function blackjack(N) % BLACKJACK. Use random numbers in Monte Carlo simulation. % Play the game of Blackjack, either one hand, or thousands of hands, % at a time, and display payoff statistics. % % In Blackjack, face cards count 10 points, aces count one or 11 points, % all other cards count their face value. The objective is to reach, % but not exceed, 21 points. If you go over 21, or "bust", before the % dealer, you lose your bet on that hand. If you have 21 on the first % two cards, and the dealer does not, this is "blackjack" and is worth % 1.5 times the bet. If your first two cards are a pair, you may "split" % the pair by doubling the bet and use the two cards to start two % independent hands. You may "double down" after seeing the first two % cards by doubling the bet and receiving just one more card. % "Hit" and "draw" mean take another card. "Stand" means stop drawing. % "Push" means the two hands have the same total. % % The first mathematical analysis of Blackjack was published in 1956 % by Baldwin, Cantey, Maisel and McDermott. Their basic strategy, which % is also described in many more recent books, makes Blackjack very % close to a fair game. With basic strategy, the expected win or loss % per hand is less than one percent of the bet. The key idea is to % avoid going bust before the dealer. The dealer must play a fixed % strategy, hitting on 16 or less and standing on 17 or more. Since % almost one-third of the cards are worth 10 points, you can compare % your hand with the dealer's under the assumption that the dealer's % hole card is a 10. If the dealer's up card is a six or less, she % must draw. Consequently, the strategy has you stand on any total over % 11 when the dealder is showing a six or less. Split aces and split 8's. % Do not split anything else. Double down with 11, or with 10 if the % dealer is showing a six or less. The complete basic strategy is % defined by three arrays, HARD, SOFT and SPLIT, in the code. % % A more elaborate strategy, called "card counting", can provide a % definite mathematical advantage. Card counting players keep track % of the cards that have appeared in previous hands, and use that % information to alter both the bet and the play as the deck becomes % depleated. Our simulation does not involve card counting. % % BLACKJACK(N) plays N hands with an initial bet of \$10 for each hand. % "Play" mode, N = 1, indicates the basic strategy with color, but allows % you to make other choices. "Simulate" mode, N > 1, plays N hands % using basic strategy and displays the evolving payoff results. % One graph shows the total return accumulated over the duration of the % simulation. Another graph shows the observed probabilities of the % ten possible payoffs for each hand. These payoffs include zero for a % push, win \$15 for a blackjack, win or lose \$10 on a hand that has not been % split or doubled, win or lose \$20 on a hand that has been split or doubled, % and win or lose \$30 or \$40 on hands that are after doubled after a split. % The \$30 and \$40 payoffs occur rarely (and may not be allowed at some % casinos), but are important in determining the expected return from the % basic strategy. The second graph also displays with 0.xxxx +/- 0.xxxx % the expected fraction of the bet that is won or lost each hand, together % with its confidence interval. Note that the expected return is usually % negative, but within the confidence interval. The total return in any % session with less than a few million hands is determined more by the luck % of the cards than by the expected return. % % From "Numerical Computing with MATLAB" % Cleve Moler % The MathWorks, Inc. % See http://www.mathworks.com/moler % March 1, 2004. Copyright 2004. % Copyright 2014 Cleve Moler % Copyright 2014 The MathWorks, Inc. clf shg set(gcf,'name','Blackjack','menu','none','numbertitle','off', ... 