libreblackjack/doc/blackjack.texi

\input texinfo @c -*-texinfo-*-
@c %**start of header
@documentencoding UTF-8

@setfilename blackjack.info
@set UPDATED December 20, 2020
@set VERSION v0.2.4-gafba392

@copying
This manual is for LibreBlackjack (version @value{VERSION}, @value{UPDATED}),
which is a completely free-as-in-freedom blackjack engine designed to
study and analyze the game statistics using different playing strategies
ranging from simple card counting up to other complex algorithms based
on artificial intelligence..

Copyright @copyright{} 2016-2020 Jeremy Theler.

@quotation
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3 or
any later version published by the Free Software Foundation; with no
Invariant Sections, with no Front-Cover Texts, and with no Back-Cover
Texts.  A copy of the license is included in the section entitled
``GNU Free Documentation License''.
@end quotation
@end copying

@ifnottex
@paragraphindent 0
@end ifnottex
@titlepage
@title Libre@ Blackjack
@subtitle A free blackjack back end, v0.2.4-gafba392
@author Jeremy Theler
December 20, 2020
@page
@vskip 0pt plus 1filll
@insertcopying
@end titlepage

@contents

@node Top
@top Libre@ Blackjack

@menu
* Overview::
* Running blackjack::
* Commands::
* Configuration file::
* Example automated players::
@end menu

@node Overview
@chapter Overview
@anchor{#overview}
@quotation
A @uref{https://www.gnu.org/philosophy/free-sw.html,free}
@uref{https://en.wikipedia.org/wiki/Blackjack,Blackjack} back end
inspired by @uref{https://www.gnu.org/software/chess/,GNU Chess}.
@end quotation
@uref{https://www.seamplex.com/blackjack,Libre@ Blackjack} is a
blackjack engine that emulates a dealer, deals (digital) cards and
understands plain-text commands such as @code{hit} or @code{stand}. The
basic idea is that one or more players can talk to Libre@ Blackjack
either in an interactive or in an automated way through

@itemize
@item
the standard input and/or output (optionally using named pipes or TCP
(web)sockets with @code{netcat} or @code{gwsocket}), or
@item
C++ methods (optionally loaded at runtime from shared objects TBD).
@end itemize

These players can be actual human players playing in real-time through a
front end (a GUI application, a web-based interface, a mobile app, etc.)
or robots that implement a certain betting and playing strategy playing
(i.e.@ card counting) as fast as possible to study and analyze game
statistics. There is an internal player that reads the strategy from a
text file and plays accordingly. It can also be used to play interactive
[ASCII blackjack].

@menu
* Why::
* How::
* A note on the C++ implementation::
@end menu

@node Why
@section Why
@anchor{#why}
@quotation
``I am often surprised that when people drive down two-lane roads, they
will trust complete strangers in the oncoming lane not to swerve into
their lane causing a head-on collision; but they will not trust
mathematicians to create the correct strategy for Blackjack.''

Norman Wattenberger, Modern Blackjack, 2009
@end quotation
With Libre@ Blackjack you do not have to trust other people anymore. You
have a free blackjack engine which you can

@enumerate 0
@item
run as you wish, to see the results of billions of blackjack hands,
@item
study to see how it works and change it if you do not like it,
@item
share it with your friends and colleagues, and
@item
distribute copies of your modified versions.
@end enumerate

If you do not know how to program, you have the @emph{freedom} to hire a
programmer to do it for you. That is why
@uref{https://www.seamplex.com/blackjack,Libre@ Blackjack} is
@uref{https://www.gnu.org/philosophy/free-sw.html,free software}.

@node How
@section How
@anchor{#how}
Once you trust the blackjack engine is fair, you can model and simulate
any blackjack situation you want, playing millions of times a certain
hand (say a sixteen against a ten) in different ways (say hitting or
standing) to obtain you own conclusions. You can even build the
@uref{https://wizardofodds.com/games/blackjack/strategy/4-decks/,basic
strategy charts} from scratch to convince yourself there is no
@uref{https://wizardofodds.com/ask-the-wizard/blackjack/,flaw}.

The main objective is research and optimization of playing and betting
strategies depending on

@itemize
@item
particular table rules (number of decks, hit on soft 17, double after
split, etc.),
@item
card counting strategies
@item
risk of ruin
@item
removal of cards
@item
arranged shoes
@end itemize

These automatic players can range from simple no-bust or
mimic-the-dealer hitters or standers, up to neural-networks trained
players taking into account every card being dealt passing through basic
strategy modified by traditional card counting mechanisms.

@node A note on the C++ implementation
@section A note on the C++ implementation
@anchor{#a-note-on-the-c-implementation}
The first Libre@ Blackjack version (v0.1) was written in C. This version
(v0.2) is a re-implementation of nearly the same functionality but
written completely from scratch in C++. I am not a fan of C++ and still
prefer old plain C for most of my programming projects, but for the
particular case of Libre@ Blackjack these advantages of C++ over C ought
to be noted:

@itemize
@item
the inheritance mechanisms of C++ and virtual methods allows to have
generic dealer and player classes from which particular games (dealers)
and strategies (players) can be instantiated. This way, Blackjack
variations like

@itemize
@item
@uref{https://wizardofodds.com/games/spanish-21/,Spanish 21}
@item
@uref{https://wizardofodds.com/games/down-under-blackjack/,Down under
Blackjack}
@item
@uref{https://wizardofodds.com/games/free-bet-blackjack/,Free Bet
Blackjack}
@item
@uref{https://wizardofodds.com/games/blackjack/switch/,Blackjack Switch}
@end itemize

or even the Spanish ``Siete y medio'' could be also implemented in the
same framework (the card deck should also be changed though). But also
playing variations like a dealer that exposes the hole card a certain
amount of the time (say 1% or 2% of the hands) could also be studied by
extending the base blackjack dealer class.

@item
the private members of the C++ classes allow information to be hidden
between the dealer and the player, so a far better separation of
information can be achieved. This also prevents ``cheating'' in players
by looking at information which is not available for them (such as the
dealer's hole card or the content of the shoe).

@item
the virtual members of derived players and even be linked to other
high-level programming language parsers (such as Python or Julia)
allowing to use the vast variety of AI/ML libraries available for these
languages to implement advanced playing strategies.

@item
the usage of STL containers, methods and algorithms allows for a faster
and cleaner implementation of cards, hands, decks and shoes.

@end itemize

@node Running blackjack
@chapter Running @code{blackjack}
@anchor{#running-blackjack}

@menu
* Invocation::
* Interactive game::
@end menu

@node Invocation
@section Invocation
@anchor{#invocation}
The format for running the @code{blackjack} program is:

@verbatim
blackjack [options] [path_to_conf_file]
@end verbatim

If no configuration file is given, a file named @code{blackjack.conf} in
the current directory is used, provided it exists. With no options and
no configuration file, @code{blackjack} starts in interactive mode and
it is ready to start a blackjack game.

The @code{blackjack} executable supports the following options:

All the options which can be given in the configuration file can be
passed as a command-line argument. For example, running

@verbatim
blackjack --decks=4 --no_insurance=true
@end verbatim

is equivalent to using a configuration file with

@verbatim
decks = 4
no_insurance = true
@end verbatim

@node Interactive game
@section Interactive game
@anchor{#interactive-game}
If @code{blackjack} is attached to an interactive TTY (i.e.@ neither the
standard input nor outputs are redirected), an interactive game is
triggered. First thing the program will do is to ask for a bet:

@verbatim
LibreBlackjack v0.2+Δ
a free & open blackjack engine


Starting new hand #1 with bankroll 0
 <-- Bet?
 > 
@end verbatim

So the user should enter a number, say ``1'' and then press Enter and
then a game will be dealt:

@verbatim
Player's card is Q♥
Dealer's up card is 10♠
Player's card is Q♣
Dealer's hole card is dealt
No blackjacks
 -- Dealer's hand:  --------
 _____    _____   
|10   |  |#####|  
|     |  |#####|  
|  ♠  |  |#####|  
|     |  |#####|  
|___10|  |#####|  
    Value: 10
 -- Player's hand --------
 _____    _____   
|Q    |  |Q    |  
|     |  |     |  
|  ♥  |  |  ♣  |  
|     |  |     |  
|____Q|  |____Q|  
    Value: 20
 <-- Play? 20 10
 > 
@end verbatim

A flat-betting game can be played by passing @code{--flat_bet=true}
through the command line.

@node Commands
@chapter Commands
@anchor{#commands}
The dealer (he) and the player (she) ``talk'' through commands, which
are ASCII strings sent through any of the different IPC mechanisms
discussed in [Automated playing through IPC]. In the most basic case, a
human player reads commands from the dealer from @code{blackjack}'s
standard output and writes her commands into the dealer's standard
input. Those commands from the dealer that require a particular action
from the player end with a quotation sign such as @code{bet?},
@code{insurance?} or @code{play?}.

All numerical values such as hand totals or bankrolls are given as
decimal ASCII strings.

@menu
* From the dealer to the player::
* From the player to the dealer::
@end menu

@node From the dealer to the player
@section From the dealer to the player
@anchor{#from-the-dealer-to-the-player}

@node From the player to the dealer
@section From the player to the dealer
@anchor{#from-the-player-to-the-dealer}
The following commands are available for the player for playing her
hand.

The following are general commands in the sense that they can be sent
from the player to the dealer at any moment of the game.

