, . . . , p∗m, 1) ∈ <m+1
constitute a TU competitive
+
equilibrium if and only if for each i ∈ N,
1. ui(xi∗) − p∗ · (xi∗ − wi) ≥ ui(xi) − p∗ · (xi − wi) ∀xi ∈ <m
+
2. ξi = p∗ · wi − p∗ · xi∗
Proof. Note first that the price for money is normalized to 1. Since (x∗, ξ) =
∏
∗
((x1∗ , ξ1), . . . , (xn∗ , ξn)) ∈ i∈N <m+1
is
a
competitive
equilibrium
under
p
, we
+
33
have for all i ∈ N that
p∗ · xi∗ + ξi ≤ p∗ · wi, and
(1)
p∗ · yi + ηi ≤ p∗ · wi implies ui(yi) + ηi ≤ ui(xi∗) + ξi
(2)
Then, since
∗
p ·
∑
i∈N
∗
wi = p ·
∑
xi∗
∗
=p ·
i∈N
∑
i∈N
xi∗
+
∑
∗
ξi ≤ p ·
i∈N
∑
wi
i∈N
we have 2, that is
p∗ · xi∗ + ξi = p∗ · wi ∀i ∈ N.
To see 1, letting ηi = p∗ · wi − p∗ · yi in (2), we have
ui(yi) − ui(xi∗) ≤ ξi − ηi
= (p∗ · wi − p∗ · xi∗) − (p∗ · wi − p∗ · yi),
or,
ui(xi∗) − p∗ · (xi∗ − wi) ≥ ui(yi) − p∗ · (yi − wi).
34
Conversely, let p∗ and (xi∗, ξi) satisfy 1 and 2 for all i ∈ N. Then, p∗ ·yi +ηi ≤
p∗ · wi implies that
ui(yi) + ηi ≤ ui(yi) + p∗ · wi − p∗ · yi
≤ ui(xi∗) − p∗ · (xi∗ − wi)
= ui(xi∗) + ξi.
Hence, the price vector (p∗, 1) and the allocation ((x1∗ , ξ1), . . . , (xn∗ , ξn)) constitute a competitive equilibrium.
Theorem 1.4.4. Let b = (b1, . . . , bn) be the competitive equilibrium payoff
vector defined by
bi = ui(xi∗) − p∗ · (xi∗ − wi) ∀i ∈ N,
where p∗ and (x1∗ , . . . , xn∗ ) constitute a TU comtetitive equilibrium. Then, b
belongs to the core of the market game.
35
∏
Proof. For any S ⊆ N, let S −allocation y ∈ i∈S <m
+ satisfy that
∑
ui(yi) = v(S )
i∈S
{∑
}
= max
ui(xi) x is an S allocation .
i∈S
Then, by the definition of bi we have that
bi ≥ ui(yi) − p∗ · (yi − wi).
Hence,
∑
i∈S
bi ≥
∑
∗
ui(yi) − p ·
i∈S
∑
=
ui(yi)
i∈S
= v(S ).
36
(∑
ı∈S
yi −
∑
i∈S
wi
)
Moreover,
∑
bi =
i∈N
∑
ui(xi∗)
≤ max
i∈N
{∑
ui(xi)
}
= v(N).
i∈N
Hence, b is in the core.
Theorem 1.4.5. Every payoff vector b = (b1, . . . , bn) in the core of a game
(N, v) is competitive in the direct market (N, e, u∗) of that game.
Remark 1.4.7. If the game v is not balanced, the theorem is vacuously true.
If the game is balanced but not totally balanced, the theorem is stil true
because it is shown in Shapley and Subik [41, 1969] that any balanced game
is d-equivalent to its cover v̄; and hence, the core coincides with each other.
Proof. By the above remark, we shall assume that the game v is a market
game. Let b be in the core of v. We show that b itself can be used as a competitive price vector in the direct market of v. Recall that u∗(eR) = v(R) for
37
all R ⊆ N, which is because a market game is totally balanced and coincides
with its cover v̄. Then,
u∗(eN ) = v(N) = b · eN
Take any x ∈ <+N and let {δS : S ⊆ N} be any nonnegative weights with
∑
S ⊆N δS eS = x. Then, by the fact that b is in the core of v,
∑
∑
∑
u∗(x) = max
δS v(S ) ≤ max
δS (b · eS ) = max b ·
δS eS = b · x.
S ⊆N
S ⊆N
S ⊆N
Now, define prices by πi = bi for all i ∈ N, and take zN satisfying zi = fieN
∑
and fi ≥ 0 for all i ∈ N with i∈N fi = 1. Then it is easy to see that the pair
(π, zN ) constitutes a competitive equilibrium. In fact, for each i ∈ N we have,
u∗(zi) − π · (zi − ei) = 0 + πi ≥ u∗(xi) − π · (xi − ei) ∀xi ∈ <+N .
38
Dominance-Equivalence. Two games are called d-equivalent (domination - equivalent) if
1. they have the same sets of imputations
2. they have the same domination relations on the imputation sets
It is known that d-equivalent games have
• the same NM-stable sets, or lack of them
• the same cores, or lack of them
• the property of being balanced is preserved under d-equivalence within
the class of games satisfying
∑
v(S ) +
v({i}) ≤ v(N) ∀S ⊆ N
i∈N\S
• every balanced game is d-equivalent to its cover.
39
The last fact may be seen as follows.
1. The imputation sets are equal since v̄(N) = v(N), and v̄({i}) = v({i}).
2. For imputations x and y, if x dom y in v̄ but not in v, we must have some
nonempty R ⊆ N such that
（a）xi > yi ∀i ∈ R, and x(R) ≤ v̄(R)
（b）x(S ) > v(S ) ∀S ⊆ R
3. Recalling the definition of v̄ that
∑
∑
v̄(R) =
δS v(S ) and
δS e S = e R ,
S ⊆R
S ⊆R
it follows from 2-(b) that
∑
∑ ∑
v̄(R) <
δS x(S ) =
δS xi = x(R).
S ⊆R
i∈R S 3i, S ⊆R
4. This contradicts 2-(a).
40
1.4.4
Glove Market Games
Recall that a market game is defined as follows.
Definition 1.4.6. A game (N, v) is a market game if there is a market
i
i
(N, <m
+ , A = (a )i∈N , W = (w )i∈N ) with