'userdata',[]) rand('state',sum(100*clock)) if nargin == 0 N = 10000; kase = 1; else if ischar(N) N = str2double(N); end bj(N) kase = 2; end while kase > 0 kase = bjbuttonclick(kase); switch kase case 0, break % Close case 1, bj(1) % Play one hand case 2, bj(N) % Simulate end end close(gcf) % ------------------------ function bj(N) % Blackjack, main program. % Play N hands. % If N == 1, show detail and allow interaction. S = get(gcf,'userdata'); n = length(S); bet = 10; detail = N==1; % Set up graphics if detail delete(get(gca,'children')) delete(findobj(gcf,'type','axes')) axes('pos',[0 0 1 1]) axis([-5 5 -5 5]) axis off bjbuttons('detail'); stake = sum(S); if stake >= 0, sig = '+'; else, sig = '-'; end str = sprintf('%6.0f hands, \$ %c%d',n,sig,abs(stake)); titl = text(-2.5,4.5,str,'fontsize',20); n0 = n+1; n1 = n0; else bjbuttons('off'); payoffs = [-4:1 1.5 2:4]*bet; % Possible payoffs counts = hist(S,payoffs); n0 = n+1; n1 = ceil((n0)/N)*N; subplot(2,1,2) h = plot(0,0); end S = [S zeros(1,n1-n0+1)]; for n = n0:n1 bet1 = bet; P = deal; % Player's hand D = deal; % Dealer's hand P = [P deal]; D = [D -deal]; % Hide dealer's hole card % Split pairs split = mod(P(1),13)==mod(P(2),13); if split if detail show('Player',P) show('Dealer',D) split = pair(value(P(1)),value(D(1))); % 0 = Keep pair % 1 = Split pair split = bjbuttonclick('split',split+1); else split = pair(value(P(1)),value(D(1))); end end if split P2 = P(2); if detail, show('Split',P2); end P = [P(1) deal]; bet2 = bet1; end % Play player's hand(s) if detail [P,bet1] = playhand('Player',P,D,bet1); show('Player',P) if split P2 = [P2 deal]; show('Split',P2) [P2,bet2] = playhand('Split',P2,D,bet2); end else [P,bet1] = playhand('',P,D,bet1); if split P2 = [P2 deal]; [P2,bet2] = playhand('',P2,D,bet2); end end % Play dealer's hand D(2) = -D(2); % Reveal dealer's hole card while value(D) <= 16 D = [D deal]; end % Payoff if detail show('Dealer',D) show('Player',P) s = payoff('Player',P,D,split,bet1); if split show('Split',P2) s = s + payoff('Split',P2,D,split,bet2); end else s = payoff('',P,D,split,bet1); if split s = s + payoff('',P2,D,split,bet2); end end S(n) = s; if detail stake = stake + s; if stake >= 0, sig = '+'; else, sig = '-'; end str = sprintf('%6.0f hands, \$ %c%d',n,sig,abs(stake)); set(titl,'string',str) end chunk = min(2000,N); if ~detail & mod(n,chunk) == 0 Schunk = S(n-chunk+1:n); subplot(2,1,2) ydata = get(h,'ydata'); ydata = ydata(end) + cumsum(Schunk); ylim = get(gca,'ylim'); if max(ydata) > ylim(1) | min(ydata) < ylim(2) ydata = cumsum(S(1:n)); h = plot(1:n,ydata); line([1 n1],[0 0],'color','black') ylim = 1000*[floor(min(min(ydata)/1000,-1)) ... ceil(max(max(ydata)/1000,1))]; axis([1 n1 ylim]) else set(h,'xdata',n-chunk+1:n,'ydata',ydata); end subplot(2,1,1) [kounts,x] = hist(S(n-chunk+1:n),payoffs); counts = counts + kounts; p = counts/n; bar(x,p) axis([-4.5*bet 4.5*bet 0 .45]) stake = ydata(end); if stake >= 0, sig = '+'; else, sig = '-'; end str = sprintf('%c%d',sig,abs(stake)); if abs(stake) < 1000, str = [' ' str]; end if abs(stake) < 100, str = [' ' str]; end if abs(stake) < 10, str = [' ' str]; end title(sprintf('%6.0f hands, \$ %s',n,str)) set(gca,'xtick',payoffs); for k = 1:length(payoffs) if payoffs(k)==15, y = -.12; else, y = -.08; end text(payoffs(k)-6.5,y,sprintf('%9.4f',p(k))); end % Mean and confidence interval, relative to unit bet r = payoffs/bet; mu = p*r'; crit = 1.96; % norminv(.975) rho = crit*sqrt((p*(r.