TRANSACTIONS OF THE
AMERICAN MATHEMATICALSOCIETY
Volume 171, September 1972
INFINITE GAMESWITHIMPERFECT INFORMATION^)
BY
MICHAEL OR KIN
ABSTRACT. We consider
an infinite,
two person
zero sum game played as follows: On the rath move, players
A, B select privately
from fixed finite sets, A ,
B , the result of their selections
being made known before the next selection
is
made. After an infinite
number of selections,
a point in the associated
sequence
space,
fl, is produced
upon which B pays A an amount determined
by a payoff
function defined on SI. In this paper we extend a result
of Blackwell
and show
that if the payoff function
is the indicator
function
of a set in the Boolean
algebra generated
by the G~'s (with respect
to a natural
topology
on fl) then the
game in question
has a value.
1. Introduction.
by several
with
writers,
of these
Let
|A
paper,
[l],
information
[2], and Shapley
we will discuss
have
been
151 Before
the structure
studied
proceeding
and admissible
games.
j, [B ! be sequences
zz'tz
^
II~=1 Z
Q be topologized
Suppose
with imperfect
Blackwell
of this
ß be the space
Z . Let
games
notably
the main result
strategies
let
Infinite
of nonempty
finite
xY J
of infinite
as follows
sequences
i.e.
Let
Z n = A n x B n , and
co = (z , z2, • ••)
(for a related
X is the set of all positions,
sets.
discussion,
finite
see
sequences,
where
x =
izx, z ,.. ■,z ), z. £ Z., 77= 0, 1, 2, • • •. Then if co £ Q, x e X, we define
be a neighborhood
as sets,
they
of co if co passes
form a base
through
for a Hausdorff,
x. If the positions
disconnected
z^ e
[3]):
x to
are thus considered
topology
for Í1 in which
fi
is compact.
In this
lection
open
topology
any open
of positions).
and closed.
collection
members
of T], which
two person
Received
game
T such
that
by a finite
by the editors
of X (a countable
collection
Baire
of positions
if G is a G g then
G = \co £ 0 | co passes
we will henceforth
/ is a bounded
G., played
by a subset
by Wolfe [7] that
through
denote
by G = T i.o.
function
on Q. Then
there
is both
exists
infinitely
we define
col-
a
many
a zero sum
as follows:
July 1, 1971.
AMS 1970 subject classifications.
Key words and
game, lower value,
(1) This paper
nia, Berkeley,
and
defined
Any set defined
It is shown
of positions
Now, suppose
set is
Primary 90D05, 90D15; Secondary 28A05, 60G45.
phrases.
Infinite
games,
imperfect
information,
two person zero sum
payoff function,
Baire function.
is part of the author's
doctoral
dissertation
at the University
of Califorwas prepared
with the partial
support
of the U. S. Air Force,
Grant
AF-AFOSR-1312-67, and the U. S. Office of Naval Research, Contract NONR N00014-66C0036.
Copyright © 1972, American Mathematical Society
501
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502
MICHAEL ORKIN
First,
£B
player
A selects
The result,
selects
a
of moves
= ia
while
is a point
A strategy
ability
z
£ A
¡7
e A
b2 £ B2, etc.
co = (z , z2, • • • ) £ Í2 and
a (/3) for A (B) gives
distribution
on .A
ß)
defines
(Bn
a probability
to A in G. when
player
B simultaneously
, b A £ Z , is announced
B selects
choice
distribution
A uses
B pays
for each
A
position
according
infinite
the amount
tz, say) a prob-
if the current
to a (ß).
A
sequence
/(oj).
x (of length
that
bj
upon which
of this
P « on fl and, hence,
a and B uses
selects
to both players,
The result
j) with the stipulation
tion is x, A (B) will make his next
(a,
while
[September
posi-
A pair of strategies
an expected
payoff
ß:
E(f,a, ß) = ff(co)dPaß(co).
The lower and upper
values
of G, ate,
respectively,
L(G.) = sup inf E(f, a, ß),
U(G.) = inf sup E(f, a, ß).
'aß
It is always
true that
L(G.) < U(G,);
'
if L(G,)
ß
a
= U(G,),
this
common
value
is called
the value of G. and will be denoted by Val (G A.
2. Main Result.
algebra
generated
result
this
We will show that if / = ¡G, where G £ B(G5) (the Boolean
by the
Cj's),
if G is a G g. Before
type
and mention
Example
proving
a related
1. On each
1. The winning
then
set
G, has
this result,
move,
players
strategy;
If B ever
still
hit.
finite
game is 1, which
by saying
says
Example
of changes
2. The winning
also
said
1, A wins;
and
A wins if there
to predict
is a strategy
least
starts
0 again,
a 0 or 1. If player
The value
set
if B said
B's
many
proved
this
of games
of
move.
saying
many
tz and
co = (l,
l) for
l) for at most
by A with a nonrandom
0 on every
If B then
back to l's,
says
etc.