∑



S
i i , ∀S ⊆ N,
v(S ) = max 
w (x ) xS ∈ X 




i∈S
}
{
∑
∑
i
m
i
i
S
i
where X := xS = (x )i∈S x ∈ <+ ∀i ∈ S and i∈S x = i∈S a .
41
Definition 1.4.7. A glove market game (N, v) is given by
{
}
v(S ) = min |S ∩ N1|, |S ∩ N2| , ∀S ⊆ N,
where
• {N1, N2} is a partition of N with N1 , ∅, N2 , ∅;
• m = 2, ai = (1, 0) for all i ∈ N1, ai = (0, 1) for all i ∈ N2;
• wi(x) = min{x1, x2} for all x ∈ <2+ and for all i ∈ N.
Proposition 1.4.2. The glove market game is a market game; and hence,
totally balanced.
42
Proof. We have to show that a glove market game satisfies the definition of a
market game. That is, we have to show




∑



i
i S
v(S ) = max 
min{x1, x2} xS ∈ X 
, ∀S ⊆ N.




i∈S
Note first that since min{x1i , x2i } ≤ x1i , x2i , we have for any S ⊆ N,
∑
∑
∑
i
i
i
min{x1, x2} ≤
x1 ,
x2i .
i∈S
Hence,
∑
i∈S
i∈S
i∈S








∑
∑
∑
∑






i
i
i
i
i
i
min{x1, x2} ≤ min 
x1 ,
x2 
= min 
a1 ,
a2 
,








i∈S
i∈S
i∈S
i∈S
that is, min{·, ·} is constant; so that we have








∑
∑
∑
∑






i
i
i
i
v(S ) ≤ max min 
x1,
x2
= min 
a1 ,
a2 
.








i∈S
i∈S
43
i∈S
i∈S
On the other hand, since the allocation yS = (yi)i∈S given by


∑
∑



i
i
i
y =  a1,
a2 , for some i ∈ S ,
i∈S
i∈S
y j = (0, 0), for all j ∈ S \ {i}
is a feasible S −allocation, we have