^2)'-mu^2)/n); pm = char(177); text(20,.3,sprintf('%6.4f %c %6.4f',mu,pm,rho)); drawnow end end set(gcf,'userdata',S); % ------------------------ function c = deal % Deal one card persistent deck ncards if isempty(deck) | ncards < 6 % Four decks deck = [1:52 1:52 1:52 1:52]; % Shuffle ncards = length(deck); deck = deck(randperm(ncards)); end c = deck(ncards); ncards = ncards - 1; % ------------------------ function v = valuehard(X) % Evaluate hand X = mod(X-1,13)+1; X = min(X,10); v = sum(X); % ------------------------ function v = value(X) % Evaluate hand X = mod(X-1,13)+1; X = min(X,10); v = sum(X); % Promote soft ace if any(X==1) & v<=11 v = v + 10; end % ------------------------ function [P,bet] = playhand(hand,P,D,bet) % Play player's hand while value(P) < 21 % 0 = stand % 1 = hit % 2 = double down if any(mod(P,13)==1) & valuehard(P)<=10 strat = soft(value(P)-11,value(D(1))); else strat = hard(value(P),value(D(1))); end if length(P) > 2 & strat == 2 strat = 1; end if ~isempty(hand) show(hand,P) show('Dealer',D) strat = bjbuttonclick('hit',strat+1,length(P)>2); end switch strat case 0 break case 1 P = [P deal]; case 2 % Double down. % Double bet and get one more card bet = 2*bet; P = [P deal]; break otherwise break end end % ------------------------ function s = payoff(who,P,D,split,bet) % Payoff detail = ~isempty(who); fs = 20; valP = value(P); valD = value(D); if valP == 21 & length(P) == 2 & ... ~(valD == 21 & length(D) == 2) & ~split s = 1.5*bet; if detail, str = ['BLACKJACK: +' int2str(s)]; end elseif valP > 21 s = -bet; if detail, str = ['BUST: ' int2str(s)]; end elseif valD > 21 s = bet; str = ['WIN: +' int2str(s)]; if detail text(min(1.5*length(D)-4.5,2.75),-2.5,'BUST','fontsize',fs) end elseif valD > valP s = -bet; if detail, str = ['LOSE: ' int2str(s)]; end elseif valD < valP s = bet; if detail, str = ['WIN: +' int2str(s)]; end else s = 0; if detail, str = 'PUSH'; end end if detail x = min(1.5*length(P)-4.5,2.75); if isequal(who,'Player') y = 2.5; else y = 0; end text(x,y,str,'fontsize',fs) end % ------------------------ function show(who,H) % Displays one hand switch who case 'Player', y = 2.5; case 'Split', y = 0; case 'Dealer', y = -2.5; end x = -4; for j = 1:length(H) card(x,y,H(j),length(H)) x = x + 1.5; end % ------------------------ function card(x,y,v,gray) % card(x,y,v) plots v-th card at position (x,y). z = exp((0:16)/16*pi/2*i)/16; edge = [z+1/2+7*i/8 i*z-1/2+7*i/8 -z-1/2-7*i/8 -i*z+1/2-7*i/8 9/16+7*i/8]; pips = {'A','2','3','4','5','6','7','8','9','10','J','Q','K'}; if v <= 0 % Hole card patch(real(edge)+x,imag(edge)+y,[0 0 2/3]) else fs = 20; s = ceil(v/13); v = mod(v-1,13)+1; x1 = x; if v==10, x1 = x1-.2; end offwhite = [1 1 1]; if y == 0 & gray == 1, offwhite = [.75 .75 .75]; end patch(real(edge)+x,imag(edge)+y,offwhite) switch s case {1,4}, redblack = [0 0 0]; case {2,3}, redblack = [2/3 0 0]; end if 0 % PC has symbol font with card suits. text(x1-.2,y,pips{v},'fontname','courier','fontsize',fs, ... 