## Blackjack

### October 17, 2014

We represent a deck as a list of integers from 0 to 51, from ace of spades, through hearts and diamonds, to the king of clubs; then we ignore the suit, and treat face cards as if they were ten-spots:

`(define (pips n)`

(let ((rank (+ (modulo n 13) 1)))

(if (= rank 1) 11

(min rank 10))))

Next is a function to count a hand, which is complicated by the fact that an ace can be either 1 or 11. We first compute the sum of the non-aces, plus 1 for each ace. If the total is greater than 21, we are bust. Otherwise, we add 10 for each ace as long as the sum doesn’t exceed 21, then return the final total.

`(define (count cards)`

(call-with-values

(lambda () (split-while (lambda (n) (< n 11)) (sort < cards)))

(lambda (numbers aces)

(let ((sum (+ (apply + numbers) (length aces))))

(if (< 21 sum) sum

(let loop ((sum sum) (n (length aces)))

(if (or (zero? n) (< 21 (+ sum 10))) sum

(loop (+ sum 10) (- n 1)))))))))

To play one game, we folow the same strategy for both play and dealer, holding on 17 or more and hitting below:

`(define (play) ; 1 player, 0 tie, -1 dealer`

(let* ((deck (map pips (shuffle (range 52))))

(pcards (take 2 deck)) (deck (drop 2 deck))

(dcards (take 2 deck)) (deck (drop 2 deck)))

;(display "deal: ")

;(display pcards) (display " ")

;(display dcards) (display " ")

;(display deck) (display " ")

;(display (count pcards)) (display " ")

;(display (count dcards)) (newline)

(if (= (count pcards) 21) (if (= (count dcards) 21) 0 1)

(let player ((pcards pcards) (deck deck))

;(display "player: ")

;(display pcards) (display " ")

;(display deck) (display " ")

;(display (count pcards)) (newline)

(if (< 21 (count pcards)) -1

(if (< (count pcards) 17)

(player (cons (car deck) pcards) (cdr deck))

(let dealer ((dcards dcards) (deck deck))

;(display "dealer: ")

;(display dcards) (display " ")

;(display deck) (display " ")

;(display (count dcards)) (newline)

(if (< 21 (count dcards)) 1

(if (< (count dcards) 17)

(dealer (cons (car deck) dcards) (cdr deck))

(if (< (count pcards) (count dcards)) -1

(if (< (count dcards) (count pcards)) 1 0)))))))))))

If you wish, you can uncomment the diplay lines and watch as a game is played. Finally we are ready to run the simulation, summing the wins, losses and ties:

`(define (simulate n)`

(let loop ((n n) (sum 0))

(if (zero? n) sum

(loop (- n 1) (+ sum (play))))))

If we run ten simulations, playing 1000 games each time, we see that the player always loses; that’s because he has to play first, and may go bust before the dealer ever plays:

`> (simulate 1000)`

-31

> (simulate 1000)

-89

> (simulate 1000)

-14

> (simulate 1000)

-46

> (simulate 1000)

-122

> (simulate 1000)

-16

> (simulate 1000)

-33

> (simulate 1000)

-113

> (simulate 1000)

-114

> (simulate 1000)

-73

Don’t play Blackjack!

You can run the program at http://programmingpraxis.codepad.org/BN3OknEY, where you will also see the functions that we used from the Standard Prelude.

I should stop, but here’s a J solution:

A sample run is

I don’t have time to work this up into a full solution right now, but it should be possible to calculate the exact percentages as follows:

Build a graph of the possible game states; a game state is a target value (initially 16), a score so far, and a deck (or maybe a shoe) that is just a bag of integers 1-10. Initially, for the player, the target is 17, the score is 0 and the deck is whatever we want to start with, eg. 1-9 map to 4, 10 maps to 16 for a single pack.

Each transition takes one element out of the bag and adds it to the score, a final state is one where the score is greater that the target, at which we stop (with the player bust or not). Each non-final state has a probability associated with each outgoing transition (just the probability of selecting that particular bag element) and we can propagate backwards to get the probabilities of the various final states.

There are a few thousand possible finishing states where the player doesn’t go bust and we need to use each one as a starting point for a simulation of the dealer, but this shouldn’t be completely intractable.

Here’s some Python then. Use a dictionary to keep track of states (just store the remaining deck as a tuple in here – dictionary keys must be immutable). Divide up the probabilities as we go and combine afterwards for an overall win/lose/draw value, we seem to be able to do this without too much problem with rounding error. Starting with a single pack, we end up with 1336 state in which the player hasn’t busted & must be simulated for the dealer. Total runtime on my i3 laptop is 25 secs and the final probabilities are: lose: 48.9%, win: 41.2%, draw: 9.8%.

