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Title
A ten-person game which has no stable set
Author(s)
Okuda, Hidesuke
Citation
経営と経済, 55(4), pp.297-301; 1976
Issue Date
1976-03-30
URL
http://hdl.handle.net/10069/27987
Right
This document is downloaded at: 2017-07-13T01:50:55Z
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A ten-person
game which
has no stable
A ten-person
set
game which
297
has no stable
set
Hidesuke
1.
Introduction
In 1944
of stable
It
von Neumann and Morgenstern
sets
has
classes
for n-person
game which
However,
2.
paper
But,
c4,---
and
used
in C3)
stable
sets
do exist
for
several
set.
paper
But,
gives
his proof
the
a ten-person
is somewhat
simple
large
compli-
and perspective
proof.
Definitions
In this
in (£l.
this
a theory
W. F. Lucas QO described
has no stable
Therefore,
C4) introduced
games.
been shown that
of games.
cated.
Okuda
in this
c10,
3.
there
are used
paper
and
the
there
the
sets
same notations
are unnecessary
FJk,
Fp,
F79,
that
were used
the vectors
F139
and
cD, c2,
F thatwere
Core
The following
relation
was obtained
by Lucas in C3)-
Theorem 1. C= {x^B:x(13579)^4}
Proof.
The
relation
C^= {x(eB:x(13579)^4}
is derived
from
the
relation
v({l,3,5,7,9))=4.
It
If
will
x&B,
be
then
(13579)^4,
then
proved
x(137)^2
that
C2{xeB:x(13579)^4}in
the
x(12)=x(34)=Ȍ =x(910)=1.
then
x(1379)^3.
and
x(179)^2.
And
And
also
Therefore,
if
x^B
x(1479)>2,
if
and
following.
x^B
and x
x(1379)^3,
etc..
In
the
THE KEIEI TO KElZAI
298
long run, it is proved the following relation that if xEB and x
(13579)"24, then
4.
x(S)~v(S)
for all SeN.
Q. E. D.
Relation of dominion
The following Theorem 2 and 3 were obtained by Lucas (3).
Theorem 2. El
n Dom. B=¢ for all S+{i, r, 7, 9},
Proof. First, observe that
AnDom s A= ¢
(1)
when S=N and when v(S)=O.
Second, note that
EnDom {h-l,h} A= ¢
(2)
for all h=2, 4, 6, 8 and 10, since
EeB and x(h-l,h)=l=v({h-l,h}) for all xEB.
Third, one can show that
ElnDomsB=¢ for all S=J={i,r,7,9}.
(3)
If xEE 1, then XJ =X k =1, and if YEB, then yh-s::.1 for all hEN.
Thus Y cannot dominate x via any S which contains j or k. This
remark along with (1) and (2) proves (3).
Q. E. D.
Theorem 3. Dom{i, r, 7, 9}C= ¢.
Proof.
If
yEC
and
(4)
Dom{i,r,7,9}Y=F¢,
then y(13579)"24. and y(ir79)-::;;'2, since v( {i, r, 7,9}) =2. So that
Y J = Yk = 1. Terefore Y r =0. This contradicts (4).
Q. E. D.
Corollary 1. En Dom sC = ¢ for all S.
Proof. This is derived from Theorem 2. and 3.
Let
Q. E. D.
A ten-person game which has no stable set
WI
=
299
{xl XEB: E I nDom{i. r. 7 .9}x=\=,0}
then
WI = B n Dom - 1 E 1 •
The abouve relation is derived from Theorem 2.
Corollary 2. WI =BnDom-1E 1.
From Corollary 1 and 2, it is derived the following proposition. '
To prove that this game has no stable set, it is sufficient to
prove that
(I)
Dom C=:l(A-B)u((Wl UW 2 UWa)-E).
(II)
There exists no stable set K' for E.
This paper gives the simple and perspective proof of ( I ).
5.
Dominion of the core
The following Theorem 4 was obtained by Lucas(3).
Theorem 4. Dom C=:lA-B
Proof. Pick xEA-B, i.e., consider any x with
EN,
:Zh&N
X
h
20 for all h
x h =5, and x(h-1,h)+1 for some h=2, 4, 6,8 or 10. It
is clear that there exists a h=2,4,6, 8 or 10 such that x(h-1,h)<1.