∑


∑ i ∑ i 

i i
v(S ) ≥
w (y ) = min 
a
,
a
.
1
2




i∈S
Hence,
i∈S
i∈S




∑
∑
{
}



i
i
v(S ) = min 
a1 ,
a2 
= min |S ∩ N1|, |S ∩ N2| .




i∈S
i∈S
44
Remark 1.4.8. The function w(x) = w(x1, x2) = min{x1, x2} is concave in
<2+.
Proof. Let h ∈ [0, 1], and take any x, y ∈ <2+. Then,
hw(x) = min{hx1, hx2} ≤ hx1, hx2,
(1 − h)w(y) = min{(1 − h)y1, (1 − h)y2} ≤ (1 − h)y1, (1 − h)y2
Then,
hw(x) + (1 − h)w(y) ≤ hx1 + (1 − h)y1, hx2 + (1 − h)y2.
Hence,
hw(x) + (1 − h)w(y) ≤ min{hx1 + (1 − h)y1, hx2 + (1 − h)y2}
= w(hx + (1 − h)y).
45
Remark 1.4.9. The function w(x1, x2) = min{x1, x2} is homogeneous of degree 1, that is,
w(tx1, tx2) = tw(x1, x2) for all t > 0.
Remark 1.4.10. Since the function w(x1, x2) = min{x1, x2} is concave and
homogeneous of degree 1, it is superadditive; that is,
min{x1 + y1, x2 + y2} ≥ min{x1, x2} + min{y1, y2} ∀x, y ∈ <2+.
The proof of the above proposition will be made more concise by exploiting
this property.
46
1.4.5
Assignment Games
Let
• N = S ∪ B, where S , B , ∅ and S ∩ B = ∅.
• S is the set of sellers of indivisible goods, and B is the set of potential
buyers.
Each seller has exactly one item, and each buyer wants at most one
item.
ai ≥ 0 is the worth of the good to the seller i ∈ S ,
bi j ≥ 0 is buyer j’s worth for the good of seller i,
c({i, j}) = max{bi j − ai, 0} is the joint net profit of {i, j} for all i ∈ S
and j ∈ B.
47
Let T ⊆ N. Then an assignment for T is a collection of pairs of a seller
and a buyer in T ; that is, a collection T ⊆ 2T such that
|P ∩ S | = |P ∩ B| = 1 and P ∩ Q = ∅ for all P, Q ∈ T with P , Q.
Example 1.4.1. Let N = {1, 2, . . . , 9}; S = {1, 2, 3, 4, 5}, B = N \ S , and let
T = {2, 4, 5, 6, 9}. Then T = {{2, 6}, {4, 9}} is an assignment for T . Note that
the seller 5 is excluded from the trade in the assignment T .
Definition 1.4.8. (N, v) is an assignment game if




∑




T
is
an
assignment
for
T
, for all T ⊆ N.
v(T ) = max 
c(P)





P∈T
By convention, the ”empty sum” is zero, so that v(T ) = 0 for all T ⊆ S
and T ⊆ B. For any T ⊆ N, let τ(T ) be the set of all assignments for T .
48
Remark 1.4.11. The glove market game
v(S ) = min{ |S ∩ L|, |S ∩ R| } for all S ⊆ N
is an assignment game, where every member in L has one left glove, and
every member in R has one right glove.
To see this, let L be the set of sellers of left gloves, and let R be the set
of buyers of left gloves. Take any set T ⊆ N and an assignment T ; and
any seller i and buyer j paired in the assignment T . Then, letting bi j =
min{1, 1} = 1, ai = 0 and 0 ≤ p ≤ 1 be the price of left gloves, the joint net
profit for P = {i, j} is 1 − p + p = 1; namely,
c({i, j}) = max{bi j − ai, 0} = 1.
Hence,






∑

v(T ) = max 
c(P)
T
∈
τ(T
)
= min{ |T ∩ L|, |T ∩ R| } for all T ⊆ N.





