FAVORABLE STRATEGY FOR TWENTY-ONE

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A
FAVORABLE
STRATEGY
FOR
TWENTY-ONE*
BY
EDWARD
THORP
DEPARTMENT
OF
MATHEMATICS,
MASSACHUSETTS
INSTITUTE
OF
TECHNOLOGY
Communicated
by
Claude
E.
Shannon,
November
21,
1960
1.
Introduction.-It
has
long
been
an
open
question
as
to
whether
those
of
the
standard
gambling
games
which
are
not
repeated
independent
trials
admit
strategies
favorablet
to
the
player.
There
have
been
numerous
implications2-4
that
favor-
able
strategies
do
not
exist.
In
this
note,
we
settle
the
issue
by
showing
that
there
is
a
markedly
favorable
mathematicali
strategy
for
one
of
the
most
widely
played
games,
twenty-one,
or
blackjack.
2.
Previous
Work.-Our
point
of
departure
is
the
work
of
Baldwin,
Cantey,
Maisel,
and
McDermott,5
6
the
only
serious
treatment
of
blackjack
that
has
been
given
to
date.
The
reader
will
find
further
references
and
a
representative
set
of
rules
in
their
paper.
Although
there
are
minor
variations
in
the
game,
we
shall
adopt
those
rules
(including
insurances).
3.
Method
and
Results.-Our
calculations
are
similar
to
those
outlined
in
Baldwin
et
al.,5
but
there
are
some
very
important
changes.
First,
a
high-speed
computer
was
programmed
to
find
the
player's
best
possible
strategy
and
the
corresponding
expectation.
The
electronic
calculator
enabled
us
to
dispense
with
many
of
the
approximations
that
were
needed
by
Baldwin
et
al.
to
reduce
the
calculations
to
desk
computer
size.
This
led
to
noticeable
improvements
in
results.
In
particular,
the
player's
expectation
for
a
complete
deck
was
found
to
be
a
startling-0.21%.
(Baldwin
et
al.
give-0.62%).
Oursecondchangeinapproach
was
to
program
the
computer
to
do
the
calculations
for
arbitrary
sets
of
cards.
This
made
it
possible
to
take
into
account
cards
that
become
visible
during
play,
a
feature
which
is
essential
for
the
determination
of
any
winning
strategy.§
A
standard
deck
of
cards
has
approximately
3.4
X
107
subsets
which
are
dis-
tinguishable
under
the
rules
of
blackjack.
It
is
thus
impractical
to
compute
the
optimal
strategy
for
each
of
these
subsets.
Instead,
we
have
studied
a
number
of
carefully
preselected
subsets,
and
from
the
information
gained,
several
favorable
strategies
are
obtained.
Some
of
our
subsets
and
results
are
given
in
Table
1
below.
Let
Q(I)
be
the
number
of
cards
of
value
I.
The
special
subsets
in
Table
1
differ
from
a
full
deck
only
in
that
the
number
of
cards
of
a
single
value
has
been
altered.
In
actual
play,
these
special
subsets
occur
infrequently,
and
some
are
even
im-
possible.
Even
so,
they
yield
a
profusion
of
winning
strategies.
For
example,
one
TABLE
1
PLAYER'S
EXPECTATION
WITH
SELECTED
SUBSETS
Description
Player's
Description
Player's
of
the
subset
expectation
of
the
subset
expectation
Complete
deck
-.0021
Q(7)
=
0
.0125
Q(1)
=
0
-.0272
Q(8)
=
0
.0005
Q(2)
=
0
.0142
Q(9)
=
0
-.0091
Q(3)
=
0
.0189
Q(10)
=
12
-
.0215
Q(4)
=
0
.0236
Q(10)
=
20**
.0189
Q(5)
=
0
.0329
Q(10)
=
24**
.0394
Q(6)
=
0
.0187
110

VOL.
47,
1961
MATHEMATICS:
E.
THORP
111
of
these
winning
strategies
may
be
obtained
by
considering
the
subset
Q(5)
=
0.
