Princeton University Press
Chapter Title: ON A GAME WITHOUT A VALUE
Chapter Author(s): Maurice Sion and Philip Wolfe
Book Title: Contributions to the Theory of Games (AM-39), Volume III
Book Author(s): C. BERGE, L. D. BERKOVITZ, L. E. DUBINS, H. EVERETT, W. H. FLEMING,
D. GALE, D. GILLETTE, O. GROSS, J. F. HANNAN, J. C. HOLLADAY, J. R. ISBELL, S.
KARLIN, J. G. KEMENY, J. MILNOR, J. C. OXTOBY, M. O. RABIN, R. RESTREPO, H. E.
SCARF, L. S. SHAPLEY, M. SION, G. L. THOMPSON, W. WALDEN and P. WOLFE
Book Editor(s): M. Dresher, A. W. Tucker, P. Wolfe
Published by: Princeton University Press. (1957)
Stable URL: http://www.jstor.org/stable/j.ctt1b9x26z.20
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Contributions to the Theory of Games (AM-39), Volume III
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ON A GAME WITHOUT A VALUE
Maurice Sion
§1 .
1
and Philip Wolfe
O
INTRODUCTION
The object of the present paper is to show that one of the main
results in the theory of infinite games, the theorem of Glicksberg [2 ] on
semi-continuous payoffs, cannot be extended in certain directions.
In §2 we present a game on the square (a form of continuous
Blotto) which does not have a value, but whose payoff function is topo­
logically even simpler than that of the classical example due to Ville [6 ].
Scarf and Shapley [5] have applied Glicksberg1s theorem to a number of in­
finite games in extensive form. It is natural to ask whether the condition
of semi-continuity of the payoff is equally important for the determinacy
of such games. To answer this question we show in §3 that any game on the
square may be transcribed into a game in extensive form with its value, or
lack of value, preserved. In §4 we find that the transcription of the ex­
ample of §2 to extensive form yields a type of "game of pursuit" without a
value.
§2 . THE GAME ON THE SQUARE
and
K
Glicksberg1s theorem states: If A and B are compact sets,
is an upper (lower) semi-continuous function on A x B, then
sup infI K df dg = inf supI K df dg
f
where
f
and
measures on
A
g
g
J J
g
f
«/,J
,
range respectively over the sets of all probability Borel
and
B.
Supported by the United States Air Force, through the Office of Scientific
Research of the Air Research and Development Command, under contract No.
AF 1 8 (6 oo)—i1 0 9 .
2 Under contract with the Office of Naval Research.
299
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SION AND WOLFE
The example delimiting possible extensions of this theorem is
the game on the unit square (o < x < 1 , 0 < y < 1) having the payoff
(see Figure 1)
-1
o
K(x, j )
+ 1
if
x < j
if
x = j
< x + j
or
,
y = x + l
otherwise .
Note that, unlike Ville's ex­
ample, this function K assumes the
values + 1 , - 1
respectively on two
open sets and vanishes on their comple
ment. It is clearly neither upper
nor lower semi-continuous. (Recall
that a function F is upper (lower)
semi-continuous if for any number c,
the set {P | f(P) < c }({P | F(P) > c}
is open.) We shall show that
sup inf
f
R
(
a
K df dg
1)
inf sup
K df dg
ii
Figure 1
Let
f
be any probability measure on
f( M )
let
yf = i.
6 > 0
T
•
i)) > i
’
so that
'([<■•*-•))
and let
yf
If
if
f ([”'
choose
-
[o, 1].
,
In either case, it is quickly checked that
- 6.
inf
>i
J J K df dg <
On the other hand, if
f
J°K(x, y ^ ) df(x) < -j
is chosen so that
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301
A GAME WITHOUT A VALUE
f((0j) = f ({i})
then for all
j
= f({i}) = j
,
y
K(x, y) df (x) = J
K(0, y) + K^
j,
y) + K(i,
i i
Thus the first equation in (1 ) is proved.
Next, let
g
g( [0, 1 )) > y , let
in this case: if
be any probability measure on
xg = 1 .
If
*([«■
let
x
choose
= 0;
[0, 1 ].
g( [0, 1 )) < i , then
*))
- T
•
s([o,l)) > i
,
If
g( {1 } ) > j
,
and
if
8 > 0
so that
g ([°> 2 ~ 6 )) - 7
and let
xg = ~ - 6.
sup
On the other hand, if
In any case it is easily checked that
J J K df dg >
g
J K(xg, j ) dg(y)
«({*})
g((!})
J K(x,
.
is chosen so that
mud
then for any
> ^
1
7
2
7
k
7
'
'
'
x
y) dg(y) =
+ 2k|x,
+ hK(x,
1)
- 7
Thus the second equation in (1 ) is proved.