1's
move,
forever,
If there
F is
E is
are an in-
move,
the players
choose
1, the game is over on that move;
If A never
many moves
says
with outcome
choice.
See [2] for a related
game is
1, but there
Blackwell,
a 0 or
E is hit.
is a G g. On each
says
n and
co = (l,
If B says
0's.
0, B wins.
for A, due to David
simultaneously,
as follows:
can be achieved
A switches
A ever
are infinitely
of this
B choose,
is defined
G is hit, otherwise
taneously
A tries
1 on each
1, A then
If B ever says
number
two examples
G = {co\ co = (0, O) for infinitely
of this
he starts
hit.
we give
A and
F
F = {cú\ co = (O, O) for at most finitely
finitely many n\.
The value
In [l], Blackwell
open question.
S of the form C.U
S = G U F where
infinitely
many 72!,
a value.
1, the game continues
(0, 0). (In other
words,
game.)
are no optimal
which,
simul-
if B
strategies
for fixed
1 - 1/N:
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for A. Here
N, guarantees
A at
1972]
INFINITE GAMES WITH IMPERFECT INFORMATION
503
Define N. = 2'N, / = 1, 2, • • • , so that
f^N.
Player
has
A divides
the trials
not yet stopped
N
into successive
the game,
i.e.
blocks
played
1 when
block
X. at random from Í1, 2, • • • , N .\. He then plays
B's
previous
X. — 1 plays
out the block.
in the
of lengths
block
N
N 2, • • • . If he
/ is reached,
he selects
1 at the X .th trial of block /' if
are all l's;
otherwise,
he plays
0 through-
Then, clearly,
PiA loses on /th block | ;'th block is reached)
< 1/zV ..
Thus, P(A loses on ;'th block) < l/N.,
and P(A loses by failing to match) <
2 (l/N.)
of this
= 1//V. However,
A will win, since
there
The following
which
are simple
payoff
fact,
Iq
— Iq
be (0, 0)'s
based
c Iß
unsolved
on sets
+ • ■ ■+ c I„
know whether
have
we will discuss
they
then
remains
functions
of the form /=
the form
would
question
we do not even
paper
by the nature
a value,
some
strategy,
in each
if the game goes
block.
in general:
in B(G ^) have
, where
where
0,,
cases
Q.
□
Do games
a value,
with payoffs
i.e.
games
with
B . £ B(G g), c . ate constants.
or not the much simpler
special
on forever,
games
ate open
of these
kinds
with payoffs
and disjoint.
of games
and
In
of
In another
show
that
have a value.
We are now ready
Lemma
to prove
1. Consider
the main result.
the class
of sets
of the form
G,1 U F.i U (G,¿ n F ¿A U • • • U (G n D F n )
where
G. e G g, F. £ F
Proof.
Fp.'s
see
is an
. This
class
By the fact that a finite
F(y, and by the standard
[4, Proposition
of sets
union
is precisely
of G As is a G g, a finite
results
I. 2.2, p. 7]) it is easily
the form \Jn{ =1 (G. C\ F A, G. £ G%, F. £ Fa.
for generating
shown
set
intersection
algebras
of
(e.g.
in B(G g) is of
Thus, every set in B(Gs) is of the
(u'<<vrvF¿>Yn^i"^
/
Boolean
that every
form
\z = l
B(G §).
z'= l
¿FOu(ñc-)ü(¿(FínCíc'
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504
MICHAEL ORKIN
which,
again
easily
seen
using
Lemma
way:
C
the fact
about
finite
to be of the required
2. Consider
Proof.
(intersections)
of Gg's
(F
's)
is
form.
the class
= G <-; if n > 1, C
of sets in C
unions
[September
of sets
= sets
C = U°°_i
of the form
^-
generated
GgU A _
in the following
where
A
= complements
(e.g. C2 = Gg U F ). We claim C = B(G A.
By using
De Morgan's
law and the fact that
GgUGs
= Gg,
FaC\F
a =
F_cr' , it is easily 7 seen that C 4. = sets of the form G,112272tz
U F, U (F. O G,), C, = sets
of the form G U F, U (JJ"_2 ^- n G-)) which by Lemma 1 gives the sets in
B(GS). a
The next
lemma
is the first
step
in an induction
which
will yield
the main
result.
Lemma
3. Let
cf> be upper
the game with payoff
Proof.