@node Configuration file
@chapter Configuration file
@anchor{#configuration-file}
Libre@ Blackjacks reads a configuration file that contains

@itemize
@item
settings about the rules of the game
@itemize
@item
number of decks,
@item
whether if the dealer has to hit soft seventeen or not,
@item
blackjack payout,
@item
maximum bet allowed,
@item
etc@dots{}
@end itemize

@item
how the player is supposed to play
@itemize
@item
number of hands
@item
whether a flat or variable bet is going to be used,
@item
etc@dots{}
@end itemize

@item
if there are any particular shoe arrangement, i.e.@ a predefined set of
cards dealt in a certain order for instance to play one million hands of
a sixteen against a dealer's ten
@item
what kind of information is shown in the interactive session
@itemize
@item
if ASCII-art cards are supposed to be shown,
@item
a real-time delay to make the game smoother,
@item
etc.
@end itemize

@item
how the automated player communicates with the dealer
@itemize
@item
using standard input/output,
@item
FIFO named pipes,
@item
POSIX message queues,
@item
POSIX shared memory,
@item
etc@dots{}
@end itemize

@end itemize

The location of the configuration file can be given in the command line.
If none is provided, a file named @code{blackjack.conf} in the current
directory is used. If such file does not exists, the defaults values of
each variable are used. Individual variables can be set from the command
line by passing one or more times the option
@code{--}configuration_variable@code{[=}@emph{value}@code{]} in the
@ref{#invocation,invocation}.

Comments can be inserted using either a hash @code{#} or a colon
@code{;}. The following configuration file is the default provided in
the main distribution tarball:

@verbatim
# uncomment the following line to arrange cards
# arranged_cards = 1 5 14 9 27

flat_bet = 1       # do not ask for bets
no_insurance = 1   # do not ask for insurance
decks = 1          # number of decks, negative means infinite
@end verbatim

@menu
* Variables and values::
* Dumb internal player::
@end menu

@node Variables and values
@section Variables and values
@anchor{#variables-and-values}

@node Dumb internal player
@section Dumb internal player
@anchor{#dumb-internal-player}

@node Example automated players
@chapter Example automated players
@anchor{#example-automated-players}
The directory @code{players} contains a few examples of automated
player, which are discussed in the following sections. A script
@code{check.sh} runs some of them and compares the expected value of the
bankroll relative to the number of hands which each player obtains with
the theoretical expected value (according to the game rules and player's
strategy) within an allowed statistical uncertainty. This scripts writes
the following table (actual values might vary depending on the random
nature of the game):

@multitable {Mimic the dealer} {Expected} {-0.0064075} {0.00927449} {Status} 
@headitem 
Case
 @tab Expected
 @tab Result
 @tab Error
 @tab Status
@item 
Internal player
 @tab -0.004
 @tab -0.0064075
 @tab 0.00348953
 @tab ok
@item 
Always stand
 @tab -0.150
 @tab -0.15539
 @tab 0.00940931
 @tab ok
@item 
No-bust strategy
 @tab -0.075
 @tab -0.082075
 @tab 0.00942851
 @tab ok
@item 
Mimic the dealer
 @tab -0.055
 @tab -0.0571
 @tab 0.00927449
 @tab ok
@end multitable

The columns are

@enumerate 
@item
Case name, as discussed below.
@item
Expected theoretical result in absolute units (i.e.@ -0.004 means -0.4%)
@item
Actual result obtained by Libre@ Blackjack in absolute units
@item
Estimated error equal to the standard deviation of the result
@item
Number of played hands per second
@item
Whether the result coincides or not with the theoretical value
@end enumerate

@menu
* Internal player::
* Always stand::
* No-bust strategy::
* Mimic the dealer::
* Derivation of the basic strategy::
@end menu

@node Internal player
@section Internal player
@anchor{#internal-player}
If @code{blackjack} is called with the @code{-i} option, it uses an
@emph{internal} player to play against itself. By default it plays basic
strategy. Run

@verbatim
blackjack -i
@end verbatim

and you will get the following report with the results of playing one
million hands with basic strategy.

@verbatim
---
result: "(-0.7 ± 0.3) %"
mean: -0.006799
error: 0.00348707
hands: 1e+06
bankroll: -6799
bustsPlayer: 0.139358
bustsDealer: 0.239722
wins: 0.448034
pushes: 0.085913
losses: 0.495532
...
@end verbatim

@node Always stand
@section Always stand
@anchor{#always-stand}
To play Blackjack as an ``always-stander'' run the following command:

@verbatim
yes stand | blackjack -n1e5 --flat_bet=true --no_insurance=true > /dev/null
@end verbatim

The UNIX command @code{yes stand} writes the string ``stand'' repeteadly
to the standard output, which is piped to the executable
@code{blackjack} (assumed to be installed system-wide). The arguments
tell Libre@ Blackjack to play one hundred thousand hands (@code{-n1e5})
using a flat bet (@code{flat_bet}, it defaults to a unit bet in each
hand) and without asking for insurance if the dealer shows an ace
(@code{no_insurance}). As there is no @code{blackjack.conf} file, the
rules are---as expected---the default ones (see the documentation for
details).

The @code{/dev/null} part is important, otherwise Libre@ Blackjack will
think that there is a human at the other side of the table and will

@enumerate 
@item
run slower (it will add explicit time delays to mimic an actual human
dealer), and
@item
give all the details of the dealt hands in the terminal as ASCII
(actually UTF-8) art
@end enumerate

This example is only one-way (i.e.@ the player ignores what the dealer
says) so it is better to redirect the standard output to
@code{/dev/null} to save execution time. The results are written as a
@uref{http://yaml.org/,YAML}-formatted data to @code{stderr} by default
once the hands are over, so they will show up in the terminal
nevertheless. This format is human-friendly (far more than JSON) so it
can be easily parsed, but it also allows complex objects to be
represented (arrays, lists, etc.).

@verbatim
---
result: "(-15.8 ± 0.9) %"
mean: -0.157675
error: 0.00940803
hands: 100000
bankroll: -15767.5
bustsPlayer: 0
bustsDealer: 0.27344
wins: 0.38585
pushes: 0.04807
losses: 0.56608
...
@end verbatim

@quotation
@strong{Exercise:} verify that the analytical probability of getting a
natural playing with a single deck (for both the dealer and the player)
is 32/663 = 0.04826546@dots{}
@end quotation

@node No-bust strategy
@section No-bust strategy
@anchor{#no-bust-strategy}
This directory shows how to play a ``no-bust'' strategy, i.e.@ not
hitting any hand higher or equal to hard twelve with Libre@ Blackjack.
The communication between the player and the back end is through
standard input and output. The player reads from its standard input
Libre@ Blackjack's commands and writes to its standard output the
playing commands. In order to do this a FIFO (a.k.a. named pipe) is
needed. So first, we create it (if it is not already created):

@verbatim
mkfifo fifo
@end verbatim

Then we execute @code{blackjack}, piping its output to the player (say
@code{no-bust.pl}) and reading the standard input from @code{fifo},
whilst at the same time we redirect the player's standard output to
@code{fifo}:

@verbatim
rm -f fifo; mkfifo fifo
blackjack -n1e5 < fifo | ./no-bust.pl > fifo
@end verbatim

As this time the player is coded in an interpreted langauge, it is far
smarter than the previous @code{yes}-based player. So the player can
handle bets and insurances, and there is not need to pass the options
@code{--flat_bet} nor @code{--no_insurance} (though they can be passed
anyway). Let us take a look at the Perl implementation:

@verbatim
#!/usr/bin/perl
# this is needed to avoid deadlock with the fifo
STDOUT->autoflush(1);

while ($command ne "bye") {
  # do not play more than a number of commands
  # if the argument -n was not passed to blackjack
  if ($i++ == 1234567) {
    print "quit\n";
  }
  
  # read and process the commands
  chomp($command = <STDIN>);
  
  if ($command eq "bet?") {
    print "1\n";
  } elsif ($command eq "insurance?") {
    print "no\n";
  } elsif ($comm eq "play?") {
    @tokens = split(/ /, $command);
    if ($tokens[1] < 12) {
      print "hit\n";
    } else {
      print "stand\n";
    }
  }
}
@end verbatim

The very same player may be implemented as a shell script:

@verbatim
#!/bin/sh

i=0
while read command
do
  i=$((i+1))
  if test ${i} -ge 12345; then
    echo "quit"
  elif test "${command}" = 'bye'; then
    exit
  elif test "${command}" = 'bet?'; then
    echo 1  
  elif test "${command}" = 'insurance?'; then
    echo "no"
  elif test "$(echo ${command} | cut -c-5)" = 'play?'; then
    count=$(echo ${command} | cut -f2 -d" ")
    if test ${count} -lt 12; then
      echo "hit"
    else
      echo "stand"
    fi
  fi
done
@end verbatim

To check these two players give the same results, make them play against
Libre@ Blackjack with the same seed (say one) and send the YAML report
to two different files:

@verbatim
blackjack -n1e5 --rng_seed=1 --report_file_path=perl.yml  < fifo | ./no-bust.pl  > fifo
blackjack -n1e5 --rng_seed=1 --report_file_path=shell.yml < fifo | ./no-bust.awk > fifo
diff perl.yml shell.yml 
@end verbatim

As expected, the reports are the same. They just differ in the speed
because the shell script is orders of magnitude slower than its
Perl-based counterpart.

@quotation
@strong{Exercise:} modify the players so they always insure aces and see
if it improves or degrades the result.
@end quotation

@node Mimic the dealer
@section Mimic the dealer
@anchor{#mimic-the-dealer}
This example implements a ``mimic-the-dealer strategy,'' i.e.@ hits if
the hand totals less than seventeen and stands on eighteen or more. The
player stands on hard seventeen but hits on soft seventeen.