#!/usr/bin/python

# Create a new deck with ith count decremented

def makenewdeck(deck,i):

newdeck = list(deck)

newdeck[i] -= 1

return tuple(newdeck)

class State:

def __init__ (self,score,ncards,aces,deck,prob):

self.score = score; self.aces = aces

self.deck = deck; self.prob = prob;

self.ncards = ncards; self.ndeck = sum(deck)

def __str__ (self):

return “[%d %d %d %g %s]”%(self.score,self.aces,

self.ncards,self.prob,self.deck)

def finalstate(self):

return self.score > 16

def getscore(self):

if self.score > 21: return None

else: return self.score

def blackjack(self):

return self.ncards == 2 and self.score == 21

def play(initdeck):

initstate = State(0,0,0,initdeck,1.0)

states = [initstate]

deckindex = { initdeck: 0 }

i = 0

while(i != len(states)):

s = states[i]; i += 1

if not s.finalstate():

for j in range(0,len(s.deck)):

if s.deck[j] > 0:

newdeck = makenewdeck(s.deck,j)

newprob = s.deck[j]*s.prob/s.ndeck

n = deckindex.get(newdeck)

if n != None:

# Seen this state already

states[n].prob += newprob

else:

newscore = s.score

newaces = s.aces

if (j == 0):

# Aces initially valued at 11

newscore += 11; newaces += 1

else:

newscore += j+1

if newscore > 21 and newaces > 0:

# If we busted with an ace left, count it as 1

newscore -= 10; newaces -= 1

newstate = State(newscore,s.ncards+1,

newaces,newdeck,newprob)

n = len(states)

states.append(newstate)

deckindex[newdeck] = n

return states

initdeck = (4,4,4,4,4,4,4,4,4,16)

pstates = play(initdeck)

dealerprob = 0; playerprob = 0; drawprob = 0 # Final probabilities

for i in range(0,len(pstates)):

pstate = pstates[i];

if pstate.finalstate():

pprob = pstate.prob

pscore = pstate.getscore() # player score

if pscore == None:

dealerprob += pprob # Player is bust

else:

dstates = play(pstate.deck)

# Accumulate subsidiary probabilities here

# to reduce rounding errors

dealerprob2 = 0; playerprob2 = 0; drawprob2 = 0

for j in range (0,len(dstates)):

dstate = dstates[j]

if dstate.finalstate():

dprob = dstate.prob

dscore = dstate.getscore() # dealer score

if dscore == None or dscore < pscore:

playerprob2 += dprob

elif dscore == pscore:

if pstate.blackjack() and not dstate.blackjack():

playerprob2 += dprob

else:

drawprob2 += dprob

else:

dealerprob2 += dprob

# Add in the subsidiary probabilities

dealerprob += pprob*dealerprob2

playerprob += pprob*playerprob2

drawprob += pprob*drawprob2

print (dealerprob,playerprob,drawprob,dealerprob+playerprob+drawprob)

And with proper indentation (it would be nice if comments were editable):

;; Interactive console Black jack

(defn cartesian-product [l & ls]

(reduce

(fn [a b] (for [x a y b] (conj x y)))

(map vector l) ls))

;; The deck of cards. The suits are not being used in the game but it’s more

;; fun if they’re printed anyway

(def deck (cartesian-product ‘(\u2665 \u2660 \u2666 \u2663) (range 13)))

(defn print-card [[color val]]

(let [v (if (== val 0) ‘A

(if ( a 9) 10 (+ a 1)))))

(defn bust? [v] (> v 21))

;; Find all the possible sums of values of the cards and choose the appropriate one

;; return 50 if bust

(defn hand-value [cards]

(let [possible-values (apply cartesian-product (map card-value cards))

possible-sums (map #(apply + %) possible-values)]

(if (bust? (apply min possible-sums)) 50

(apply max (filter #( v 16))

;; Interactive player’s logic

(defn players-logic [hand value]

(if (= 21 value) true

(do

(println (map print-card hand) “\n” value ” – More?”)

(= “n” (read-line)))))

;; The two game phases:

(defn dealers-game [hand pile value]

(let [[d-hand pile d-value] (deal-hand pile dealers-logic)]

(println “Dealer’s hand” (map print-card d-hand))

(if (or (bust? d-value) (> value d-value)) :win :lose)))

(defn game [pile]

(let [[hand pile value] (deal-hand pile players-logic)]

(println “Your hand” (map print-card hand))

(cond (= 21 value) :win

(bust? value) :lose

:else (dealers-game hand pile value))))

;; Let’s play!

(loop [won 0 lost 0]

(let [result (game (shuffle deck))

win (if (= :win result) 1 0)

won (+ win won)

lost (- (+ 1 lost) win)]

(println ({:win “You win” :lose “Dealer wins”} result))

(if ((fn [] (println “Play another game?”) (= “n” (read-line))))

(do

(println “You won” (int (* 100 (/ won (+ won lost)))) “% of the games”)

(vector won lost))

(recur won lost))))

my previous post is about clojure.

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