[l].
let
part
G = T i.o.,
G*X be the game,
time after
of this
where
starting
upper
value
uous,
so, by [6], G* has
of the original
game
for fixed
such
what
Let
B, starting
A's method
starting
B plays,
of play
G £ G «. Then
is the same
of positions.
UÍG-)
t
(p
B has
as in
For any position
x,
if T is hit for the first
hit after
from r. This
and player
e > 0, we present
A does,
from x, play
x, say at t. Then
and
0 if T is never
a value
Val (G* ) > Ux(G-);
after
proof
T is a collection
from x with payoff
x at t, with payoff
that no matter
0 < cp < 1. Suppose
<tj = min (cp, lA) has a value.
The first
Suppose
semicontinuous,
x, where
payoff
is lower
an optimal
a strategy
UÁG-)
is the
semicontin-
strategy.
We claim
for B starting
from x
E (d/) < Val (G* ) + c
optimally
starting
in G* until
T is hit for the first
from t, to keep
E (d,) < U (G-)
z
cp
time
+ e, so
EXQ>)=Y P^t)Eticp)<Y p(t)Ut(Gj)+ c< Val(G*)+ e.
zeT
zeT
Now, for f > 0, we describe
a strategy
for A such
E(<7j)> UÍG-) - e, and the lemma will be proved.
G* (e denotes
the empty
e/8 optimally
in G* , etc.
e/2n+
optimally
(z y z2,-
times.
(1)
If T is hit after
that no matter
A plays
what
B does,
<r/4 optimally
e, say at 7,
in G* .) Let the resulting
a sequence
of random
sequence
variables:
if T was hit for the &th time at t,,
Thus,
in
A then plays
(If T is hit for the Tzth time at t , A then plays
of moves
be denoted
by z =
•• ).
We define
c/¿ (G-)
position).
First,
X
= U(G~);
fot
k > 1, X, =
X, = 0 if T was hit less than &
we have
E(Xk\Xk_1,...,X0)>Xk_l-e/2^.
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1972]
INFINITE GAMES WITH IMPERFECT INFORMATION
505
This is obvious
if Xfe_j = 0. If not, T was hit for the (k - l)st
say,
after which A played
e/2k+ 1 optimally
II
> X._x
- e/2*+
on both
sides,
Val (G* ) - e/2k+ J
k
. Since
the payoff
in G*
rZ
is Xfe, (1) follows.
Taking
expectations
we get
(2)
EiXk)>EiXk_A-£/2k
Now, by the definition
on ii, for every
any point
time at tk_v
in G* to get at least
point
+1 => E(Xk)>
of upper
z = iz
and the nature
, z2, • ■• ) and every
co = icox, co2, • • A with
cpico) < cpiz) + c/2
semicontinuity
UiGj) - c/2.
e > 0, there
of the topology
exists
k such
that
co ■= z. for i < k has the property
=£> if z £ G, epico) < cpiz) + c/2
=£> (still
if z £ G) l/(Zl>. . . ,,. ) (Gj)
=^" (for any
z) lim sup X
< cjiz) + c/2
< cpiz) + e/2.
n
The last
implication
so for 22> N, X
is obvious
=0.
Using
it z £ G. If z d. G, T is only hit say
Fatou's
lemma
on the last
inequality,
N times,
we get
E icj) > lim sup F (Xn) - e/2 > UÍG¿) - c. □
zz
Theorem 1. Suppose H £ £ , i.e. H = G U S, G £ G g, Sc £ C _,,
also,
that
cp is upper
Then
the game with payoff
Proof.
theorem
Lemma
(assume,
Ut(G-)
We claim
loss
of positions
continues
the theorem
happens.
a value
is true for sets
time
that
G ,= 0).
Suppose
the
G £ G 0>
s, Sc £
Suppose
x, let
T is hit after
Otherwise,
(the payoff
in C
H £ (c 72', H = G U S, where
of generality,
the first
Suppose,
0 < cp < 1, cp = 1 072 S.
has a value.
T. For any position
until
when this
H* has
that
in C 72—1,. Let
without
some collection
at x which
with the property
cp = min (fp, /,,)
3 shows
is true for sets
C
ting
semicontinuous
G = T i.o.
for
W* be the game starting
x, say at
the game continues
in H* may be neither
t, with
and
upper
A get-
A gets
nor lower
Is.
semi-
continuous).
Observe that if C is closed, /eGg=4>Cn/eGg;
therefore
/ £ C^ =#> C D / £ C . Let
of positions
Cn-1
passing
since
through
Sc is.
Define
the upper
semicontinuous
satisfies
the conditions
tion hypothesis
has a value
negation
it clearly
be the open
hit
starting
since
allows
is
T.
Let
/ is the
set defined
C
= Gc.
payoff
by the collection
Then
in f/*.