Suppose
that
just
before
a
particular
deal
the
player
sees
that
all
fives
have
been
used
(so
that
the
unseen
cards
are
a
subset
of
that
subset
which
we
describe
by
Q(5)
=
0)
and
that
the
unused
portion
of
the
deck
is
ample
for
that
deal.
If
the
player
does
not
take
into
account
the
cards
other
than
fives
that
he
has
seen
on
previous
deals,
then
as
far
as
he
is
concerned,
his
probabilities
for
success
are
the
same
as
for
the
subset
Q(5)
=
0.
Using
Table
1,
it
follows
that
the
player
who
adopts
the
strategy
for
Q(5)
=
0
(see
Table
2)
when
there
are
no
fives
remaining
has
an
expectation
of
0.0329
at
those
times.
A
winning
strategy
may
now
be
defined
as
follows.
If
Q(5)
P
0,
the
player
bets
the
minimum
allowed
amount,
m,
merely
to
remain
in
the
game
and
follows
the
complete
deck
strategy
given
by
Baldwin
et
al.
When
Q(5)
=
0
(and
the
re-
mainder
of
the
deck
will
suffice
for
the
next
deal),
the
situation
has
turned
in
favor
of
the
player.
He
now
bets
a
large
amount,
M,
and
uses
the
computed
strategy
for
Q(5)
=
0,
which
is
given
in
Table
2.tt
TABLE
2
THE
STRATEGY
WHEN
Q(5)
=
0
Pair
Splitting
Doubling
Down
Dealer shows:
Dealer
shows:
Pair
23
4
6
7
8
9
10
A
Total2
3
4
6
7
8
9
10
A
A
X X
X
X
X
X
X
X
X
20
S
10
X
19
S
S
S
9
X
X
X
X
X
X
X
18
S
S
S
S
8
X
X
X
X
X
X
X
X X
17
S
S
S
S
S
7
X X
X X
X
X X
15
S
S
6
X
X
X
X
14
S
S
S
4
X
13
S
S
S
3
X X
X
X
X X
11
H
H
H
H
E
H
H H
H
2
X
X
X X
X
X
10
H
H
H
H
E
H
H
H H
9
H H
H
H
H
8
H
H
Minimum
Standing
Numbers
Dealer
shows:
Total2
3
4
6
7
8
9
10
A
19
S
S
18
S
S
S.S
S S
S
17
H
16
H
H
15
H
H
12
H
H
H
H
Legend:
X:
split
the
pair
S:
soft
total
only
H:
hard
total
only
A
disadvantage
of
this
strategy
is
that
the
event
Q(5)
0
occurs
only
in
about
3.5
per
cent
to
10
per
cent
of
the
deals
(depending
on
the
number
of
players).
A
similar
remark
applies
to
the
other
Q(I)
=
0
type
strategies.
However,
careful
scrutiny
has
disclosed
another
strategy
which
partially
overcomes
this
disad-
vantage
and
gives
the
player
a
greater
expectation
as
well.
This
strategy
depends
on
the
somewhat
surprising
fact
that
all
the
crucial
quantities
are
almost
linearly
dependent
on
the
proportion
of
tens
in
the
deck
and
nearly
independent
of
the
absolute
number
of
tens.
The
details
are
too
extensive
to
be
given
here
and
will
appear
elsewhere.
The
main
characteristics
are
that
the
player
has.
an
advantage

112
MATHEMATICS:
E.
THORP
PROC.
N.
A.
S.
almost
half
the
time;
his
expectation
exceeds
0.04
about
a
tenth
of
the
time,
and
occasionally
(probability
1/5,000-1/10,000)
exceeds
0.86.
Since
this
strategy
offers
a
"spectrum"
of
favorable
expectations,
it
seems
reason-
able
to
let
the
size
of
the
bets
increase
with
the
expectation.
The
advantages
of
this
procedure
have
not
yet
been
studied
in
detail.
Nor
have
we
considered
in
detail
the
question
of
letting
the
player's
bet
vary
with
the
size
of
his
fortune.