For this h let Yh-l=xh-1+e, and Yh=xh+e where 2e=1-x h_l
-xh>O. And let
Yk-l=1, and Yk=O where k=2, 4,6,8 or 10 but
=th. Then yEC, and Y dom{h-l,h} x, since Yh-l=xh-1+e>x h- 1,
Yh=xh+e>x h, and y(h-1,h)=x(h-1,h)+2e=1. Therefore,
Dom C=:lA-B.
Q. E. D.
The following Theorem 5 was obtained by Lucas (3).
Theorem 5. El nDom{i,r,7,9}(E I UE k )=,0.
Proof. Because if yEE I UEk then YJ=l, and thus YJ+l=Yr=O.
Q. E. D.
Theorem 6. W1-E=W1-E J .
Proof. It is easy to derive Theorem 6 from Theorem 5.
THE KElEl TO KElZAl
300
Thcorem 7. Dom C=:JW1-E J •
Proo/. We prove only for the case i = 1. For the other cases, it
is proved similarly. It is derived from definition of W I that
WI = {x=-B: O<xI-::;;'1,O<x 4 -::;;'Z ,O<X7 ,O<X9 ,Xl +X 4 +X7 +x 9-::;;'2},
and
2} .
Therefore, if xI=l, and xs=1, then WlcE a . And if
x5~1,
xI~l
or
then WI nEa = ¢.
Therefore,
WI-E a = {X/XEW I , xI~1 or xs~1}.
We will use the following inequalities.
+X7 +x g <2
(5)
Xl
(6)
O<x 4
(7)
xa<1.
Pick XE(WI -Ea).
i)
If x 1 <1,x s <1,x 7 <1,and x9<1, then there exist CI>O,CS>O,
'e 7>O, e9>O such that(xl+e l )
and (Xl
+CI ):::;:1, (xs +c s ):::;:1,
+ (xs+c s ) + (x 7 +e 7 ) +(x 9 +e 9 ) =3,
(x 7 + C7 ):::;:1 and (X9 +e 9):::;:1, since from
(5) Xl +X5 +X7 +x 9<3.
And so, there exists y such that yEB, and YI =X I +C I , Ya =1,
Ys =X5 +C 5, Y7 =X7 +c 7 and Y9 =Xg +c g • It is easy to check that yEC,
and YI>X I , Ys>x 5, Y7>X 7 and Y9>X 9 , and v({1,5,7,9})=3=YI+
Y5
+ Y7 + Y9.
ii)
Therefore, Y dom x, and so xEDom C.
If Xl =1, then X5 <1, and it follows from (5) that X 7<1, X9
<1. Also, it follows from (5) that X5 +X7 +X9 <2. Therefore,
Xa
+
x S+X 7 +x 9<3 and x a<1,x s <1,x 7 <1 and x9<1. Similarly as in case
i), it is proved that Xc Dom C.
iii)
If
X 7
=1, then it follows from (5) that
Xl
<1, X9 <1 and Xl +
A ten-person game which has no stable set
301
x9<1. Therefore,x t +x a +xg<2,and x t <l,x a <l and xg<l.
Similarly as in case i), it is proved that xEDom C.
If
Xg
= 1, then it is proved similarly as in case
X
1
= 1 that xE
Q. E. D.
DomC.
It is derived from Theorem 4, 6 and 7 that
(I)
Dom C:::JCA-B)UC(W 1 UWa UW5)-E).
REFERENCES
1.
D. B. Gillies, solutins to general non-zero games, Ann. of
Math. studies, No. 40, princeton Univ. Press, Princeton, N. J.,
1959, pp. 47-85.
2.
W. F. Lucas, A game with no solution, Bull, Amer, Math.
Soc. 74(1968), 237-239.
3.
W. F. Lucas, The Proof That a Game May Not Have a Solu-
tion, Trans, Amer. Math. Soc., Vol. 137, pp. 219-229, 1969.
4.
J. von Neumann and O. Morgenstern, Theory of games and
economic behavior, Princeton Dniv. Press, Princeton, N. ]., 1944.