P∈T
49
1.4.6
The LP Problem and its Dual of the Assignment Game
Denoting by ci j = c({i, j}) for all i ∈ S and j ∈ B, the LP problem to obtain
v(N) is given as follows.
∑∑
∑
∑
max
ci j xi j s.t.
xi j ≤ 1, i ∈ S ;
xi j ≤ 1, j ∈ B,
i∈S j∈B
j∈B
i∈S
xi j ≥ 0, i ∈ S , j ∈ B.
Remark 1.4.12. Clearly, all the pairs {i, j} in the objective function do not
constitute the assignment in N. But it is known that the optimum is attained
with every xi j taking 0 or 1, yielding an assignment in N *1. Hence, the
optimum attains v(N).
*1
See, e.g., Danzig,G.B.:Linear Programming and Extensions, Princeton University Press, Princeton, 1963. Any optimum is
generically attained at an extreme point, i.e., at a point that cannot be expressed by a proper convex combination in the convex
set.
50
The primal problem was :
∑∑
∑
∑
max
ci j xi j s.t.
xi j ≤ 1, i ∈ S ;
xi j ≤ 1, j ∈ B,
i∈S j∈B
j∈B
i∈S
xi j ≥ 0, i ∈ S , j ∈ B.
The dual of this LP problem is then given by:


∑ 
∑
v j s.t. ui + v j ≥ ci j, i ∈ S ,
min  ui +
i∈S
j∈B
j∈B
ui ≥ 0, i ∈ S , v j ≥ 0 j ∈ B.
That the dual is given as above can be seen from a simple case. Consider
the case: S = {1, 2, 3}, B = {4, 5, 6}. The matrix form of the LP problem is as
follows:
51
Letting c = (c14, c15, c16, c24, . . . , c36) and x = (x14, x15, x16, x24, . . . , x36),
max cx
s.t.
x≥0
and

 1

 1

 1

 0

(x14, x15, x16, x24, . . . , x36)  0

 0

 0

 0

0
0
0
0
1
1
1
0
0
0
52
0
0
0
0
0
0
1
1
1
1
0
0
1
0
0
1
0
0
0
1
0
0
1
0
0
1
0

0 

0 

1 

0 

0  ≤ (1, 1, 1, 1, 1, 1)

1 

0 

0 

1
Then, denoting by F and w = (u, v) the matrix and the dual variables, respectively, the dual problem can be obtained as follows.
min u1 + u2 + u3 + v4 + v5 + v6 s.t. w ≥ 0, and
FwT ≥ cT
The constraint is thus
ui + v j ≥ ci j, i = 1, 2, 3; and j = 4, 5, 6.
1.4.7
The Core of the Assignment Game
Theorem 1.4.6. The core of the assignment game (N, v) coincides with the
set of payoff vectors w = (u, v) achieving the optimum of the corresponding
dual LP problem.
53
Proof. Let (u, v) = ((ui)i∈S , (v j) j∈B) be a vector of optimal solution of the dual.
By the duality theorem, we have
∑
∑
ui +
v j = v(N).
i∈S
j∈B
Hence, (u, v) is an imputation of the assignment game. Moreover, by the
constraints
ui + v j ≥ ci j, ∀i ∈ S , j ∈ B,
we have that for any optimal assignment T in T ⊆ N,
∑
∑
∑
(uP∩S + vP∩B) ≥
cP∩S ,P∩B =
c(P) = v(T ).
P∈T
P∈T
P∈T
Since this is true for all T ⊆ N, the vector (u, v) is in the core of the assignment game (N, v).
Conversely, by definition, any core element satisfies the constraints in the
dual problem.
54
Theorem 1.4.7. The core of the assignment game coincides with the set of
competitive equilibrium allocations.
Proof. Outline of the proof *2 : Take any payoff vector (u, v) in the core, and
let
pi = ai + ui ∀i ∈ S
be the price at which seller i ∈ S sells i’s item. Then, the typical buyer j ∈ B
will face a choice among the |S | possible net gains:
bi j − pi, i ∈ S .
If these numbers are all negative, then j will stay out of the market, and obtain
a profit of zero; otherwise j will seek to maximize the net profit. Then, since
ci j = bi j − ai, this is equivalent to maximizing
ci j − ui i ∈ S .
*2
see, e.g. Suzuki,M and S.Muto: Theory of Cooperative Game Theory (Japanese), University of Tokyo Press. 1985
55
But, since the core coincides with the dual optimal solutions, any of these
values cannot exceed v j; and, at least one of them is equal to v j. Hence,
buyer j’s maximum profit is precisely v j by trading with the seller i for which
ci j − ui = v j.
The converse can be seen from the fact that the assignment game is a market game, so that the core contains competitive allocations.
56
1.5
Simple Games
Definition 1.5.1. A pair (N, W) is a simple game if N is the set of players
and W is a set of subsets of N satisfying
1. N ∈ W
2. ∅ < W
3. If S ⊆ T ⊆ N, then S ∈ W =⇒ T ∈ W
The collection W of coalitions is the set of winning coalitions.
Definition 1.5.2. A simple game (N, W) is
• proper if S ∈ W ⇒ N \ S < W
• strong if S < W ⇔ N \ S ∈ W
∩
• weak if V = S ∈W S , ∅
57
The members of V are called veto players or vetoers. The simple game is
dictatorial if there exists j ∈ N such that S ∈ W ⇐⇒ j ∈ S . The player
j ∈ N is called the dictator.
Remark 1.5.1. Consider the simple game with a unique vetoer that is not the
dictator. Then this game is weak and proper, but is not strong.
Definition 1.5.3. Given a simple game g = (N, W), the associated coalitional game (N, v) is defined by




