Games with convex weight in restricted cooperation
by
E. Marchi*) and G. Simonetti
*)
Founder and Ex Director of the Instituto de Matemática Aplicada San Luis, CONICET. Universidad
Nacional de San Luis. Ejército de los Andes 950. San Luis. Argentina.
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1. Introduction and results
It is supposed a given matrix F   s A  of order s x 2s 1 with non negative
rational elements  s A .
2. A Sandwich Theorem
Be N a finite conjunct of n elements and F a non empty conjunct of N’s subconjuncts. The elements of F will be called feasible. It is not supposed that   F . If
that happens it will be written F0  F  .
Two arbitrary functions, f and g, which are defined over F are supposed given,
with values in the real numbers that verify the followings relations
g  A  f  A
g    f    0
for all A  F
if   F
It is denote by R N to the Euclidean space of dimension N .
Definition 1:
X , X  R N , X  x1 , x2 ,, x N 
X  : F  R is defined by X   A    i A xi
i A
Also it is necessary an indicator function 1U : N  R to the sub-conjuncts U  N
1 if i  U
1U i   
0 if i  U
Theorem 1:
The following sentences are equivalent taking into account the previous
suppositions.
a)
i) There exists X  R N such a
g  A  X   A  f  A
A  F
2
ii) For all
s
m
j 1
k 1
A1 , A2 , , As , B1 , B2 , Bm  F
that verifies the equality
 i A j 1A j   i Bk 1Bk for each i of N, it is fulfilled
 g Aj    f Bk 
s
m
j 1
k 1
b) The followings statements are equivalent
i) There exists X  R N , X  0 such a
g  A  X   A  f  A
ii)
For
s
m
j 1
k 1
A  F
A1 , A2 , , As , B1 , B2 , Bm  F0
all
that
verifies
 i A j 1A j   i Bk 1Bk for each i of N, it is fulfilled
 g Aj    f Bk 
s
m
j 1
k 1
Demonstration: i  ii 
Be A1 , A2 , , An , B1 , , Bm  F .

 s 
 g A j      i A j xi      xi  i A j
j 1
j 1  iA j
 i 1  j / iA j 
s
 
s
m

   i B
k 1  iBk
k

 s 

xi    i A j
 

 i 1  A j
 iA j

 s 
   xi    i B
k
 i 1  Bk

 iBk






 m
xi    f Bk 
 k 1

ii  i  .
The condition i) is valid if the following linear program has a feasible solution.
max O.X
subject to
X   A  f  A
 X   A   g  A
A  F
A  F
X  R N without restrictions.
The dual associated program is given by
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

min   f  A y A   g  A z A 
YA ,Z A   AF
AF

subject to

AF
iA
 i A yA     i A zA
AF
iA
yA  0; z A  0
i  1, 2, , n
A  F
and it results feasible. Proving that also it is bounded, it is obtained the feasibility of the
primal program.
If it is supposed that it is not bounded, there exist vectors Y   y A  , Z  z A  , in
R
F
, Y , Z  0 such a
 i A
AF
iA
and
yA     i A zA
AF
iA
i  1, 2, , n
 f  A y A   g  A z A .
AF
AF
We can suppose without losing generality that the components of Y, Z are
rational numbers. Eliminating denominators it can be chosen Y, Z with entire
components. That contradicts the condition ii), then the dual program has to be
bounded.
In the following part, the necessary elements, which permit to use a
particularization of the Theorem 1 in the analysis of the existence of the -core of a
game, are defined.
Be L  F a sub-collection formed by conjuncts of F that satisfy the following
conditions:
i) N  L
ii) For all A, B  L such a there exists some F  F and F  A  B , it is fulfilled that
A B  L.
Suppose that the collection exists.
For each F  F , be F   L  L : F  L.
Corollary 2:
If  i A   i A for each A  F , the following statements are equivalent:
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i) There exists X  R N such a g  A  X  A
A  F
g L  X L
L  L
and
ii) For all A1 ,, As  F0 , L1 ,, Lm  L that verifies
fulfilled
s
 i A
j 1
m
j
1A j    i Lk 1Lk it is
k 1
 g Aj    f Lk .
s
m
j 1
k 1
Demonstration: take f  X  in the Theorem 1.
(the part i )  ii ) is evident)
Demonstration to the reciprocal one
 
