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, SAll calculations from Casino point of view

1 2 3 4 5 6 7 8 9 10

1 0 +1 +1 +1 +1 +1 +1 +1 +1 +1

2 -1 0 +1 +1 +1 +1 +1 +1 +1 +1

3 -1 -1 0 +1 +1 +1 +1 +1 +1 +1

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

8 -1 -1 -1 -1 -1 -1 -1 0 +1 +1

9 -1 -1 -1 -1 -1 -1 -1 -1 0 +1

10 -1 -1 -1 -1 -1 -1 -1 -1 -1 0

PL

AY

ER

DEALER

-0.9 -0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 0.9

0.9

0.7

0.5

0.3

0.1

-0.1

-0.3

-0.5

-0.7

-0.9

0.1

0

Score matrix, expectations and game value

• Score matrix, S

• Card probability vectorp = (0.1, 0.1, 0.1, 0.1, 0.1, 0.1 , 0.1, 0.1, 0.1, 0.1)

• Player expectations

• Dealer expectations

• Expected game value

pS

SpT

pSpT

0.0%

Dealer totalP

laye

r to

tal

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10Player wins

Equal

Dealer wins

Score matrix, expectations and game valueBlackjack 1.0

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 MatrixAn example of a Markov Matrix

1 2 3 4 5 6 7 8 9 10 Bust

1 0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

2 0 0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2

3 0 0 0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3

4 0 0 0 0 0.1 0.1 0.1 0.1 0.1 0.1 0.4

5 0 0 0 0 0 0.1 0.1 0.1 0.1 0.1 0.5

6 0 0 0 0 0 0 0.1 0.1 0.1 0.1 0.6

7 0 0 0 0 0 0 0 0.1 0.1 0.1 0.7

8 0 0 0 0 0 0 0 0 0.1 0.1 0.8

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

TO

TA

L B

EF

OR

ETOTAL AFTER

New TotalO

ld T

otal

1 2 3 4 5 6 7 8 9 10 Bust

1

2

3

4

5

6

7

8

9

10

Bust

Draw Matrix, D

Drawing Two CardsMultiply Markov matrices D.D

New TotalO

ld T

otal

1 2 3 4 5 6 7 8 9 10 Bust

1

2

3

4

5

6

7

8

9

10

Bust

Draw one card if total < n

New TotalO

ld T

otal

1 2 3 4 5 6 7 8 9 10 Bust

1

2

3

4

5

6

7

8

9

10

Bust

Keep hitting while total < 6Multiply Markov matrices indefinitely

Final TotalO

rigin

al T

otal

1 2 3 4 5 6 7 8 9 10 Bust

1

2

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

• Player expectations

• Dealer expectations

• Expected game value

SDT

pSDT

pSDpT

T

SDpT

T

Blackjack 1.1: Player hits when < 620.8%

Dealer totalP

laye

r to

tal

1 2 3 4 5 6 7 8 9 10 Bust

1

2

3

4

5

6

7

8

9

10

BustPlayer wins

Equal

Dealer wins

Blackjack 1.1: Player hits when < 523.0%

Dealer totalP

laye

r to

tal

1 2 3 4 5 6 7 8 9 10 Bust

1

2

3

4

5

6

7

8

9

10

BustPlayer wins

Equal

Dealer wins

Blackjack 1.1: Player hits when < 421.1%

Dealer totalP

laye

r to

tal

1 2 3 4 5 6 7 8 9 10 Bust

1

2

3

4

5

6

7

8

9

10

BustPlayer wins

Equal

Dealer 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

• Dealer’s Markov matrix

• Expected game value

pDSDp dealer

T

playerT

playerD

dealerD

Game MatrixCompare opposing strategies

• 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

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

Pla

yer

Str

ateg

ies

Dealer Strategies

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

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%

Game Matrix for Version 2.0

• MINIMUM entry in each column

• MAXIMUM entry in each row

• There is a Saddle-point

Pla

yer

Str

ateg

ies

Dealer Strategies

Optimal strategies for Decimal Blackjack 2.0

• Dealer should HIT if total < 7

• Player should HIT if total < 5

• Advantage to Casino = 7.3%

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%

Game Matrix for Version 2.1New goal: Closest to 21 without busting

• MINIMUM entry in each column

• MAXIMUM entry in each row

• There is NO saddle-point! NO equilibrium!

Pla

yer

Str

ateg

ies

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%

Movement diagrams

• Either person will change strategies if unhappy with current situation.

Pla

yer

Str

ateg

ies

Dealer Strategies

hit when n<17

29% n<17

and

71% n<18

n<18

n<15 13.2% 8.3%

n<17 7.1% 10.8%

43% n<15

and

57% n<17

9.7% 9.7% 9.7%

Mixed Strategies

• Randomly change between 2 or more different strategies

Pla

yer

Str

ateg

ies

Dealer Strategies29% n<17

and

71% n<18

9.7%

9.7%

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

134

131

131

131

131

131

131

131

131

131p

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%

Game Matrix for Version 2.13

• 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%

Pla

yer

Str

ateg

ies

Dealer Strategies

Effect of card distribution

• Blackjack 2.10

– 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

– Dealer advantage 7.6%– Dealer mixes hitting with n<16 and n<17– Player mixes hitting with n<14 and n<16

101

101

101

101

101

101

101

101

101

101p

134

131

131

131

131

131

131

131

131

131p

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

134

131

131

131

131

131

131

131

131

131p

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%

Game Matrix for

• If dealer cannot mix, outcome will be one of the blue entries

• Dealer settles for 7.2%

Pla

yer

Str

ateg

ies

Dealer Strategies

134

131

131

131

131

131

131

131

131

131p

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%

Game Matrix for

• 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%

Pla

yer

Str

ateg

ies

Dealer Strategies

101

101

101

101

101

101

101

101

101

101p

Blackjack 3.0 with card counting and fixed dealer strategy n<17

– Hit with n<14 (dealer advantage 7.2%)

– Hit with n<17 (dealer advantage 7.1%)

– Hit with n<18 (player advantage 4.3%)

– Hit with n<12 (player advantage 27%)

101

101

101

101

101

101

101

101

101

101p

134

131

131

131

131

131

131

131

131

131p

0000051

51

51

51

51p

5151

51

51

5100000p

Blackjack 4.0 Can see the dealer’s first card

• Instead of using a game value of

use instead

pDSDp ndealer

T

playerT

)17(

cardndealer

T

playerT eDSDp

)17(

00010000007 e

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%