Sc O C
Also,
g: g = cp* on Q , g = 1 elsewhere.
of the theorem
its payoff
of the payoff
the players;
0x
later
cp* = 1 - /, where
function
the game
since
x which
J £ F a =&■C C\J £ F a;
and
cp* = min(g,
at x with payoff
proof can be used
the addition
of a constant
of proof
g
Therefore,
in Lemma
by reversing
to the payoff).
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define
Thus
nçC), so by the induc-
cp* has a value.
/ = 1 — cp* (the method
the same
lc
£
H*
3 allows
the role of
506
MICHAEL ORKIN
By reasoning
U (G-).
similar
tothat
Now, for fixed
in Lemma
[September
3, it can be shown
t > 0, we will exhibit
a strategy
that
Val (H* ) >
for player
A which
guaran-
tees that E(0) > U(G-) - c.
As in Lemma
3, A starts
out playing
for the Tzth time at t , A then
play
plays
e/4
optimally
e/2" + l optimally
in //*,
etc.
(If T is hit
in H* .) Let the resulting
be z = (zy, z2,- ■• ).
Define the random variables:
hit for the zkh time at t.,
similar
X = U(G-): fot k > 1, X, = U.AG-)
0
rb '
X, = ¡s if T is hit less
to that in Lemma
-
k
tk
than
k times.
<t>
if T is
By reasoning
3, we get
(3)
E(Xk)>U(G¿)-c/2.
Again,
by the definition
point
of upper
semicontinuity,
(ú = ico., co~, ••• ) agreeing
1
with
¿
there
exists
z up to z,
R ( £ ,z )
k(e
. such
that
any
has the property
cpico) < 0(z) + f/2 =£> if z £ H, 0(<u) < 0(z) + e/2
=^> if z £ H, U,Zi¡...tZk)(G^)<cJ(z)
*•■ for any
z,
lim sup X
+ e/2
< cf>(z) + e/2.
72
Again, the last step is obvious
if z £ G. If not, S Cl Gc is hit and 0=1
Gc is hit and lim supn X^ = 0. Thus, by Fatou,
EQ>) > lim supn £(^n)
or Sc O
> !/(Gj
-
<r. D
Corollary 1. If H £ B(G A, the game with payoff /„ has a value.
Proof.
Let
cf>= 1 and use
Corollary
2. G, has a value
(a)
is a collection
There
f is constant
and Lemma
if f satisfies
segment
of any other
on all sequences
passing
2.
the following
T of nonoverlapping
x £ T =^ x is not an initial
then
the theorem
conditions:
positions
member
through
(nonoverlapping
of T) such
that
means
if x £ T
x.
(b) 0 < / < 1 072 T.
(c)
There
Proof.
exists
H £ B(G g) such
The function
0 = 1 on //, and /=
for suggesting
the course
f = ¡H if T is never
off T, cp = f otherwise,
min (0, ¡H) so the theorem
Acknowledgement.
well
0=1
that
I wish
the problem
to express
herein
is upper
applies.
my appreciation
hit.
semicontinuous,
O
to Professor
and for his encouragement
David
and advice
Blackduring
of my research.
REFERENCES
1.
99-101.
D. Blackwell,
infinite
G „-games
with
imperfect
information,
MR 39 #5158.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
Zastos.
Mat.
10 (1969),
1972]
INFINITE GAMES WITH IMPERFECT INFORMATION
2. D. Blackwell
159-163.
and T. S. Ferguson,
507
The big match, Ann. Math. Statist.
39 (1968),
MR 36 #6211.
3. D. Gale and F. M. Stewart,
Infinite
games with perfect
information,
Contributions
to the Theory of Games,
vol. 2, Ann. of Math. Studies,
no. 28, Princeton
Univ. Press,
Princeton, N. J., 1953, pp. 245-266.
4.
J. Neveu,
Bases
MR 14, 999.
mathématiques
du calcul
des probabilités,
Masson,
Paris,
1964;
English transi., Holden-Day, San Francisco, Calif., 1965. MR 33 #6659; #6660.
5. L. S. Shapley, Stochastic games, Proc. Nat. Acad. Sei. U. S. A. 39 (1953), 10951100. MR 15, 887.
6. M. Sion, On general
minimax theorems,
Pacifie
J. Math. 8 (1958), 171 —176.
MR 20
#3506.
7.
P. Wolfe,
(1955), 841-847.
The strict
determinateness
of certain
infinite
games,
Pacifie
J. Math.
5
MR 17, 506.
DEPARTMENT OF STATISTICS, UNIVERSITY OF CALIFORNIA, BERKELEY, CALIFORNIA
94720
Current
address:
Department
of Mathematics,
Case
Western
land, Ohio 44106
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Reserve
University,
Cleve-