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 PLAYERDEALER 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 PLAYER DEALER

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 Dealer expectations Expected game value

Score matrix, expectations and game value Blackjack 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. PLAYERDEALER

Decimal Blackjack (Version 1.1) PLAYERDEALER

The Draw Matrix An example of a Markov Matrix Bust Bust TOTAL BEFORE TOTAL AFTER

Draw Matrix, D

Drawing Two Cards Multiply Markov matrices D.D

Draw one card if total < n

Keep hitting while total < 6 Multiply Markov matrices indefinitely This matrix above is an idempotent matrix

Score matrices for Decimal Blackjack 1.1 Player’s drawing Markov matrix, D Score matrix Player expectations Dealer expectations Expected game value

Blackjack 1.1: Player hits when < 6

Blackjack 1.1: Player hits when < 5

Blackjack 1.1: Player hits when < 4

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) PLAYERDEALER

Scoring for Decimal Blackjack 1.2 Player’s Markov matrix Dealer’s Markov matrix Expected game value

Game Matrix Compare 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 whenn<4n<5n<6n<7n<8 n<406.4%8.8%6.0%-3.6% n<5-6.4%04.5%3.8%-3.7% n<6-8.8%-4.5%01.8%-3.2% n<7-6.0%-3.8%-1.8%0-2.0% n<83.6%3.7%3.2%2.0%0 Player Strategies 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 whenn<4n<5n<6n<7n<8 n<44.7%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<86.4%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 Player Strategies 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 whenn<14n<15n<16n<17n<18 n<142.6%12.8%18.7%17.5%9.1% n< %1.1%10.7%13.2%8.3% n< %-7.1%3.1%9.7%8.9% n< %-7.7%-0.3%7.1%10.8% n<18-5.3%-0.5%4.4%9.3%14.3% Game Matrix for Version 2.1 New goal: Closest to 21 without busting MINIMUM entry in each column MAXIMUM entry in each row There is NO saddle-point! NO equilibrium! Player Strategies Dealer Strategies

hit whenn<14n<15n<16n<17n<18 n<142.6%12.8%18.7%17.5%9.1% n< %1.1%10.7%13.2%8.3% n< %-7.1%3.1%9.7%8.9% n< %-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. Player Strategies Dealer Strategies

hit whenn<17 29% n<17 and 71% n<18 n<18 n<1513.2%8.3% n<177.1%10.8% 43% n<15 and 57% n<17 9.7% Mixed Strategies Randomly change between 2 or more different strategies Player Strategies Dealer Strategies 29% n<17 and 71% n<18 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

hit whenn<14n<15n<16n<17n<18 n<140.8%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<186.6%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% Player Strategies 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

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

hit whenn<14n<15n<16n<17n<18 n<140.8%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<186.6%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% Player Strategies Dealer Strategies

hit whenn<14n<15n<16n<17n<18 n<142.6%12.8%18.7%17.5%9.1% n< %1.1%10.7%13.2%8.3% n< %-7.1%3.1%9.7%8.9% n< %-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% Player Strategies Dealer Strategies

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%)

Blackjack 4.0 Can see the dealer’s first card Instead of using a game value of use instead

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%