For each A  F it is defined f  A  g A .
For each A  F , g  A  f  A .
That is because if A  F , A  L ; considering the collections A  F , A  L and
verifying that
s
 i A 1A
j
j 1
j
  i A 1A   i A 1A when it is taken s=m=1
 
it is obtained g  A  g A  f  A .
Now, be any collection A1 , A2 ,, As , B1 , B2 ,, Bm  F0 that verifies
s
 i A
j 1
m
j
1A j   i Bk 1Bk for each i of N
k 1
that says that
s
from that
s
m
j 1
k 1
 i A j 1A j   i Bk 1Bk
 
m
m
k 1
k 1
 g Aj   g Bk    f Bk 
j 1
now the result is deduced form the Theorem 1.
Corollary 3:
3. Games with restricted cooperation
In this type of games it is not permitted any collection T  N .
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Definition 2:
A finite game with restricted cooperation consists of
* A finite conjunct of players, N.
* A conjunct of feasible coalitions, F.
* A function defined over F, to real values
V : F  R with v   0 vN   v0
and it is denoted by J  N , F ,v ,v0  .
In the following part, it is supposed that N  F .
Definition 3:
The -core of a game J is the conjunct of all the X  R N such a
n
i)
 i N
xi  v0
 i A
xi  v  A
i 1
ii)
iA
A  F
Definition 4:
One collection A1 , A2 ,, As is said -balanced if for any natural number m and
for each i  N
1
m
s
 i A
k 1
k
1Ak   i N 1N
Definition 5:
One game J is -balanced if for all the -balanced collection A1 , , As ,
1
m
s
 i Ak 1Ak   i N 1N implies
k 1
s
1
  v Ak   v0 .
m k 1
An extension of the Bondareva’s Theorem results from the Corollary 3, taking
L  N  and g  v .
Theorem from 2:
The -core of a game J is not empty if and only if J is -balanced.
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Definition 6:
A game J is exact if for all A  F
v A 
min
X   Core
X   A .
Taking g  v and L  N , A in the Corollary 3, it is obtained the following theorem.
Theorem 3:
J  N , F ,v ,v0  is exact if and only if for all A1 , A2 ,, As  F there exist
natural numbers m, r such a
s
  i Ak 1Ak  m  i N 1N  r  i A 1A implies
k 1
s
 v Ak   m v0  r v A
k 1
Definition 7:
  C  v   X    Corev  such a x  0 will denote to the -positive core
of the game J  N , F ,v ,v0  .
Definition 8:
J  N , F ,v ,v0  is completely balanced if   C  v    .
Be J  N , F ,v ,v0  a positive game v  0  , -balanced , with the following
property: if A, B are in F also A  B  F .
Theorem 4:
If for all A  F such a a  A,  b A  a N   a A  s N
Then
  C v     C  v  if and only if 
i  F for all i  N .
Demonstration:
(only if)
i    ii xi  0 from where xi  0 , for all i.
for i  N , X  
(if)
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~
Be a  N such a, a F and A   A  F / a  F .
~
~
As A  a  0 there exists b  A  a.
Be X    0  , which is defined in the following way:


 xi  
aN



X    xi 
 aN

 xi


if i  a
if i  b
in another case
It is easy to verify for X  that
n
 iN
i 1
To verify the condition
 i A
iA
xi  v A
xi  v0
A  F , different cases must be analyzed,
the most interesting one is that where a A and b 

 

aN
 i A xi   i A xi   a A  xa  
i A
i A
i  a ,b


   b A  xb 
bN




 
  v A   b A  a A   v A

bN a N 
Then X     C v  . However for a big enough  , X     C  v  .
An example showing that   C v     C  v  and the singles do not belong to F.
Be S  1,2,3 F  S ,1,2
, 1,3
, 2,3
 1S 
4
2
1
 2S   3S 
3
3
3
A  1,2
 11,2 
1
1
 21,2 
2
2
 31,3 
1
3
 22 ,3  1  32 ,3 
1
4
 11,3  1
3
x  C v     s S xs  vS 
s 1
8
  sA xs  v A
A  F
sA
We define
2
v  A  
1
if A  S
if A  S, A  F
then x  C v  
4
2
1
x1  x2  x3  2
3
3
3
x
x2
2) 1 
1
2
2
x3
x1 
1
3
x3
x2 
1
4
1)
 x  1,1,0
  C  v   ?  1,1,0 
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