This time, the configuration file @code{blackjack.conf} is used. If a
file with this name exists in the directory where @code{blackjack} is
executed, it is read and parsed. The options should be fairly self
descriptive. See the [configuration file] section of the manual for a
detailed explanation of the variables and values that can be entered. In
particular, we ask to play one hundred thousand hands at a six-deck game
where the dealer hits soft seventeens. If the random seed is set to a
fixed value so each execution will lead to the very same sequence of
cards.

Now, there are two options that tell Libre@ Blackjack how the player is
going to talk to the backend: @code{player2dealer} and
@code{dealer2player}. The first one sets the communication mechanism
from the player to the dealer (by default is @code{blackjack}'s standard
input), and the second one sets the mechanism from the dealer to the
player (by default @code{blackjack}'s standard output). In this case,
the configuration file reads:

@verbatim
h17 = true
@end verbatim

This means that two FIFOs (a.k.a. named pipes) are to be used for
communication, @code{player2dealer} from the player to the dealer and
@code{dealer2player} for the dealer to the player. If these FIFOs do not
exist, they are created by @code{blackjack} upon execution.

The player this time is implemented as an awk script, whose input should
be read from @code{dealer2player} and whose output should be written to
@code{player2dealer}. To run the game, execute @code{blackjack} in one
terminal making sure the current directory is where the
@code{blackjack.conf} file exists. It should print a message telling
that it is waiting for someone to be at the other side of the named
pipes:

@verbatim
$ blackjack
[...]
waiting for dealer2player buffered fifo 'dealer2player'...
@end verbatim

In another terminal run the player

@verbatim
$ ./mimic-the-dealer.awk < dealer2player > player2dealer
@end verbatim

Both dealer and player may be run in the same terminal putting the first
one on the background:

@verbatim
rm -f d2p p2d; mkfifo d2p p2d
gawk -f mimic-the-dealer.awk < d2p > p2d &
blackjack -n1e5 > d2p < p2d 
@end verbatim

To understand the decisions taken by the player, we have to remember
that when Libre@ Blackjack receives the command @code{count} asking for
the current player's count, it returns a positive number for hard hands
and a negative number for soft hands. The instructions @code{fflush()}
are needed in order to avoid deadlocks on the named pipes:

@verbatim
#!/usr/bin/gawk -f
function abs(x){return ( x >= 0 ) ? x : -x } 

/bet\?/ {
  print "1";
  fflush();
}

/insurance\?/ {
  print "no";
  fflush();
}

/play\?/ {
  # mimic the dealer: hit until 17 (hit soft 17)
  if (abs($2) < 17 || $2 == -17) {   # soft hands are negative
    print "hit";
  } else {
    print "stand";
  }
  fflush();  
}

/bye/ {
  exit;
}
@end verbatim

@verbatim
---
result: "(-5.7 ± 0.9) %"
mean: -0.05716
error: 0.00926292
hands: 100000
bankroll: -5716
bustsPlayer: 0.27064
bustsDealer: 0.18905
wins: 0.41088
pushes: 0.09888
losses: 0.49024
...
@end verbatim

@quotation
@strong{Exercise:} modify the player and the configuration file so both
the dealer and the player may stand on soft seventeen. Analyze the four
combinations (player h17 - dealer h17, player h17 - dealer s17, player
s17 - dealer h17, player s17 - dealer s17)
@end quotation

@node Derivation of the basic strategy
@section Derivation of the basic strategy
@anchor{#derivation-of-the-basic-strategy}

@menu
* Quick run::
* Full table with results::
* Detailed explanation::
* Implementation::
@end menu

@node Quick run
@subsection Quick run
@anchor{#quick-run}
Execute the @code{run.sh} script. It should take a few minutes:

@verbatim
$ ./run.sh
h20-2 (10 10)   8.0e+04 +63.23  (1.1)   -171.17 (1.1)   -85.32  (0.5)   stand
h20-3 (10 10)   8.0e+04 +64.54  (1.1)   -171.50 (1.1)   -85.50  (0.5)   stand
h20-4 (10 10)   8.0e+04 +65.55  (1.1)   -170.33 (1.1)   -85.50  (0.5)   stand
h20-5 (10 10)   8.0e+04 +66.65  (1.1)   -171.25 (1.1)   -85.51  (0.5)   stand
h20-6 (10 10)   8.0e+04 +67.80  (1.1)   -171.07 (1.1)   -85.59  (0.5)   stand
h20-7 (10 10)   8.0e+04 +77.44  (1.1)   -170.53 (1.1)   -85.44  (0.5)   stand
h20-8 (10 10)   8.0e+04 +79.11  (1.1)   -170.08 (1.1)   -85.02  (0.6)   stand
h20-9 (10 10)   8.0e+04 +75.77  (1.1)   -170.31 (1.1)   -84.87  (0.6)   stand
[...]
p2-6            8e+04   +24.78  (2.9)   +3.07   (1.0)   yes
p2-7            8e+04   +1.48   (2.0)   -8.90   (1.0)   yes
p2-8            8e+04   -17.57  (2.0)   -16.33  (1.0)   uncertain
p2-8            3e+05   -17.88  (1.0)   -16.10  (0.5)   no
p2-9            8e+04   -38.73  (2.0)   -24.38  (1.0)   no
p2-T            8e+04   -54.45  (1.8)   -34.92  (0.9)   no
p2-A            8e+04   -67.11  (1.5)   -51.59  (0.9)   no
@end verbatim

A new text file called @code{bs.txt} with the strategy should be created
from scratch:

@verbatim
#    2  3  4  5  6  7  8  9  T  A
h20  s  s  s  s  s  s  s  s  s  s  
h19  s  s  s  s  s  s  s  s  s  s  
h18  s  s  s  s  s  s  s  s  s  s  
h17  s  s  s  s  s  s  s  s  s  s  
h16  s  s  s  s  s  h  h  h  h  h  
h15  s  s  s  s  s  h  h  h  h  h  
h14  s  s  s  s  s  h  h  h  h  h  
h13  s  s  s  s  s  h  h  h  h  h  
h12  h  h  s  s  s  h  h  h  h  h  
h11  d  d  d  d  d  d  d  d  d  d  
h10  d  d  d  d  d  d  d  d  h  h  
h9   h  d  d  d  d  h  h  h  h  h  
h8   h  h  h  h  h  h  h  h  h  h  
h7   h  h  h  h  h  h  h  h  h  h  
h6   h  h  h  h  h  h  h  h  h  h  
h5   h  h  h  h  h  h  h  h  h  h  
h4   h  h  h  h  h  h  h  h  h  h  
#    2  3  4  5  6  7  8  9  T  A
s20  s  s  s  s  s  s  s  s  s  s  
s19  s  s  s  s  d  s  s  s  s  s  
s18  d  d  d  d  d  s  s  h  h  h  
s17  h  d  d  d  d  h  h  h  h  h  
s16  h  h  d  d  d  h  h  h  h  h  
s15  h  h  d  d  d  h  h  h  h  h  
s14  h  h  h  d  d  h  h  h  h  h  
s13  h  h  h  h  d  h  h  h  h  h  
s12  h  h  h  h  d  h  h  h  h  h  
#    2  3  4  5  6  7  8  9  T  A
pA   y  y  y  y  y  y  y  y  y  y  
pT   n  n  n  n  n  n  n  n  n  n  
p9   y  y  y  y  y  n  y  y  n  n  
p8   y  y  y  y  y  y  y  y  y  y  
p7   y  y  y  y  y  y  n  n  n  n  
p6   y  y  y  y  y  n  n  n  n  n  
p5   n  n  n  n  n  n  n  n  n  n  
p4   n  n  n  y  y  n  n  n  n  n  
p3   y  y  y  y  y  y  n  n  n  n  
p2   y  y  y  y  y  y  n  n  n  n  
@end verbatim

@node Full table with results
@subsection Full table with results
@anchor{#full-table-with-results}
The script computes the expected value of each combination

@enumerate 
@item
Player's hand (hard, soft and pair)
@item
Dealer upcard
@item
Hit, double or stand (for hard and soft hands) and splitting or not (for
pairs)
@end enumerate

The results are given as the expected value in percentage with the
uncertainty (one standard deviation) in the last significant digit.

