Probability, matrices and
game theory
The mathematics of Blackjack (21)
Basic Idea
• Draw cards to get as close as possible to
a total of 21 without going over
PLAYER
DEALER
Highest total wins!
Complications of Blackjack
•
•
•
•
•
•
•
•
Ace = 1 or 11
Blackjack (Ace+Ten) vs. 21
Splitting pairs
Doubling down
Insurance
When cards are shuffled
Whose cards you can see
Casino-dependent special cases
Decimal Blackjack (Version 1.0)
• Remove all the face-cards; let Ace=1
• Each number from 1 to 10 occurs with
equal probability of 0.1
• One card each: high card wins
Not very exciting ... no decisions to make
The Score Matrix, S
All calculations from Casino point of view
0
-0.9
-0.7 -0.5
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
DEALER
1
2
3
4
5
6
7
8
9
10
1
0
+1 +1
+1
+1
+1
+1
+1
+1
+1
0.7
2
-1
0
+1
+1
+1
+1
+1
+1
+1
+1
0.5
3
-1
-1
0
+1
+1
+1
+1
+1
+1
+1
0.3
4
-1
-1
-1
0
+1
+1
+1
+1
+1
+1
5
-1
-1
-1
-1
0
+1
+1 +1
+1
+1
6
-1
-1
-1
-1
-1
0
+1 +1
+1
+1
7
-1
-1
-1
-1
-1
-1
0
+1 +1
+1
-0.5
8
-1
-1
-1
-1
-1
-1
-1
0
+1
+1
-0.7
9
-1
-1
-1
-1
-1
-1
-1
-1
0
+1
-0.9
10
-1
-1
-1
-1
-1
-1
-1
-1
-1
0
0.1
0.1
-0.1
-0.3
PLAYER
0.9
Score matrix, expectations and game value
• Score matrix, S
• Card probability vector
p = (0.1, 0.1, 0.1, 0.1, 0.1, 0.1 , 0.1, 0.1, 0.1, 0.1)
• Player expectations
Sp
• Dealer expectations
T
• Expected game value
p S
T
p Sp
Score matrix, expectations and game value
Blackjack 1.0
0.0%
Dealer total
1
2
3
4
5
6
7
8
9
10
Dealer wins
1
Player total
2
3
4
5
Equal
6
7
8
9
10
Player wins
Decimal Blackjack (Version 1.1)
• Player (only) has option to HIT
– drawing additional cards to improve total
before seeing Dealer’s card
– must not go over 10.
PLAYER
DEALER
Decimal Blackjack (Version 1.1)
PLAYER
DEALER
The Draw Matrix
An example of a Markov Matrix
TOTAL AFTER
1
2
3
4
5
6
7
8
9
10 Bust
0
2
0
0
3
0
0
0
4
0
0
0
0
5
0
0
0
0
0
6
0
0
0
0
0
0
7
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
9
0
0
0
0
0
0
0
0
0
0.1
0.9
10
0
0
0
0
0
0
0
0
0
0
1.0
Bust
0
0
0
0
0
0
0
0
0
0
1.0
TOTAL BEFORE
1
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.1
0.1 0.1 0.1
0.1 0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Draw Matrix, D
New Total
1
1
2
3
Old Total
4
5
6
7
8
9
10
Bust
2
3
4
5
6
7
8
9
10
Bust
Drawing Two Cards
Multiply Markov matrices D.D
New Total
1
1
2
3
Old Total
4
5
6
7
8
9
10
Bust
2
3
4
5
6
7
8
9
10
Bust
Draw one card if total < n
New Total
1
1
2
3
Old Total
4
5
6
7
8
9
10
Bust
2
3
4
5
6
7
8
9
10
Bust
Keep hitting while total < 6
Multiply Markov matrices indefinitely
Final Total
1
2
3
4
5
6
7
8
9
10
Bust
1
2
Original Total
3
4
5
6
7
8
9
10
Bust
This matrix above is an idempotent matrix
D

Score matrices for Decimal Blackjack 1.1
• Player’s drawing Markov matrix, D
• Score matrix
T
D S
• Player expectations
• Dealer expectations
• Expected game value
D
T
T
Sp
p D
T
T
S
T
p D Sp
Blackjack 1.1: Player hits when < 6
20.8%
Dealer total
1
2
3
4
5
6
7
8
9
10
Bust
Dealer wins
1
2
Player total
3
4
5
6
Equal
7
8
9
10
Bust
Player wins
Blackjack 1.1: Player hits when < 5
23.0%
Dealer total
1
2
3
4
5
6
7
8
9
10
Bust
Dealer wins
1
2
Player total
3
4
5
6
Equal
7
8
9
10
Bust
Player wins
Blackjack 1.1: Player hits when < 4
21.1%
Dealer total
1
2
3
4
5
6
7
8
9
10
Bust
Dealer wins
1
2
Player total
3
4
5
6
Equal
7
8
9
10
Bust
Player wins
Summary: Blackjack 1.1
• Player hits when total n is
–
–
–
–
–
–
–
–
–
–
–
n<1, zero advantage (same as dealer strategy)
n<2, 8.45% advantage
n<3, 14.67% advantage
n<4, 21.1% advantage
n<5, 22.98% advantage (OPTIMAL STRATEGY)
n<6, 20.79% advantage
n<7, 13.38% advantage
n<8, 6.2% disadvantage
n<9, 22.83% disadvantage
n<10, 55.2% disadvantage
n<11, 100% disadvantage
Decimal Blackjack (Version 1.2)
• Both player and dealer can HIT
– but let’s assume for now that whoever goes
second cannot see the first player’s cards
(Las Vegas style Blackjack)
PLAYER
DEALER
Scoring for Decimal Blackjack 1.2
• Player’s Markov matrix
D player
• Dealer’s Markov matrix
Ddealer
• Expected game value
T
p D
T