if S ∈ W
1
v(S ) = 


0 otherwise
58
Remark 1.5.2. Let g = (N, W) be a simple game and let G = (N, v) be the
associated coalitional game. Then:
1. G is monotonic.
2. G is constant-sum if and only if g is strong.
3. G is superadditive if and only if g is proper.
proof : S , N ∈ W and the superadditivity imply that v(N \ S ) = 0 i.e., g
is proper. Conversely, if v(S ) + v(T ) > v(S ∪ T ) for some S and T with
S ∩ T = ∅, then S , T and S ∪ T are all winning. But, then N \ S is also
winning because T ⊆ N \ S and g is monotone, which contradicts the
assumption that g is proper.
4. Any monotonic coalitional game (N, v) satisfying v(S ) ∈ {0, 1} for all
S ⊆ N and v(N) = 1 is the associated coalitional game of some simple
game.
proof: Just take W = {S ⊆ N| v(S ) = 1}.
59
Definition 1.5.4. A simple game (N, W) is a weighted majority game if
there exist a quota q > 0 and weights wi ≥ 0 for all i ∈ N such that for all
S ⊆N
S ∈ W ⇐⇒ w(S ) ≥ q
The (|N| + 1) − tuple (q; (wi)i∈N ) is called a representation of g.
Definition 1.5.5. Let g = (N, W) be a weighted majority game. The representation (q; (wi)i∈N) of g is a homogeneous representation of g if
S ∈ Wmin ⇒ w(S ) = q,
where Wmin is the set of minimal winning coalitions. A weighted majority
game is homogeneous if it has a homogeneous representation.
60
Remark 1.5.3. A symmetric simple game g = (N, W) has the homogeneous
representation (k; 1, . . . , 1), where k = minS ∈W |S |. Such a game is also
denoted by (|N|, k). The simple majority game is then given by (|N|, [ n2 ] + 1),
where [z] is the integer satisfying z − 1 < [z] ≤ z for all real numbers z.
Example 1.5.1. The UN Security Council is a weighted majority game with
the representation
g := (39; 7, 7, 7, 7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1).
This game is weak and homogeneous. The vetoers are the Big Five.
61
Problem 1.5.1. Prove the following assertions.
1. A weak simple game is proper.
2. A simple game is dictatorial if and only if it is both weak and strong.
3. A dictatorial simple game gives rise to an inessential game.
62
1.5.1
Social Choice Games
• The Social Choice Game (N, v) is the coalitional game defined by
∑
v(S ) = max
ui(y) ∀S ∈ W,
y∈Y
i∈S
where
N = {1, ..., n}, n ≥ 3
Y is a finite set of social alternatives
W is a nonempty set of winning coalitions with the monotonicity condition
S ∈ W, S ⊆ T =⇒ T ∈ W
63
• The core C(W) is given by
∑
∑
{
}
N C(W) = x ∈ <+ xi = v(N),
xi ≥ v(S ) ∀S ∈ W
i∈N
i∈S
Theorem 1.5.1. (Kaneko *3). Let (N, v) be the social choice game with the
simple majority rule Wm:
Wm = {S ⊆ N | |S | > n/2}.
Then, x ∈ C(Wm) if and only if there exists yN ∈ Y with v(N) =
satisfying
∑
i∈N
ui(yN )
1. xi = ui(yN ) ∀i ∈ N
∑
2. v(S ) =
ui(yN ) ∀S ∈ Wm
i∈S
*3
Kaneko, M.: Necessary and sufficient conditions for the existence of nonempty core of a majority game, International Journal
of Game Theory 4 (1974), 215-219
64
Proof. Sufficiency is obvious. To see (1), suppose to the contrary that there
exist x ∈ C(Wm) and i ∈ N such that xi > ui(yN ). Then, for this i we have
v(N) − xi ≥ v(N \ {i}).
Hence,
∑
v(N) − ui(y ) > v(N) − xi ≥ v(N \ {i}) or
N
u j(yN ) > v(N \ {i}).
j∈N\{i}
Since N \ {i} ∈ Wm, this inequality contradicts the definition of v(N \ {i}) that
∑
v(N \ {i}) ≥ j∈N\{i} u j(yN ). Hence (1).
∑
Next, suppose that there existed S ∈ Wm with v(S ) , j∈S u j(yN ). Then,
by definition of v and (1)
∑
∑
N
v(S ) >
u j(y ) =
x j,
j∈S
j∈S
which contradicts the assumption that x ∈ C(Wm).
65
This theorem indicates that it is rather an exceptional case to obtain a
nonempty core of a simple majority game. If it happened that there existed a
social alternative yN ∈ Y in the core C(W), then the core consists of a unique
payoff vector, and every member i of the society would acquire exactly the
payoff equal to ui(yN ). In this sense, the majority rule would yield a fair
social state involving no sidepayments at all.
In real world social decision making situations, however, illegal sidepayments such as bribery or vote purchases are frequently observed. The next
theorem clarifies the structural relation between bribery and power.
66
Theorem 1.5.2. (Nakayama *4 ). Suppose there exist a payoff vector x and
i ∈ N satisfying that