This game can be considered as a continuous Blotto game as follows
Player A must assign a force x to the attack of one of two mountain passes
and 1 - x to the other. Player B must assign a force y to the defense
of the first pass, and 1 - y to the other, at which is also located an
extra stationary defense force of 1 /2 . A player receives from the other
a payment of 1 at each pass if his force at that pass exceeds his op­
ponent ’s, and receives nothing if they are equal there. The payoff is thus
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SION AND WOLFE
B(x, y ) = sgn (x - y) + sgn ([ 1 - x] - [3/ 2 -
y] ).
It iseasily checked that 1 + B(x, y) = K(x, y ), so that this' game has
minorant
value - 2 / 3 and majorant value - 4/7. Itis interesting that
the continuous Blotto game played with forces x, a - x for playerA and
y, b - y
for B (withno reserves) always has a value
[3 ].
§3 . TRANSCRIPTION TO EXTENSIVE FORM
In this section, we wish to translate the game on the square dis­
cussed above into a game in extensive form. To this end, we consider a
certain class of zero-sum two person infinite games in extensive form and
show, in general, how to transcribe any game on the square into a game of
this class.
We first give the structure of the games
r
in question.
Let players A and B choose alternately either.a zero or a one,
in ignorance of the other1s choices. A completed play p
of r
thus
consists of a sequence of zeroes and ones, the odd-numbered terms having
been chosen by A and the even-numbered terms by B. A pure strategy for a
player is a sequence of zeroes and ones. If
=
(x , x2, • • • ) ,
=
the play
(2 )
ape Pure strategies used by A, B respectively, then
P = pU^
«2 ) =
y,, x2, j 2 ,
...)
results.
An Initial portion of the representation of this game as a tree
(see [4] ) is given in Figure 2.
Now let P denote the set of all plays of r and, for p € P,
h(p) represent the payment from player B to player A as a result of the
play p. Let H denote the set of all pure strategies for either play­
er. In view of (2 ), a payoff h on P is determined by any function K
on n x it, by setting h(z^ zg, Zy z^, ...) = K((z^ z^, •*.), (zg, z^,
Let i as in C1 ]) a base for a topology on P be the family of those sets
which consist of all plays passing through a given vertex in the tree rep­
resentation of r, I.e., of all sequences having a given initial segment.
This is equivalent to topologizing P by coordinatewise convergence. If
we topologize n also by coordinatewise convergence and use the product
topology for n x n, then the map from n x n to P defined by (2 ) is
a homeomorphism, and hence any topological property of the function K on
n x IT Is Inherited by the payoff h on P. Consequently, in order to
describe a game r with certain topological properties for the payoff h,
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A GAME WITHOUT A VALUE
we need only specify these properties
for its "normal form” payoff K.
We proceed to transcribe a
game on the square, having a bounded
Borel measurable payoff function K,
into a game on n x n. For any such
K, we construct K on ir x ,n as
follows:
Let
Inform ation sets
for p lay e r A
T
map each * =
into that point of
the unit interval having it as its
°°
—1 \
dyadic expansion (namely,
s i = i x i 2
}
n let
For
*2
e
( x ]f x2, ... ) € n
K( it 1,
ic2 ) = K ( T « 1 . t *2 )
T is clearly continuous, hence the
mapping of n x n onto the unit
square defined by (^ , *2 ) — *
(Tjt1, Tit2 ) is also. Thus, K_ is
Borel measurable on n x n. Further­
more, If K is continuous or semicontinuous, so is K.
Information sets
fo r p lay e r B —
Figure 2
In what follows, \ i \ x
will be understood to range over all
probability measures over the Borel sets of ir; and f 1
to range
over all probability measures over the Borel sets of the unit interval.
THEOREM.
sup inf
J J
K_ di-i-jdiig = sup inf
1
2
inf sup
/1 1/
K dp.ndfj-2 = inf sup
J J * K df 1 df2
and
same way.
II
11
K df 11 df,2
PROOF. We prove only the first equality; the second goes the
Let L, R denote respectively its left and right side.
We first show
inf
*2
fK(V
L < R.
n
)
Given
€ > 0,
) = Inf
choose
C_K dn°d|ig
\±
>L
so that
- e
^2
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3 04
SION AND WOLFS
Define f° by f°(A) = h-°(T“ 1 A) for all Borel sets A. Now for any y,
let it2 be such that T*2 = y. Then (see e.g., Halmos, Measure Theory,
page 1 63, Theorem C)
/ K(x, y ) d f ° ( x ) =
=
J K(x,
J KU,,
Tn2 ) d f ° ( x ) -
J
K fT *,, T«2 )dn °(It 1)
n2 )dn°(It1 ) > L - €
.