player

dealer
S D
p
Game Matrix
Compare opposing strategies
Player Strategies
Dealer Strategies
hit when
n<4
n<5
n<6
n<7
n<8
n<4
0
6.4%
8.8%
6.0%
-3.6%
n<5
-6.4%
0
4.5%
3.8%
-3.7%
n<6
-8.8%
-4.5%
0
1.8%
-3.2%
n<7
-6.0%
-3.8%
-1.8%
0
-2.0%
n<8
3.6%
3.7%
3.2%
2.0%
0
• For any given dealer strategy, player would
choose the row which gives the MINIMUM
• For any given player strategy, dealer would
choose the column which gives the MAXIMUM
Saddle-points
• An entry in the game matrix which is a
minimum in its column and a maximum in
its row is called a Saddle Point
• Pairs of strategies that form saddle points
are “optimal” and in “equilibrium”: neither
person has an incentive to play differently
Decimal Blackjack 1.2
• Both player and dealer should choose the
strategy
– HIT on all totals less than 7
Decimal Blackjack 2.0
The game is no longer SYMMETRIC
Game Matrix for Version 2.0
Player Strategies
Dealer Strategies
hit when
n<4
n<5
n<6
n<7
n<8
n<4
4.7%
7.2%
10.1%
8.0%
-0.7%
n<5
-5.5%
1.4%
6.8%
7.3%
1.3%
n<6
-7.5%
-2.1%
3.8%
7.4%
4.9%
n<7
-4.0%
-0.2%
3.9%
8.5%
10.1%
n<8
6.4%
8.8%
11.3%
14.2%
17.2%
• MINIMUM entry in each column
• MAXIMUM entry in each row
• There is a Saddle-point
Optimal strategies for
Decimal Blackjack 2.0
• Dealer should HIT if total < 7
• Player should HIT if total < 5
• Advantage to Casino = 7.3%
Game Matrix for Version 2.1
New goal: Closest to 21 without busting
Player Strategies
Dealer Strategies
hit when
n<14
n<15
n<16
n<17
n<18
n<14
2.6%
12.8%
18.7%
17.5%
9.1%
n<15
-11.8%
1.1%
10.7%
13.2%
8.3%
n<16
-16.9%
-7.1%
3.1%
9.7%
8.9%
n<17
-14.8%
-7.7%
-0.3%
7.1%
10.8%
n<18
-5.3%
-0.5%
4.4%
9.3%
14.3%
• MINIMUM entry in each column
• MAXIMUM entry in each row
• There is NO saddle-point! NO equilibrium!
Movement diagrams
Player Strategies
Dealer Strategies
hit when
n<14
n<15
n<16
n<17
n<18
n<14
2.6%
12.8%
18.7%
17.5%
9.1%
n<15
-11.8%
1.1%
10.7%
13.2%
8.3%
n<16
-16.9%
-7.1%
3.1%
9.7%
8.9%
n<17
-14.8%
-7.7%
-0.3%
7.1%
10.8%
n<18
-5.3%
-0.5%
4.4%
9.3%
14.3%
• Either person will change strategies if
unhappy with current situation.
Mixed Strategies
Player Strategies
Dealer Strategies
hit when
n<17
29% n<17
and
71% n<18
n<15
13.2%
9.7%
8.3%
43% n<15
and
57% n<17
9.7%
9.7%
9.7%
n<17
7.1%
9.7%
10.8%
n<18
• Randomly change between 2 or more
different strategies
Minimax theorem
Every two-person zero-sum game
with a finite number of pure strategies
has a minimax equilibrium
if
mixed strategies are allowed
Borel, Fisher, Von Neumann, Morgenstern, Nash
proved in a various ways between 1920 and 1950
Blackjack 2.13
• Let’s put the face cards back in the deck
– Face cards count as 10
• Probability distribution is now
p
1
13
1
13
1
13
1
13
1
13
1
13
1
13
1
13
1
13
4
13