@ Hand | @math{n} | Stand [%] | Double [%] | Hit [%] | Play |@*
h20-2 | 8.0e+04 | +63.23 (1.1) | -85.32 (0.5) | -171.17 (1.1) | stand
|@*
h20-3 | 8.0e+04 | +64.54 (1.1) | -85.50 (0.5) | -171.50 (1.1) | stand
|@*
h20-4 | 8.0e+04 | +65.55 (1.1) | -85.50 (0.5) | -170.33 (1.1) | stand
|@*
h20-5 | 8.0e+04 | +66.65 (1.1) | -85.51 (0.5) | -171.25 (1.1) | stand
|@*
h20-6 | 8.0e+04 | +67.80 (1.1) | -85.59 (0.5) | -171.07 (1.1) | stand
|@*
h20-7 | 8.0e+04 | +77.44 (1.1) | -85.44 (0.5) | -170.53 (1.1) | stand
|@*
h20-8 | 8.0e+04 | +79.11 (1.1) | -85.02 (0.6) | -170.08 (1.1) | stand
|@*
h20-9 | 8.0e+04 | +75.77 (1.1) | -84.87 (0.6) | -170.31 (1.1) | stand
|@*
h20-T | 8.0e+04 | +43.47 (1.1) | -85.86 (0.5) | -164.14 (1.1) | stand
|@*
h20-A | 8.0e+04 | +10.32 (1.0) | -89.73 (0.5) | -149.01 (1.0) | stand
|@*
h19-2 | 8.0e+04 | +37.61 (1.4) | -73.52 (0.7) | -146.58 (1.4) | stand
|@*
h19-3 | 8.0e+04 | +39.34 (1.4) | -72.58 (0.7) | -145.41 (1.4) | stand
|@*
h19-4 | 8.0e+04 | +41.60 (1.4) | -72.88 (0.7) | -145.28 (1.4) | stand
|@*
h19-5 | 8.0e+04 | +43.51 (1.4) | -72.98 (0.7) | -145.12 (1.4) | stand
|@*
h19-6 | 8.0e+04 | +45.41 (1.4) | -72.56 (0.7) | -144.94 (1.4) | stand
|@*
h19-7 | 8.0e+04 | +61.77 (1.5) | -71.07 (0.7) | -142.94 (1.5) | stand
|@*
h19-8 | 8.0e+04 | +59.05 (1.5) | -71.22 (0.7) | -143.04 (1.5) | stand
|@*
h19-9 | 8.0e+04 | +28.52 (1.5) | -71.93 (0.7) | -143.09 (1.5) | stand
|@*
h19-T | 8.0e+04 | -1.96 (1.4) | -75.09 (0.7) | -142.54 (1.4) | stand |@*
h19-A | 8.0e+04 | -17.95 (1.2) | -80.99 (0.6) | -131.78 (1.2) | stand
|@*
h18-2 | 8.0e+04 | +10.97 (1.6) | -61.93 (0.8) | -124.23 (1.6) | stand
|@*
h18-3 | 8.0e+04 | +13.15 (1.6) | -62.50 (0.8) | -124.98 (1.6) | stand
|@*
h18-4 | 8.0e+04 | +16.93 (1.6) | -61.94 (0.8) | -123.60 (1.6) | stand
|@*
h18-5 | 8.0e+04 | +19.25 (1.6) | -62.22 (0.8) | -123.74 (1.6) | stand
|@*
h18-6 | 8.0e+04 | +22.23 (1.6) | -61.47 (0.8) | -122.43 (1.6) | stand
|@*
h18-7 | 8.0e+04 | +39.60 (1.7) | -58.93 (0.8) | -118.77 (1.7) | stand
|@*
h18-8 | 8.0e+04 | +11.02 (1.7) | -59.17 (0.8) | -118.63 (1.7) | stand
|@*
h18-9 | 8.0e+04 | -18.50 (1.6) | -61.67 (0.8) | -122.96 (1.6) | stand
|@*
h18-T | 8.0e+04 | -24.23 (1.5) | -67.40 (0.8) | -126.70 (1.5) | stand
|@*
h18-A | 8.0e+04 | -46.34 (1.3) | -74.66 (0.7) | -118.34 (1.3) | stand
|@*
h17-2 | 8.0e+04 | -16.04 (1.7) | -53.84 (0.9) | -107.43 (1.7) | stand
|@*
h17-3 | 8.0e+04 | -11.84 (1.7) | -53.28 (0.9) | -107.23 (1.7) | stand
|@*
h17-4 | 8.0e+04 | -7.85 (1.8) | -52.66 (0.9) | -105.90 (1.8) | stand |@*
h17-5 | 8.0e+04 | -4.57 (1.8) | -52.56 (0.9) | -103.96 (1.8) | stand |@*
h17-6 | 8.0e+04 | -0.04 (1.8) | -51.63 (0.9) | -104.55 (1.8) | stand |@*
h17-7 | 8.0e+04 | -10.94 (1.8) | -48.70 (0.9) | -96.41 (1.8) | stand |@*
h17-8 | 8.0e+04 | -37.84 (1.8) | -50.88 (0.9) | -101.70 (1.8) | stand
|@*
h17-9 | 8.0e+04 | -42.50 (1.7) | -55.60 (0.8) | -109.70 (1.7) | stand
|@*
h17-T | 8.0e+04 | -46.71 (1.6) | -61.73 (0.8) | -116.13 (1.6) | stand
|@*
h17-A | 8.0e+04 | -66.37 (1.4) | -70.97 (0.7) | -111.14 (1.4) | stand
|@*
h16-2 | 8.0e+04 | -28.77 (1.8) | -47.30 (0.9) | -95.43 (1.8) | stand |@*
h16-3 | 8.0e+04 | -24.29 (1.8) | -46.54 (0.9) | -92.83 (1.8) | stand |@*
h16-4 | 8.0e+04 | -20.04 (1.8) | -46.18 (0.9) | -91.87 (1.8) | stand |@*
h16-5 | 8.0e+04 | -16.86 (1.8) | -45.19 (0.9) | -89.29 (1.8) | stand |@*
h16-6 | 8.0e+04 | -11.67 (1.9) | -44.23 (0.9) | -88.78 (1.9) | stand |@*
h16-7 | 8.0e+04 | -47.23 (1.9) | -41.86 (0.9) | -83.70 (1.9) | hit |@*
h16-8 | 8.0e+04 | -51.20 (1.8) | -46.31 (0.9) | -91.59 (1.8) | hit |@*
h16-9 | 8.0e+04 | -54.18 (1.7) | -51.19 (0.9) | -102.23 (1.7) | hit |@*
h16-T | 2.0e+07 | -57.57 (0) | -57.55 (0) | -107.33 (0) | hit |@*
h16-A | 8.0e+04 | -72.19 (1.4) | -68.30 (0.7) | -105.39 (1.4) | hit |@*
h15-2 | 8.0e+04 | -28.22 (1.9) | -41.63 (0.9) | -82.97 (1.9) | stand |@*
h15-3 | 8.0e+04 | -24.40 (1.9) | -41.04 (0.9) | -80.75 (1.9) | stand |@*
h15-4 | 8.0e+04 | -20.61 (1.9) | -39.80 (0.9) | -79.58 (1.9) | stand |@*
h15-5 | 8.0e+04 | -16.64 (1.9) | -38.18 (1.0) | -76.95 (1.9) | stand |@*
h15-6 | 8.0e+04 | -12.54 (1.9) | -37.21 (1.0) | -75.91 (1.9) | stand |@*
h15-7 | 8.0e+04 | -47.73 (1.9) | -37.35 (0.9) | -73.94 (1.9) | hit |@*
h15-8 | 8.0e+04 | -50.75 (1.9) | -41.05 (0.9) | -84.62 (1.9) | hit |@*
h15-9 | 8.0e+04 | -54.44 (1.8) | -47.05 (0.9) | -95.42 (1.8) | hit |@*
h15-T | 8.0e+04 | -57.26 (1.7) | -54.23 (0.9) | -100.92 (1.7) | hit |@*
h15-A | 8.0e+04 | -72.42 (1.4) | -66.47 (0.8) | -101.91 (1.4) | hit |@*
h14-2 | 8.0e+04 | -28.04 (1.9) | -36.34 (1.0) | -71.61 (1.9) | stand |@*
h14-3 | 8.0e+04 | -24.76 (1.9) | -34.74 (1.0) | -70.47 (1.9) | stand |@*
h14-4 | 8.0e+04 | -20.69 (1.9) | -33.14 (1.0) | -68.04 (1.9) | stand |@*
h14-5 | 8.0e+04 | -16.68 (2.0) | -31.75 (1.0) | -64.69 (2.0) | stand |@*
h14-6 | 8.0e+04 | -12.21 (2.0) | -30.33 (1.0) | -60.55 (2.0) | stand |@*
h14-7 | 8.0e+04 | -47.78 (1.9) | -31.73 (1.0) | -66.56 (1.9) | hit |@*
h14-8 | 8.0e+04 | -50.59 (1.9) | -36.85 (0.9) | -75.91 (1.9) | hit |@*
h14-9 | 8.0e+04 | -54.02 (1.8) | -43.78 (0.9) | -87.29 (1.8) | hit |@*
h14-T | 8.0e+04 | -57.53 (1.7) | -50.98 (0.9) | -94.12 (1.7) | hit |@*
h14-A | 8.0e+04 | -71.95 (1.5) | -63.33 (0.8) | -97.81 (1.5) | hit |@*
h13-2 | 3.2e+05 | -28.71 (1.0) | -30.93 (0.5) | -61.09 (1.0) | stand |@*
h13-3 | 8.0e+04 | -24.71 (2.0) | -29.25 (1.0) | -57.88 (2.0) | stand |@*
h13-4 | 8.0e+04 | -20.47 (2.0) | -27.32 (1.0) | -55.06 (2.0) | stand |@*
h13-5 | 8.0e+04 | -16.25 (2.0) | -25.88 (1.0) | -52.01 (2.0) | stand |@*
h13-6 | 8.0e+04 | -11.78 (2.0) | -24.12 (1.0) | -47.83 (2.0) | stand |@*
h13-7 | 8.0e+04 | -47.69 (2.0) | -26.36 (1.0) | -58.47 (2.0) | hit |@*
h13-8 | 8.0e+04 | -51.19 (1.9) | -32.25 (1.0) | -69.55 (1.9) | hit |@*
h13-9 | 8.0e+04 | -54.03 (1.9) | -38.96 (0.9) | -81.88 (1.9) | hit |@*
h13-T | 8.0e+04 | -57.83 (1.8) | -46.80 (0.9) | -88.48 (1.8) | hit |@*
h13-A | 8.0e+04 | -72.42 (1.5) | -60.37 (0.8) | -93.03 (1.5) | hit |@*
h12-2 | 8.0e+04 | -29.07 (2.0) | -25.61 (1.0) | -49.41 (2.0) | hit |@*
h12-3 | 3.2e+05 | -24.62 (1.0) | -23.31 (0.5) | -47.37 (1.0) | hit |@*