Thus
R > inf
fj
Since
e
f
y)df°(x)df2 (y) = inf
K(x,
f2
K(x, y)df°(x) > L
y
is arbitrary,
To show
L < R.
L > R,
inf
choose
f
f^
so that
K(x, y)df°(x) > R - e.
We shall define 11° so that |i^(T~ ’A) = f^(A) for all Borel sets A. The
only difficulty arises
from the fact that T is not one-to-one. However,
only for the countable
set of pointsof the form
x = m/2n does T~ Cx]
have more than one element, in which case it has
two. For each such x,
we select one element
it in T~ 1 [x] and denote the countable set of all
it thus selected by ft.
For any Borel set A C n, let n°(A) = f°(T(A - ft)).
We see
first that the map A ---►- T(A - ft) sends Borel sets into Borel
sets:for,
if A is closed (hence compact1 ) then *T(A) is also closed, and since
T(A) - T(ft) C T(A - ft) C T(A),
theset T(A - ft) differs from
T(A) by
at most a countable set, and is thus Borel; and it is easy to check that
this mapping preserves differences and countable unions. It is immediate
that
is a probability measure on n, and that ii°(T~ 1 (A)) = f°(A)
for all A Borel, since T(T~1 (A) - ft) = A. We then see as before:
L > inf
f K(* , * Jdn^*.,) = inf f K(T* , Tir )dn°(it )
jt^ ^
= inf
7
1
The space
n
/ K(x, y)df°(x) > R - €
J
is compact, being the product of doubletons.
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305
A GAME WITHOUT A VALUE
§4.
THE GAME IN EXTENSIVE FORM
Referring to the sequence games of §3 , let Y 1, Y2 be disjoint
sets of finite sequences of zeroes and ones such that no member of Y 1
is an initial segment of a member of Y2, and vice-versa. Let P 1 (P2 )
be the set of all plays containing some member of Y 1 (Y2 ) as initial seg
ment.
(In the tree representation of the game, Y 1 and Y2 are disjoint
sets of vertices such that no vertex in one set covers any in the other,
and P 1 (P2 ) Is the set of all plays passing through some vertex in
Y 1 (Y2 ).) The sets P 1, P2 are disjoint by construction, and, being
unions of neighborhoods, are open. Let the payoff h on P be defined by
(3)
h(p!
+1
if
p e P ,
- 1
if
p € P2,
0 otherwise .
Looking at each member of Y 1 or Y2 as a position reached in some finite
length of time, this payoff makes a play of the game essentially ended as
soon as It passes through a position of Y 1 or Y2, in which case either
player A or player B has won the play. Every other case Is a tie. This
formulation serves as a model for a two-sided game of pursuit, in which
player A may destroy B, or B destroy A, or both escape.
One may well feel that the vanishing of the payoff on all "non­
ending11 plays makes this game so nearly finite as to ensure its deter­
minateness. Indeed, if Y2 is finite, then both P - P 1 and P2 are
closed, so that (p | h(p) < c} is closed for any number c . Thus h,
and hence the associated normal-form payoff K on n x n, Is lower semicontinuous. It follows by Glicksberg’s theorem [2 ] (since n is compact)
that the extensive game in this case has a value (similarly if
finite).
Y1
is
However, the transcription of the game in §2 to extensive form
yields a payoff of the type (3 ), since the inverse Images of the open sub­
sets of the square bearing payoffs + 1 , - 1 , are open in P. Application
of the theorem of §3 shows that this game does not have a value.
BIBLIOGRAPHY
[1 ] GALE, D., and STEWART, F. M., "infinite games with perfect informa­
tion," Annals of Mathematics Study No. 28 (Princeton, 1953),
pp. 245-266.
[2 ]
GLIOKSBERG, I. L., "Minimax theorem for upper and lower semicontinuous payoffs," The RAND Corporation, Research Memorandum RM-478,
October, 1950.
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306
SION AND WOLFE
[3 ] GROSS, 0. A., and WAGNER, R. A., "A continuous Colonel Blotto game,n
The RAND Corporation, Research Memorandum RM-408, June, 1 9 5 0.
Ik]
KUHN, H. W., "Extensive games and the problem of information, 11
Annals of Mathematics Study No. 28 (Princeton, 1953), pp. 193-2 1 6 .
[5 ]
SCARF, H., and SHAPLEY, L. S., "Games with information lag," The
RAND Corporation Research Memorandum RM-1 3 2 0 , August, 195?*
[6]
VILLE, J., nSur la theorie generale des jeux ou intervient l !habilit6
des joueurs,M Traite du Calcul des Probabilites et de ses Applica­
tions, par E. Borel et collaborateurs, Paris (1 9 3 8 ), Vol. 2 , No. 5,
pp. 105-113-
Maurice Sion
Philip Wolfe
The Institute for Advanced Study
Princeton University
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