4.
Remarks.-With
only
minor
modifications,
our
program
can
be
used
by
a
high-speed
computer
to
play
blackjack
directly.
The
computer
would
play
a
near-perfect
game.
If
the
bets
were
of
constant
size,
the
player
would
have
a
decided
advantage.
If
the
bet
size
were
varied,
the
player's
advantage
would
be
overwhelming.
The
rules
variations
in
Nevada
casinos
have
been
tabulated
and
analyzed.
The
expectations
never
vary
more
than
0.005
from
our
figures.
Consequently
our
strategies
are
advantageous
regardless
of
these
variations
in
rules.
The
"home"
game
of
blackjack
differs
from
the
casino
game
principally
in
that
the
dealer's
strategy
is
fixed
in
the
latter
and
arbitrary
in
the
former.
Using
our
methods,
it
is
an
easy
matter
to
find
the
optimal
player
strategy
against
each
possible
fixed
dealer
strategy.
The
methods
of
game
theory
then
apply
to
the
case
of
mixed
dealer
strategies
and
yield
a
complete
solution
for
the
home
game.
The
author
is
indebted
to
R.
Baldwin,
W.
Cantey,
H.
Maisel,
and
J.
McDermott
for
making
available
their
detailed
calculations
and
to
the
M.I.T.
Computation
Center
for
making
the
IBM
704
available.
*
This
research
was
supported
in
part
by
the
United
States
Air
Force
under
contract
No.
AF49(638)-42,
monitored
by
the
Air
Force
Office
of
Scientific
Research
of
the
Air
Research
and
Development
Command.
t
A
strategy
is
favorable
if,
for
some
unifoim
bound
on
the
player's
bets,
his
fortune
converges
with
probability
1
to
plus
infinity.
|
By
saying
"mathematical
strategy,"
we
mean
to
exclude
such
time-honored
approaches
to
winning
strategies
as
physical
strategies
(defective
roulette
wheel,
defective
dice)
or
the
large
class
of
strategems
(sleight
of
hand
with
the
cards,
collusion
with
the
dealers,
etc.).
§
Further
detailed
results,
together
with
the
lengthy
computer
routine,
the
methods
used
to
insure
that
it
is
correct,
and
the
discussion
of
the
errors
introduced
by
certain'
simplifying
as-
sumptions,
will
appear
elsewhere.
**
Insurance
contributes
0.0032
to
this
value
when
Q(10)
=
20
and
0.0073
when
Q(10)
=
24.
tt
At
the
casinos,
M/m
generally
is
from
100
to
500.
With
these
values,
the
overall
"expecta-
tion"
E
(i.e.,
the
expected
value
of
the
amount
won,
in
units
of
M,
divided
by
the
number
of
times
that
M
was
bet)
is
greater
than
0.03.
In
fact,
M/m
>
15
insures
E
>
0.025
and
if
M/m
_
3,
E
>
0.
1
Feller,
W.,
An
Introduction
to
Probability
Theory
and
Its
Applications
(New
York:
John
Wiley
and
Sons,
Inc.,
1957).
2
Huff,
Darrell,
"The
Mathematics
of
Sex,
Gambling,
and
Insurance,"
Harper's,
Sept.
1959.
3
Fox,
Philip
G.
(as
told
to
Stanley
Fox),
"A
Primer
for
Chumps,"
Sat.
Eve.
Post,
Nov.
21,
1959,
pp.
31ff.
4Von
Mises,
Richard,
Probability,
Statistics
and
Truth,
(London:
Allen
and
Unwin,
1957).
5
Baldwin,
R.,
W.
Cantey,
H.
Maisel,
and
J.
McDermott,
"The
Optimum
Strategy
in
Black-
jack,"
J.
Am.
Stat.
Assn.,
51,
429-439
(1956).
6
Baldwin,
R.,
W.
Cantey,
H.
Maisel,
and
J.
McDermott,
Playing
Blackjack
to
Win:
A
New
Strategy
for
the
Game
of
21
(New
York:
M.
Barrows,
1957).