Game Matrix for Version 2.13
Player Strategies
Dealer Strategies
hit when
n<14
n<15
n<16
n<17
n<18
n<14
0.8%
6.9%
9.2%
7.2%
0.7%
n<15
-4.3%
2.2%
6.9%
7.4%
3.3%
n<16
-5.2%
-0.2%
5.0%
8.2%
6.9%
n<17
-1.7%
2.0%
5.8%
9.8%
11.5%
n<18
6.6%
9.1%
11.7%
14.4%
17.2%
• Both player and dealer want to HIT less often
when the proportion of 10’s are higher.
• Mixed equilibrium has dealer advantage of 7.6%
Effect of card distribution
• Blackjack 2.10
p  101
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
13
1
13
4
13

– Dealer advantage 9.7%
– Dealer mixes hitting with n<17 and n<18
– Player mixes hitting with n<15 and n<17
• Blackjack 2.13
p  131
1
13
1
13
1
13
1
13
1
13
1
13
– Dealer advantage 7.6%
– Dealer mixes hitting with n<16 and n<17
– Player mixes hitting with n<14 and n<16

Blackjack 3.0
• In legal casinos the dealer plays a fixed
and publicly known strategy!
• Let’s FIX this strategy to be
HIT if n<17
• This is the best FIXED strategy for the
casino with an initial distribution
1

p  13
1
13
1
13
1
13
1
13
1
13
1
13
1
13
1
13
4
13

p  131
Game Matrix for
1
13
1
13
1
13
1
13
1
13
1
13
1
13
1
13
Player Strategies
Dealer Strategies
hit when
n<14
n<15
n<16
n<17
n<18
n<14
0.8%
6.9%
9.2%
7.2%
0.7%
n<15
-4.3%
2.2%
6.9%
7.4%
3.3%
n<16
-5.2%
-0.2%
5.0%
8.2%
6.9%
n<17
-1.7%
2.0%
5.8%
9.8%
11.5%
n<18
6.6%
9.1%
11.7%
14.4%
17.2%
• If dealer cannot mix, outcome will be one
of the blue entries
• Dealer settles for 7.2%
4
13

p  101
Game Matrix for
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Player Strategies
Dealer Strategies
hit when
n<14
n<15
n<16
n<17
n<18
n<14
2.6%
12.8%
18.7%
17.5%
9.1%
n<15
-11.8%
1.1%
10.7%
13.2%
8.3%
n<16
-16.9%
-7.1%
3.1%
9.7%
8.9%
n<17
-14.8%
-7.7%
-0.3%
7.1%
10.8%
n<18
-5.3%
-0.5%
4.4%
9.3%
14.3%
• Out of all fixed strategies dealer would prefer to
settle for n<18 with 8.3%
• Legal casino dealers can’t change strategies
based on p so must play n<17 and get only
7.1%
1
10

Blackjack 3.0 with card counting
and fixed dealer strategy n<17
1
1
1
1
1
1
1
1
1
4

p  13 13 13 13 13 13 13 13 13 13 
– Hit with n<14 (dealer advantage 7.2%)
p  101
1
10
p
1
5
1
10
1
10
1
10
1
10
1
10
1
10
1
10
– Hit with n<17 (dealer advantage 7.1%)
1
5
1
5
1
5
1
5
0 0 0 0 0
– Hit with n<18 (player advantage 4.3%)
p  0 0 0 0 0
1
5
1
5
1
5
1
5
– Hit with n<12 (player advantage 27%)
1
5

1
10

Blackjack 4.0
Can see the dealer’s first card
• Instead of using a game value of
T
p D
T

player

dealer( n17)
S D
p
use instead
T
p D
T

player

dealer( n17)
S D
ecard
e7  0 0 0 0 0 0 1 0 0 0
Strategy for Blackjack 4.0
can see the dealer’s first card
•
•
•
•
•
•
•
•
•
•
Dealer shows 1, HIT if n < 14 (Dealer advantage 11.0%)
Dealer shows 2, HIT if n < 13 (Player advantage 1.0%)
Dealer shows 3, HIT if n < 13 (Player advantage 3.7%)
Dealer shows 4, HIT if n < 12 (Player advantage 6.6%)
Dealer shows 5, HIT if n < 12 (Player advantage 9.6%)
Dealer shows 6, HIT if n < 12 (Player advantage 12.7%)
Dealer shows 7, HIT if n < 17 (Player advantage 3.7%)
Dealer shows 8, HIT if n < 17 (Dealer advantage 3.2%)
Dealer shows 9, HIT if n < 17 (Dealer advantage 11.2%)
Dealer shows 10, HIT if n<16 (Dealer advantage 21.6%)
Overall dealer advantage 5.7%
Real Blackjack factoids
• Optimal non-card-counting strategy
– 1-deck game Player advantage 0.04%
– 4-deck game Dealer advantage 0.49%
– Infinite decks Dealer advantage 0.65%
• Observation in 1987 of 11,000 actual hands
played in Nevada/New Jersey
– Non-optimal average-Joe play Dealer advantage 2%
– Players who tried to count cards made a mistake
once every 7 hands Dealer advantage 9%