x ∈ C(W) and xi > ui(y∗),
∗
where y ∈ Y is one that attains v(N), i.e., v(N) =
player i is a vetoer.
∑
i∈N
ui(y∗). Then, the
Proof. We must have N \ {i} < W, because we would otherwise have
∑
∑
∗
u j(y ) =
u j(y∗) − ui(y∗) > v(N) − xi ≥ v(N \ {i}),
j∈N\{i}
j∈N
contradicting the definition of v(N\{i}). It then follows from the monotonicity
that
S ⊆ N \ {i} =⇒ S < W.
Hence, this i is a vetoer.
*4
Nakayama, M.: Note on the core and compensation in collective choice, Mathematical Social Sciences 2 (1982), 323-327.
67
This theorem states that if some player obtains a payoff in the core greater
than the amount obtainable under a Pareto efficient alternative y∗, then the
player is necessarily a veto player. While two payoff vectors (ui(y∗))i∈N and
x with x , (ui(y∗))i∈N cannot be both in the core under the majority rule, this
is possible under a weak social choice game, bribing all veto players.
1.6
The Information Trading Game
Technological information may have a negative externality in the sense that
the profit from the information decreases as the the number of the holders
increases. A typical example will be the information on a new technology
that reduces the production cost of a firm in an oligopolistic industry. Below,
we consider the trade of such an information.
68
• E(k) is the profit or benefit of the information holder when the number
of holders of information is k ≥ 1
• E(k) is decreasing in k.
• Player 1 is the sole initial holder of the information, and faces the choice
whether or not to sell the information.
• Player 1 may sell to nonholders under the agreement not to resell the
acquired information.
• This agreement will be generally not self-enforcing, since resale is costless to the reseller.
• We show, however, that the information can be traded without any resale
occuring (the resale-proof trading of information).
69
Example １ N = {1, 2, 3}: E(1) = 30, E(2) = 16, E(3) = 9
1. E(1) < 2E(2)
2. E(2) < 2E(3)
3. E(1) ≥ 3E(3)
4. E(1) ≥ 2E(3) + E(3) > E(2) + E(3).
Thus, by 3, the initial holder cannot sell to the two remaining nonholders.
But, even if the holder sells to just one player at price E(2), 2 implies that the
no-resale agreement will be violated and the information will be necessarily
resold to the last player. Then, 4 indicates that the profit of the initial holder
will go down to E(2) + E(3), which is less than E(1). Hence, anticipating
this, the initial holder will not sell the information to any nonholder.
70
Example 2 N = {1, 2, 3, 4}: E(1) = 30, E(2) = 16, E(3) = 9, E(4) = 5
1. E(1) < 2E(2)
2. E(2) < 2E(3)
3. E(2) ≥ 3E(4)
4. E(3) < 2E(4)
5. E(2) ≥ 2E(4) + E(4) > E(3) + E(4).
Thus:
• By 2 and 3, when there are two holders, any one of holders may try to
resell to just one player.
• But, 4 indicates that when the number of the holders becomes 3, one of
the holders surely deviates from the agreement and secures the profit by
reselling to the last nonholder, thereby allowing the information disseminated to all.
71
• Then, the profit of the first reseller will be reduced less than E(2).
• That is, any one of the two holders cannot resell to anyone: the no-resale
agreement is enforcing in this case. Hence, the original holder can sell
the information to just one player.
Definition 1.6.1. (Dissemination-Proof Coalition M) Let M ⊆ N be any
coalition with 1 ∈ M.
1. If M = N, then we say M is dissemination-proof.
2. Suppose the definition is completed for all M ⊆ N with n ≥ |M| > m ≥ 1.
Then, for all M with |M| = m, we say M is dissemination-proof if
E(|M|) ≥ (1 + |T |)E(|M ∪ T |)
for all T ⊆ N − M such that M ∪ T is dissemination-proof.
72
Theorem 1.6.1 (Nakayama et al. [31]). Let M and M 0 be any two
dissemination-proof coalitions satisfying 1 ≤ |M| < |M 0|. Then,
|M|E(|M|) ≥ |M 0|E(|M 0|)
where the inequality is strict if |M| > 1.
Proof. By the symmetry we may assume that M ( M 0. Since M and M 0 are
both dissemination-proof, it follows by definition that
E(|M|) ≥ (1 + |M 0 − M|)E(|M 0|).
Then, summing over all i ∈ M, we have
|M|E(|M|) ≥ |M|E(|M 0|) + |M|(|M 0 − M|)E(|M 0|)
= |M 0|E(|M 0|) + (|M| − 1)(|M 0 − M|)E(|M 0|)
≥ |M 0|E(|M 0|),
with a strict inequality if |M| > 1.
73
1.7
Dissemination-Proof Information Trading Game v