'fontweight','bold','color',redblack) text(x,y+.025,char(166+s),'fontname','symbol','fontsize',fs, ... 'color',redblack) else text(x1-.1,y,pips{v},'fontname','courier','fontsize',fs, ... 'fontweight','bold','color',redblack) end end % ------------------------ function val = bjbuttonclick(kase,basic,disable) bjb = bjbuttons(kase); if nargin == 3 & disable set(bjb(3),'enable','off') end if nargin >= 2 set(bjb(basic),'fore','red') end while all(cell2mat(get(bjb,'val')) == 0) drawnow end val = find(cell2mat(get(bjb,'val')))-1; % ------------------------ function bjb = bjbuttons(kase) bjb = findobj(gcf,'style','toggle'); if isempty(bjb) for b = 3:-1:1 bjb(b,1) = uicontrol('units','normal','style','toggle', ... 'pos',[.95-.18*b .02 .16 .08],'fontweight','bold'); end end set(bjb,'fore','black') switch kase case {1,2} switch kase case 1 fs = 12; y = .02; dy = .08; case 2 fs = 10; y = .01; dy = .06; end for b = 1:3 set(bjb(b),'pos',[.95-.18*b y .16 dy]) end set(bjb,'val',0,'vis','on','enable','on','fontsize',fs) set(bjb(1),'string','Close') set(bjb(2),'string','Play') set(bjb(3),'string','Simulate') set(bjb(kase+1),'fore','red') case 'detail' set(bjb(1:2),'vis','on') set(bjb(3),'vis','off') for b = 1:3 set(bjb(b),'pos',[.95-.18*b .02 .16 .08]) end set(bjb,'val',0,'fontsize',12) case 'off' set(bjb,'vis','off') case 'split' set(bjb,'val',0,'fontsize',12) set(bjb(1),'string','Keep') set(bjb(2),'string','Split') case 'hit' set(bjb,'val',0,'vis','on','fontsize',12) set(bjb(1),'string','Stand') set(bjb(2),'string','Hit') set(bjb(3),'string','Double') end % ------------------------ function strat = hard(p,d) % Strategy for hands without aces. % strategy = hard(player's_total,dealer's_upcard) % 0 = stand % 1 = hit % 2 = double down persistent HARD if isempty(HARD) n = NaN; % Not possible % Dealer shows: % 2 3 4 5 6 7 8 9 T A HARD = [ ... 1 n n n n n n n n n n 2 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 1 1 1 7 1 1 1 1 1 1 1 1 1 1 8 1 1 1 1 1 1 1 1 1 1 9 2 2 2 2 2 1 1 1 1 1 10 2 2 2 2 2 2 2 2 1 1 11 2 2 2 2 2 2 2 2 2 2 12 1 1 0 0 0 1 1 1 1 1 13 0 0 0 0 0 1 1 1 1 1 14 0 0 0 0 0 1 1 1 1 1 15 0 0 0 0 0 1 1 1 1 1 16 0 0 0 0 0 1 1 1 1 1 17 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 0 0 20 0 0 0 0 0 0 0 0 0 0]; end strat = HARD(p,d); % ------------------------ function strat = soft(p,d) % Strategy array for hands with aces. % strategy = soft(player's_total,dealer's_upcard) % 0 = stand % 1 = hit % 2 = double down persistent SOFT if isempty(SOFT) n = NaN; % Not possible % Dealer shows: % 2 3 4 5 6 7 8 9 T A SOFT = [ ... 1 n n n n n n n n n n 2 1 1 2 2 2 1 1 1 1 1 3 1 1 2 2 2 1 1 1 1 1 4 1 1 2 2 2 1 1 1 1 1 5 1 1 2 2 2 1 1 1 1 1 6 2 2 2 2 2 1 1 1 1 1 7 0 2 2 2 2 0 0 1 1 0 8 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0]; end strat = SOFT(p,d); % ------------------------ function strat = pair(p,d) % Strategy for splitting pairs % strategy = pair(paired_card,dealer's_upcard) % 0 = keep pair % 1 = split pair persistent PAIR if isempty(PAIR) n = NaN; % Not possible % Dealer shows: % 2 3 4 5 6 7 8 9 T A PAIR = [ ... 1 n n n n n n n n n n 2 1 1 1 1 1 1 0 0 0 0 3 1 1 1 1 1 1 0 0 0 0 4 0 0 0 1 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 6 1 1 1 1 1 1 0 0 0 0 7 1 1 1 1 1 1 1 0 0 0 8 1 1 1 1 1 1 1 1 1 1 9 1 1 1 1 1 0 1 1 0 0 10 0 0 0 0 0 0 0 0 0 0 11 1 1 1 1 1 1 1 1 1 1]; end strat = PAIR(p,d);