h12-4 | 1.3e+06 | -20.56 (0.5) | -21.52 (0.3) | -42.79 (0.5) | stand |@*
h12-5 | 8.0e+04 | -16.46 (2.0) | -19.95 (1.0) | -38.43 (2.0) | stand |@*
h12-6 | 8.0e+04 | -12.05 (2.0) | -17.28 (1.0) | -33.47 (2.0) | stand |@*
h12-7 | 8.0e+04 | -47.30 (2.0) | -21.83 (1.0) | -50.18 (2.0) | hit |@*
h12-8 | 8.0e+04 | -51.08 (2.0) | -27.00 (1.0) | -61.37 (2.0) | hit |@*
h12-9 | 8.0e+04 | -54.15 (1.9) | -34.39 (0.9) | -73.59 (1.9) | hit |@*
h12-T | 8.0e+04 | -57.87 (1.8) | -42.68 (0.9) | -81.48 (1.8) | hit |@*
h12-A | 8.0e+04 | -72.38 (1.5) | -56.80 (0.8) | -89.31 (1.5) | hit |@*
h11-2 | 8.0e+04 | -28.87 (2.0) | +23.35 (1.0) | +47.39 (2.0) | double
|@*
h11-3 | 8.0e+04 | -24.76 (2.0) | +25.98 (1.0) | +52.29 (2.0) | double
|@*
h11-4 | 8.0e+04 | -20.67 (2.0) | +28.62 (1.0) | +57.21 (2.0) | double
|@*
h11-5 | 8.0e+04 | -16.73 (1.9) | +30.55 (1.0) | +62.32 (1.9) | double
|@*
h11-6 | 8.0e+04 | -11.69 (1.9) | +33.02 (1.0) | +66.25 (1.9) | double
|@*
h11-7 | 8.0e+04 | -47.69 (2.0) | +29.68 (1.0) | +45.78 (2.0) | double
|@*
h11-8 | 8.0e+04 | -52.13 (2.0) | +23.74 (1.0) | +35.93 (2.0) | double
|@*
h11-9 | 8.0e+04 | -54.25 (2.0) | +16.06 (1.0) | +22.66 (2.0) | double
|@*
h11-T | 8.0e+04 | -57.92 (2.0) | +3.19 (1.0) | +9.06 (2.0) | double |@*
h11-A | 8.2e+07 | -72.22 (0) | -23.66 (0) | -23.64 (0) | double |@*
h10-2 | 8.0e+04 | -28.12 (2.0) | +18.19 (1.0) | +36.23 (2.0) | double
|@*
h10-3 | 8.0e+04 | -24.40 (2.0) | +20.90 (1.0) | +40.69 (2.0) | double
|@*
h10-4 | 8.0e+04 | -20.09 (2.0) | +22.58 (1.0) | +46.54 (2.0) | double
|@*
h10-5 | 8.0e+04 | -16.16 (2.0) | +25.57 (1.0) | +49.92 (2.0) | double
|@*
h10-6 | 8.0e+04 | -12.23 (2.0) | +28.35 (1.0) | +56.55 (2.0) | double
|@*
h10-7 | 8.0e+04 | -47.69 (2.0) | +26.37 (1.0) | +39.07 (2.0) | double
|@*
h10-8 | 8.0e+04 | -50.60 (2.0) | +19.50 (1.0) | +27.72 (2.0) | double
|@*
h10-9 | 8.0e+04 | -54.75 (2.0) | +11.43 (1.0) | +15.10 (2.0) | double
|@*
h10-T | 8.0e+04 | -57.90 (1.9) | -5.37 (1.0) | -9.06 (1.9) | hit |@*
h10-A | 8.0e+04 | -71.92 (1.7) | -28.02 (1.0) | -34.12 (1.7) | hit |@*
h9-2 | 3.2e+05 | -28.52 (1.0) | +7.07 (0.5) | +5.21 (1.0) | hit |@*
h9-3 | 3.2e+05 | -24.80 (1.0) | +9.97 (0.5) | +11.98 (1.0) | double |@*
h9-4 | 8.0e+04 | -20.79 (2.0) | +12.80 (1.0) | +18.47 (2.0) | double |@*
h9-5 | 8.0e+04 | -16.32 (2.0) | +15.69 (1.0) | +23.70 (2.0) | double |@*
h9-6 | 8.0e+04 | -12.80 (2.0) | +18.20 (1.0) | +30.52 (2.0) | double |@*
h9-7 | 8.0e+04 | -48.00 (2.0) | +17.03 (1.0) | +10.60 (2.0) | hit |@*
h9-8 | 8.0e+04 | -51.36 (2.0) | +10.40 (1.0) | -1.84 (2.0) | hit |@*
h9-9 | 8.0e+04 | -54.51 (1.9) | -4.82 (1.0) | -29.67 (1.9) | hit |@*
h9-T | 8.0e+04 | -56.87 (1.9) | -22.03 (1.0) | -50.65 (1.9) | hit |@*
h9-A | 8.0e+04 | -72.07 (1.6) | -39.82 (0.9) | -62.20 (1.6) | hit |@*
h8-2 | 8.0e+04 | -28.53 (2.0) | -2.26 (1.0) | -20.42 (2.0) | hit |@*
h8-3 | 8.0e+04 | -24.15 (2.0) | +0.65 (1.0) | -13.16 (2.0) | hit |@*
h8-4 | 8.0e+04 | -20.23 (2.1) | +3.90 (1.0) | -5.21 (2.1) | hit |@*
h8-5 | 8.0e+04 | -16.14 (2.1) | +7.09 (1.0) | -0.54 (2.1) | hit |@*
h8-6 | 3.2e+05 | -12.14 (1.0) | +10.20 (0.5) | +7.84 (1.0) | hit |@*
h8-7 | 8.0e+04 | -47.05 (2.0) | +8.66 (1.0) | -19.70 (2.0) | hit |@*
h8-8 | 8.0e+04 | -50.79 (1.9) | -5.83 (1.0) | -45.53 (1.9) | hit |@*
h8-9 | 8.0e+04 | -54.13 (1.9) | -21.18 (1.0) | -71.69 (1.9) | hit |@*
h8-T | 8.0e+04 | -57.28 (1.8) | -30.79 (1.0) | -76.30 (1.8) | hit |@*
h8-A | 8.0e+04 | -71.71 (1.5) | -49.20 (0.9) | -88.72 (1.5) | hit |@*
h7-2 | 8.0e+04 | -28.98 (2.0) | -10.50 (1.0) | -43.26 (2.0) | hit |@*
h7-3 | 8.0e+04 | -24.01 (2.0) | -7.75 (1.0) | -35.53 (2.0) | hit |@*
h7-4 | 8.0e+04 | -20.99 (2.1) | -3.92 (1.0) | -26.25 (2.1) | hit |@*
h7-5 | 8.0e+04 | -16.47 (2.1) | +0.08 (1.0) | -20.59 (2.1) | hit |@*
h7-6 | 8.0e+04 | -12.61 (2.1) | +2.82 (1.0) | -10.05 (2.1) | hit |@*
h7-7 | 8.0e+04 | -48.34 (1.9) | -6.79 (1.0) | -59.20 (1.9) | hit |@*
h7-8 | 8.0e+04 | -51.35 (1.8) | -21.21 (1.0) | -84.58 (1.8) | hit |@*
h7-9 | 8.0e+04 | -54.69 (1.8) | -28.72 (1.0) | -94.97 (1.8) | hit |@*
h7-T | 8.0e+04 | -57.65 (1.7) | -37.00 (0.9) | -96.37 (1.7) | hit |@*
h7-A | 8.0e+04 | -72.44 (1.4) | -55.11 (0.8) | -106.65 (1.4) | hit |@*
h6-2 | 8.0e+04 | -28.57 (2.0) | -15.43 (1.0) | -54.59 (2.0) | hit |@*
h6-3 | 8.0e+04 | -24.56 (2.0) | -11.04 (1.0) | -47.59 (2.0) | hit |@*
h6-4 | 8.0e+04 | -20.00 (2.1) | -8.47 (1.0) | -38.90 (2.1) | hit |@*
h6-5 | 8.0e+04 | -16.08 (2.1) | -4.01 (1.0) | -31.24 (2.1) | hit |@*
h6-6 | 8.0e+04 | -11.99 (2.1) | -0.87 (1.0) | -21.55 (2.1) | hit |@*
h6-7 | 8.0e+04 | -47.56 (1.9) | -16.78 (1.0) | -89.83 (1.9) | hit |@*
h6-8 | 8.0e+04 | -51.12 (1.8) | -24.33 (1.0) | -99.55 (1.8) | hit |@*
h6-9 | 8.0e+04 | -54.73 (1.8) | -31.31 (1.0) | -106.21 (1.8) | hit |@*
h6-T | 8.0e+04 | -57.42 (1.7) | -41.13 (0.9) | -105.18 (1.7) | hit |@*
h6-A | 8.0e+04 | -72.60 (1.4) | -56.24 (0.8) | -112.72 (1.4) | hit |@*
h5-2 | 8.0e+04 | -29.05 (2.0) | -14.61 (1.0) | -56.03 (2.0) | hit |@*
h5-3 | 8.0e+04 | -24.45 (2.1) | -11.41 (1.0) | -50.65 (2.1) | hit |@*
h5-4 | 8.0e+04 | -20.23 (2.1) | -7.78 (1.0) | -40.43 (2.1) | hit |@*
h5-5 | 8.0e+04 | -16.31 (2.1) | -3.84 (1.0) | -33.26 (2.1) | hit |@*
h5-6 | 8.0e+04 | -11.79 (2.1) | -0.37 (1.0) | -25.05 (2.1) | hit |@*
h5-7 | 8.0e+04 | -47.60 (1.9) | -15.39 (1.0) | -95.71 (1.9) | hit |@*
h5-8 | 8.0e+04 | -51.23 (1.8) | -22.53 (1.0) | -101.98 (1.8) | hit |@*
h5-9 | 8.0e+04 | -54.04 (1.8) | -29.29 (1.0) | -108.81 (1.8) | hit |@*
h5-T | 8.0e+04 | -57.95 (1.7) | -39.00 (0.9) | -107.75 (1.7) | hit |@*
h5-A | 8.0e+04 | -72.13 (1.4) | -54.85 (0.8) | -112.28 (1.4) | hit |@*
h4-2 | 8.0e+04 | -28.50 (2.0) | -14.23 (1.0) | -56.99 (2.0) | hit |@*
h4-3 | 8.0e+04 | -24.78 (2.1) | -10.80 (1.0) | -49.65 (2.1) | hit |@*
h4-4 | 8.0e+04 | -20.49 (2.1) | -6.87 (1.0) | -41.52 (2.1) | hit |@*
h4-5 | 8.0e+04 | -16.77 (2.1) | -3.61 (1.0) | -32.45 (2.1) | hit |@*
h4-6 | 8.0e+04 | -11.80 (2.1) | +1.55 (1.0) | -24.32 (2.1) | hit |@*
h4-7 | 8.0e+04 | -47.75 (1.9) | -13.39 (1.0) | -95.12 (1.9) | hit |@*
h4-8 | 8.0e+04 | -51.73 (1.8) | -19.55 (1.0) | -101.77 (1.8) | hit |@*
