blackjack.m

有趣的可视的数值方法 出自网站http://www.mathworks.com/moler

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.

clf
shg
set(gcf,'name','Blackjack','menu','none','numbertitle','off', ...
   'double','on','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,'erasemode','none');
         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

% ------------------------