∗
∗


|M
|E(|M
|)





v(S ) = 
E(1)






 0
if 1 ∈ S and |M ∗| ≤ |S |
if 1 ∈ S and |M ∗| > |S |
if 1 < S
where, M ∗ is the minimal-size dissemination-proof coalition.
Theorem 1.7.1. Let C(v) be the core of the game v, and let x∗ be the imputa(
)
∗
∗
∗
tion given by x = |M |E(|M |), 0, . . . , 0 . Then, C(v) = {x∗} ⇐⇒ M ∗ , N
Proof. (⇐=): Take any S with 1 ∈ S ⊆ N. Then, x∗(S ) = |M ∗|E(|M ∗|) ≥
v(S ); so thatx∗ ∈ C(v)．If an imputation x satisfies that x1 < x1∗ , some j ∈
N \ {1} has x j > 0．Hence, whenever x(M ∗) = |M ∗|E(|M ∗|), xk = 0 for any
(
)
∗
∗
k ∈ N \ M . Then, x (M \ { j}) ∪ {k} < v(|M ∗|). Hence, x < C(v).
(=⇒): M ∗ = N implies that v(S ) = E(1) or 0 for any S , N, it follows
that any imputation x satisfies x(S ) ≥ v(S ) for any S ⊆ N.
74
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