h4-9 | 8.0e+04 | -54.22 (1.8) | -27.76 (1.0) | -108.31 (1.8) | hit |@*
h4-T | 8.0e+04 | -58.04 (1.7) | -37.09 (0.9) | -107.42 (1.7) | hit |@*
h4-A | 8.0e+04 | -72.31 (1.4) | -53.45 (0.9) | -114.02 (1.4) | hit |@*
s20-2 | 8.0e+04 | +63.62 (2.0) | +18.06 (1.0) | +35.23 (2.0) | stand |@*
s20-3 | 8.0e+04 | +64.71 (2.0) | +20.31 (1.0) | +39.62 (2.0) | stand |@*
s20-4 | 8.0e+04 | +65.86 (2.0) | +23.05 (1.0) | +45.11 (2.0) | stand |@*
s20-5 | 8.0e+04 | +67.00 (2.0) | +25.96 (1.0) | +51.24 (2.0) | stand |@*
s20-6 | 8.0e+04 | +68.16 (2.0) | +28.27 (1.0) | +55.63 (2.0) | stand |@*
s20-7 | 8.0e+04 | +77.47 (2.0) | +25.35 (1.0) | +40.40 (2.0) | stand |@*
s20-8 | 8.0e+04 | +79.10 (2.0) | +20.23 (1.0) | +28.37 (2.0) | stand |@*
s20-9 | 8.0e+04 | +76.12 (2.0) | +11.22 (1.0) | +14.64 (2.0) | stand |@*
s20-T | 8.0e+04 | +43.51 (1.9) | -5.46 (1.0) | -8.53 (1.9) | stand |@*
s20-A | 8.0e+04 | +10.77 (1.7) | -28.26 (1.0) | -33.48 (1.7) | stand |@*
s19-2 | 8.0e+04 | +38.20 (2.0) | +11.75 (1.0) | +23.54 (2.0) | stand |@*
s19-3 | 8.0e+04 | +39.82 (2.0) | +15.29 (1.0) | +29.62 (2.0) | stand |@*
s19-4 | 8.0e+04 | +42.18 (2.0) | +17.78 (1.0) | +35.08 (2.0) | stand |@*
s19-5 | 3.2e+05 | +43.54 (1.0) | +20.12 (0.5) | +40.89 (1.0) | stand |@*
s19-6 | 5.1e+06 | +45.35 (0.2) | +23.10 (0.1) | +46.20 (0.2) | double
|@*
s19-7 | 8.0e+04 | +61.59 (2.0) | +21.74 (1.0) | +31.70 (2.0) | stand |@*
s19-8 | 8.0e+04 | +59.31 (2.0) | +15.31 (1.0) | +20.15 (2.0) | stand |@*
s19-9 | 8.0e+04 | +28.43 (2.0) | +0.63 (1.0) | -8.21 (2.0) | stand |@*
s19-T | 8.0e+04 | -1.71 (1.9) | -16.07 (1.0) | -28.88 (1.9) | stand |@*
s19-A | 8.0e+04 | -18.13 (1.7) | -35.40 (0.9) | -46.73 (1.7) | stand |@*
s18-2 | 2.0e+07 | +11.03 (0) | +5.99 (0) | +11.48 (0) | double |@*
s18-3 | 3.2e+05 | +13.61 (1.0) | +8.83 (0.5) | +17.60 (1.0) | double |@*
s18-4 | 8.0e+04 | +16.17 (2.0) | +11.32 (1.0) | +22.57 (2.0) | double
|@*
s18-5 | 8.0e+04 | +19.92 (2.0) | +14.49 (1.0) | +28.95 (2.0) | double
|@*
s18-6 | 8.0e+04 | +22.64 (2.0) | +15.59 (1.0) | +34.71 (2.0) | double
|@*
s18-7 | 8.0e+04 | +39.60 (2.0) | +16.92 (1.0) | +22.37 (2.0) | stand |@*
s18-8 | 8.0e+04 | +10.50 (2.0) | +4.49 (1.0) | -3.59 (2.0) | stand |@*
s18-9 | 8.0e+04 | -18.75 (2.0) | -9.78 (1.0) | -29.28 (2.0) | hit |@*
s18-T | 8.0e+04 | -24.01 (1.9) | -20.91 (1.0) | -38.83 (1.9) | hit |@*
s18-A | 8.0e+04 | -46.67 (1.7) | -41.65 (0.9) | -59.24 (1.7) | hit |@*
s17-2 | 2.0e+07 | -15.66 (0) | -0.55 (0) | -0.87 (0) | hit |@*
s17-3 | 3.2e+05 | -11.98 (1.0) | +2.36 (0.5) | +5.48 (1.0) | double |@*
s17-4 | 8.0e+04 | -8.31 (2.0) | +5.04 (1.0) | +11.89 (2.0) | double |@*
s17-5 | 8.0e+04 | -4.91 (2.0) | +8.85 (1.0) | +17.72 (2.0) | double |@*
s17-6 | 8.0e+04 | -0.74 (2.0) | +10.12 (1.0) | +24.25 (2.0) | double |@*
s17-7 | 8.0e+04 | -10.35 (2.0) | +5.08 (1.0) | -0.47 (2.0) | hit |@*
s17-8 | 8.0e+04 | -38.37 (2.0) | -7.07 (1.0) | -26.13 (2.0) | hit |@*
s17-9 | 8.0e+04 | -41.81 (2.0) | -15.16 (1.0) | -40.21 (2.0) | hit |@*
s17-T | 8.0e+04 | -45.89 (1.9) | -25.72 (1.0) | -50.87 (1.9) | hit |@*
s17-A | 8.0e+04 | -66.48 (1.6) | -46.33 (0.9) | -69.24 (1.6) | hit |@*
s16-2 | 8.0e+04 | -28.80 (2.1) | -2.39 (1.0) | -6.32 (2.1) | hit |@*
s16-3 | 3.2e+05 | -24.88 (1.0) | +0.60 (0.5) | -0.97 (1.0) | hit |@*
s16-4 | 8.0e+04 | -21.20 (2.1) | +3.23 (1.0) | +6.64 (2.1) | double |@*
s16-5 | 8.0e+04 | -15.99 (2.1) | +6.72 (1.0) | +13.54 (2.1) | double |@*
s16-6 | 8.0e+04 | -12.50 (2.1) | +8.31 (1.0) | +19.84 (2.1) | double |@*
s16-7 | 8.0e+04 | -47.17 (2.0) | -1.23 (1.0) | -18.63 (2.0) | hit |@*
s16-8 | 8.0e+04 | -50.91 (2.0) | -6.98 (1.0) | -31.09 (2.0) | hit |@*
s16-9 | 8.0e+04 | -54.43 (2.0) | -14.67 (1.0) | -46.42 (2.0) | hit |@*
s16-T | 8.0e+04 | -57.58 (1.9) | -26.70 (1.0) | -55.38 (1.9) | hit |@*
s16-A | 8.0e+04 | -72.33 (1.6) | -44.35 (0.9) | -71.31 (1.6) | hit |@*
s15-2 | 8.0e+04 | -29.03 (2.1) | -0.58 (1.0) | -5.65 (2.1) | hit |@*
s15-3 | 8.0e+04 | -24.71 (2.1) | +3.31 (1.0) | -1.28 (2.1) | hit |@*
s15-4 | 1.3e+06 | -20.58 (0.5) | +5.60 (0.3) | +6.44 (0.5) | double |@*
s15-5 | 3.2e+05 | -16.62 (1.0) | +8.86 (0.5) | +12.74 (1.0) | double |@*
s15-6 | 8.0e+04 | -12.09 (2.1) | +10.13 (1.0) | +19.13 (2.1) | double
|@*
s15-7 | 8.0e+04 | -47.61 (2.0) | +4.21 (1.0) | -19.48 (2.0) | hit |@*
s15-8 | 8.0e+04 | -51.16 (2.0) | -2.81 (1.0) | -30.90 (2.0) | hit |@*
s15-9 | 8.0e+04 | -54.31 (2.0) | -11.18 (1.0) | -46.93 (2.0) | hit |@*
s15-T | 8.0e+04 | -57.49 (1.9) | -23.85 (1.0) | -54.93 (1.9) | hit |@*
s15-A | 8.0e+04 | -71.98 (1.6) | -42.68 (0.9) | -71.58 (1.6) | hit |@*
s14-2 | 8.0e+04 | -28.59 (2.1) | +1.42 (1.0) | -6.73 (2.1) | hit |@*
s14-3 | 8.0e+04 | -24.89 (2.1) | +4.04 (1.0) | -0.41 (2.1) | hit |@*
s14-4 | 8.0e+04 | -20.84 (2.1) | +7.58 (1.0) | +5.13 (2.1) | hit |@*
s14-5 | 3.2e+05 | -16.36 (1.0) | +10.61 (0.5) | +12.72 (1.0) | double
|@*
s14-6 | 8.0e+04 | -12.19 (2.1) | +12.30 (1.0) | +19.62 (2.1) | double
|@*
s14-7 | 8.0e+04 | -47.42 (2.1) | +7.66 (1.0) | -18.33 (2.1) | hit |@*
s14-8 | 8.0e+04 | -50.95 (2.0) | +0.96 (1.0) | -30.90 (2.0) | hit |@*
s14-9 | 8.0e+04 | -54.73 (2.0) | -7.82 (1.0) | -46.11 (2.0) | hit |@*
s14-T | 8.0e+04 | -57.51 (1.9) | -21.19 (1.0) | -55.00 (1.9) | hit |@*
s14-A | 8.0e+04 | -72.28 (1.6) | -40.00 (0.9) | -71.64 (1.6) | hit |@*
s13-2 | 8.0e+04 | -28.76 (2.1) | +3.98 (1.0) | -7.66 (2.1) | hit |@*
s13-3 | 8.0e+04 | -25.07 (2.1) | +7.23 (1.0) | -0.04 (2.1) | hit |@*
s13-4 | 8.0e+04 | -20.02 (2.1) | +9.31 (1.0) | +4.88 (2.1) | hit |@*
s13-5 | 2.0e+07 | -16.49 (0) | +12.84 (0) | +12.70 (0) | hit |@*
s13-6 | 8.0e+04 | -12.39 (2.1) | +13.78 (1.0) | +18.69 (2.1) | double
|@*
s13-7 | 8.0e+04 | -47.43 (2.1) | +12.59 (1.0) | -18.10 (2.1) | hit |@*
s13-8 | 8.0e+04 | -50.86 (2.0) | +5.47 (1.0) | -31.53 (2.0) | hit |@*
s13-9 | 8.0e+04 | -54.25 (2.0) | -4.04 (1.0) | -46.51 (2.0) | hit |@*
s13-T | 8.0e+04 | -57.70 (1.9) | -16.62 (1.0) | -54.45 (1.9) | hit |@*
s13-A | 8.0e+04 | -71.53 (1.7) | -37.98 (0.9) | -69.91 (1.7) | hit |@*
s12-2 | 8.0e+04 | -28.39 (2.1) | +7.36 (1.0) | -6.07 (2.1) | hit |@*
s12-3 | 8.0e+04 | -25.70 (2.1) | +9.50 (1.0) | -0.03 (2.1) | hit |@*
s12-4 | 8.0e+04 | -20.48 (2.1) | +12.65 (1.0) | +6.24 (2.1) | hit |@*
s12-5 | 3.2e+05 | -16.42 (1.0) | +15.18 (0.5) | +13.22 (1.0) | hit |@*
s12-6 | 8.0e+04 | -11.66 (2.1) | +15.42 (1.0) | +19.62 (2.1) | double
|@*
s12-7 | 8.0e+04 | -47.27 (2.1) | +16.31 (1.0) | -16.99 (2.1) | hit |@*
s12-8 | 8.0e+04 | -51.51 (2.0) | +9.36 (1.0) | -30.70 (2.0) | hit |@*
s12-9 | 8.0e+04 | -54.23 (2.0) | -0.10 (1.0) | -44.47 (2.0) | hit |@*
s12-T | 8.0e+04 | -57.77 (1.9) | -14.11 (1.0) | -57.03 (1.9) | hit |@*
s12-A | 8.0e+04 | -72.70 (1.6) | -34.54 (0.9) | -70.64 (1.6) | hit |
@ Hand | @math{n} | Yes [%] | No [%] |@*
pA-2 | 8.0e+04 | +47.20 (1.7) | +6.91 (1.0) | yes |@*
pA-3 | 8.0e+04 | +53.01 (1.7) | +8.95 (1.0) | yes |@*
pA-4 | 8.0e+04 | +55.98 (1.7) | +11.49 (1.0) | yes |@*
pA-5 | 8.0e+04 | +60.89 (1.7) | +14.66 (1.0) | yes |@*
pA-6 | 8.0e+04 | +66.26 (1.7) | +20.80 (2.1) | yes |@*
pA-7 | 8.0e+04 | +46.06 (1.6) | +16.94 (1.0) | yes |@*
pA-8 | 8.0e+04 | +35.38 (1.6) | +9.52 (1.0) | yes |@*
pA-9 | 8.0e+04 | +22.97 (1.6) | -0.20 (1.0) | yes |@*
pA-T | 8.0e+04 | +8.41 (1.6) | -14.49 (1.0) | yes |@*
pA-A | 8.0e+04 | -23.73 (1.5) | -34.77 (0.9) | yes |@*
pT-2 | 8.0e+04 | -7.22 (3.8) | +63.33 (0.7) | no |@*
pT-3 | 8.0e+04 | +1.85 (3.8) | +64.34 (0.7) | no |@*
pT-4 | 8.0e+04 | +15.20 (4.0) | +65.82 (0.7) | no |@*
pT-5 | 8.0e+04 | +24.91 (4.0) | +66.75 (0.7) | no |@*
pT-6 | 8.0e+04 | +37.30 (4.0) | +67.83 (0.7) | no |@*
pT-7 | 8.0e+04 | +9.41 (2.6) | +76.75 (0.6) | no |@*
pT-8 | 8.0e+04 | -23.72 (2.5) | +79.31 (0.6) | no |@*
pT-9 | 8.0e+04 | -59.32 (2.6) | +75.92 (0.6) | no |@*
pT-T | 8.0e+04 | -77.52 (2.6) | +43.30 (0.7) | no |@*
pT-A | 8.0e+04 | -85.16 (2.2) | +10.82 (1.0) | no |@*
p9-2 | 8.0e+04 | +20.02 (2.2) | +11.24 (1.0) | yes |@*
p9-3 | 8.0e+04 | +25.25 (2.2) | +13.40 (1.0) | yes |@*
p9-4 | 8.0e+04 | +33.27 (2.2) | +16.87 (1.0) | yes |@*
p9-5 | 8.0e+04 | +39.23 (2.2) | +19.21 (1.0) | yes |@*
p9-6 | 8.0e+04 | +45.68 (2.2) | +22.91 (1.0) | yes |@*
p9-7 | 8.0e+04 | +36.00 (1.9) | +40.30 (0.9) | no |@*
p9-8 | 8.0e+04 | +23.61 (1.9) | +10.59 (0.8) | yes |@*
p9-9 | 8.0e+04 | -6.82 (1.8) | -18.28 (1.0) | yes |@*
p9-T | 8.0e+04 | -36.80 (1.9) | -23.83 (1.0) | no |@*
p9-A | 3.2e+05 | -48.13 (0.8) | -46.28 (0.4) | no |@*
p8-2 | 8.0e+04 | +5.68 (2.4) | -28.12 (1.0) | yes |@*
p8-3 | 8.0e+04 | +13.29 (2.4) | -24.45 (1.0) | yes |@*
p8-4 | 8.0e+04 | +20.18 (2.5) | -20.57 (1.0) | yes |@*
p8-5 | 8.0e+04 | +28.86 (2.5) | -15.92 (1.0) | yes |@*
p8-6 | 8.0e+04 | +38.58 (2.7) | -12.18 (1.1) | yes |@*
p8-7 | 8.0e+04 | +32.62 (2.1) | -41.09 (0.9) | yes |@*
p8-8 | 8.0e+04 | -1.93 (2.1) | -46.30 (0.9) | yes |@*
p8-9 | 8.0e+04 | -39.65 (2.2) | -50.97 (0.9) | yes |@*
p8-T | 8.0e+04 | -51.78 (1.9) | -57.43 (0.8) | yes |@*
p8-A | 3.2e+05 | -66.56 (0.8) | -68.26 (0.4) | yes |@*
p7-2 | 8.0e+04 | -14.09 (2.6) | -28.55 (1.0) | yes |@*
p7-3 | 8.0e+04 | -5.00 (2.8) | -25.59 (1.0) | yes |@*
p7-4 | 8.0e+04 | +5.68 (2.9) | -20.86 (1.0) | yes |@*
p7-5 | 8.0e+04 | +15.57 (2.9) | -16.37 (1.0) | yes |@*
p7-6 | 8.0e+04 | +25.52 (2.9) | -12.25 (1.1) | yes |@*
p7-7 | 8.0e+04 | -5.23 (2.1) | -31.97 (1.0) | yes |@*
p7-8 | 3.2e+05 | -39.50 (1.0) | -37.19 (0.5) | no |@*
p7-9 | 8.0e+04 | -57.79 (2.1) | -42.92 (0.9) | no |@*
p7-T | 8.0e+04 | -68.47 (1.8) | -50.66 (0.9) | no |@*
p7-A | 8.0e+04 | -81.50 (1.5) | -63.51 (0.8) | no |@*
p6-2 | 8.0e+04 | -17.82 (2.5) | -26.09 (1.0) | yes |@*
p6-3 | 8.0e+04 | -11.19 (2.9) | -23.68 (1.0) | yes |@*
p6-4 | 8.0e+04 | -0.23 (2.9) | -20.51 (1.0) | yes |@*
p6-5 | 8.0e+04 | +8.19 (2.9) | -16.81 (1.0) | yes |@*
p6-6 | 8.0e+04 | +18.21 (2.9) | -12.51 (1.1) | yes |@*
p6-7 | 8.0e+04 | -26.43 (2.0) | -21.52 (1.0) | no |@*
p6-8 | 8.0e+04 | -41.81 (2.0) | -27.45 (1.0) | no |@*
p6-9 | 8.0e+04 | -61.03 (1.9) | -33.82 (1.0) | no |@*
p6-T | 8.0e+04 | -74.39 (1.7) | -43.03 (0.9) | no |@*
p6-A | 8.0e+04 | -82.19 (1.4) | -57.43 (0.8) | no |@*
p5-2 | 8.0e+04 | -28.56 (2.3) | +36.16 (2.0) | no |@*
p5-3 | 8.0e+04 | -19.64 (2.5) | +40.18 (2.0) | no |@*
p5-4 | 8.0e+04 | -12.32 (2.8) | +45.78 (2.0) | no |@*
p5-5 | 8.0e+04 | -5.05 (2.8) | +52.02 (2.0) | no |@*
p5-6 | 8.0e+04 | +8.49 (2.8) | +57.75 (2.0) | no |@*
p5-7 | 8.0e+04 | -29.66 (1.8) | +38.20 (2.0) | no |@*
p5-8 | 8.0e+04 | -45.66 (1.8) | +29.62 (2.0) | no |@*
p5-9 | 8.0e+04 | -63.47 (1.7) | +14.03 (2.0) | no |@*
p5-T | 8.0e+04 | -75.39 (1.7) | -5.08 (1.0) | no |@*
p5-A | 8.0e+04 | -83.27 (1.4) | -28.59 (1.0) | no |@*
p4-2 | 8.0e+04 | -18.11 (2.5) | -2.44 (1.0) | no |@*
p4-3 | 8.0e+04 | -11.18 (2.7) | +0.19 (1.0) | no |@*
p4-4 | 8.0e+04 | -0.79 (2.9) | +3.43 (1.0) | no |@*
p4-5 | 1.3e+06 | +8.28 (0.7) | +7.07 (0.3) | yes |@*
p4-6 | 8.0e+04 | +18.98 (3.0) | +8.70 (1.0) | yes |@*
p4-7 | 8.0e+04 | -16.47 (2.0) | +7.80 (1.0) | no |@*
p4-8 | 8.0e+04 | -32.77 (2.0) | -6.00 (1.0) | no |@*
p4-9 | 8.0e+04 | -50.81 (2.0) | -20.81 (1.0) | no |@*
p4-T | 8.0e+04 | -65.67 (1.7) | -30.72 (1.0) | no |@*
p4-A | 8.0e+04 | -74.66 (1.5) | -49.18 (0.9) | no |@*
p3-2 | 2.0e+07 | -13.64 (0) | -13.80 (0) | yes |@*
p3-3 | 8.0e+04 | -5.32 (2.6) | -10.45 (1.0) | yes |@*
p3-4 | 8.0e+04 | +4.08 (2.7) | -7.45 (1.0) | yes |@*
p3-5 | 8.0e+04 | +12.53 (2.9) | -3.04 (1.0) | yes |@*
p3-6 | 8.0e+04 | +22.61 (2.9) | +0.47 (1.0) | yes |@*
p3-7 | 8.0e+04 | -4.91 (2.0) | -14.91 (1.0) | yes |@*
p3-8 | 3.2e+05 | -23.20 (1.0) | -21.69 (0.5) | no |@*
p3-9 | 8.0e+04 | -43.61 (2.0) | -29.14 (1.0) | no |@*
p3-T | 8.0e+04 | -58.65 (1.7) | -38.95 (0.9) | no |@*
p3-A | 8.0e+04 | -70.06 (1.5) | -54.59 (0.8) | no |@*
p2-2 | 3.2e+05 | -8.69 (1.1) | -11.47 (0.5) | yes |@*
p2-3 | 8.0e+04 | -1.93 (2.5) | -8.55 (1.0) | yes |@*
p2-4 | 8.0e+04 | +6.19 (2.7) | -4.95 (1.0) | yes |@*
p2-5 | 8.0e+04 | +15.78 (2.7) | -1.24 (1.0) | yes |@*
p2-6 | 8.0e+04 | +24.78 (2.9) | +3.07 (1.0) | yes |@*
p2-7 | 8.0e+04 | +1.48 (2.0) | -8.90 (1.0) | yes |@*
p2-8 | 3.2e+05 | -17.88 (1.0) | -16.10 (0.5) | no |@*
p2-9 | 8.0e+04 | -38.73 (2.0) | -24.38 (1.0) | no |@*
p2-T | 8.0e+04 | -54.45 (1.8) | -34.92 (0.9) | no |@*
p2-A | 8.0e+04 | -67.11 (1.5) | -51.59 (0.9) | no |

@node Detailed explanation
@subsection Detailed explanation
@anchor{#detailed-explanation}
We want to derive the basic strategy from scratch, i.e.@ without making
any assumption. What we are going to do is to play a large (more on what
@emph{large} means below) number of hands by fixing our first two cards
and the dealer upcard and sequentially standing, doubling or hitting the
first card. Then we will compare the results for the three cases and
select as the proper strategy the best one of the three possible
choices.

Standing and doubling are easy plays, because after we stand or double
down then the dealer plays accordingly to the rules: she hits until
seventeen, possibly hitting soft seventeen. But if we hit on our hand,
we might need to make another decision whether to stand or hit again. As
we do not want to assume anything, we have to play in such an order that
if we do need to make another decision, we already know which is the
best one.

@menu
* Hard hands::
* Soft hands::
* Pairs::
* Number of hands::
@end menu

@node Hard hands
@subsubsection Hard hands
@anchor{#hard-hands}
So we start by arranging the shoe so that the user gets hard twenty
(i.e.@ two faces) and the dealer gets successively upcards of two to
ace. So we play each combination of dealer upcard (ten) three times each
playing either

@enumerate 
@item
always standing
@item
always doubling
@item
always hitting
@end enumerate

In general the first two plays are easy, because the game stops either
after standing or after receiving only one card. The last one might lead
to further hitting, but since we are starting with a hard twenty, that
would either give the player twenty one or a bust. In any case, the game
also ends. So we play a certain number of hands (say one thousand hands)
each of these three plays for each of the ten upcard faces and record
the outcome. The correct play for hard twenty against each of the ten
upcards is the play that gave the better result, which is of course
standing.

Next, we do the same for a hard nineteen. In this case, the hitting play
might not end after one card is drawn (i.e.@ we hit on nineteen and get
and ace). But if that was the case, we would already know what the best
play is from the previous step so we play accordingly and we stand.
Repeating this procedure down to hard four we can build the basic
strategy table for any hard total against any dealer upcard.

@node Soft hands
@subsubsection Soft hands
@anchor{#soft-hands}
We can now switch to analyze soft hands. Starting from soft twenty
(i.e.@ an ace and a nine) we do the same we did for the hard case. The
only difference is that when hitting, we might end either in another
soft hand which we would already analyzed because we start from twenty
and go down, or in a hard hand, which we also already analyzed so we can
play accordingly.

@node Pairs
@subsubsection Pairs
@anchor{#pairs}
When dealing with pairs, we have to decide whether to split or not. When
we do not split, we end up in one of the already-analyzed cases: either
a soft twelve of any even hard hand. When we split, we might end in a
hard or soft hand (already analyzed) or in a new pair. But since the new
pair can be only the same pair we are analyzing, we have to treat it
like we treated the first pair: either to split it or not, so we know
how to deal with it.

@node Number of hands
@subsubsection Number of hands
@anchor{#number-of-hands}
The output is the expected value@ @math{e} of the bankroll, which is a
random variable with an associated uncertainty@ @math{\Delta e} (i.e.@ a
certain numbers of standard deviations). For example, if we received
only blackjacks, the expected value would be 1.5 (provided blackjacks
pay@ 3 to@ 2 of course). If we busted all of our hands without doubling
or splitting, the expected value would be -1. In order to say that the
best strategy is, let's say stand and not hitting or doubling down, we
have to make sure that @math{e_h-\Delta e_h > e_s+\Delta e_s} and
@math{e_h-\Delta e_h > e_d+\Delta e_d}. If there is no play that can
give a better expected value than the other two taking into account the
uncertainties, then we have to play more hands in order to reduce the
random uncertainty.

@node Implementation
@subsection Implementation
@anchor{#implementation}
The steps above can be written in a
@uref{https://en.wikipedia.org/wiki/Bash_%28Unix_shell%29,Bash} script
that

@itemize
@item
loops over hands and upcards,
@item
creates a strategy file for each possible play hit, double or stand (or
split or not),
@item
runs @uref{https://www.seamplex.com/blackjack,Libre@ Blackjack},
@item
checks the results and picks the best play,
@item
updates the strategy file
@end itemize

@verbatim
#!/bin/bash

   n0=80000
n_max=9000000

RED="\033[0;31m"
GREEN="\033[0;32m"

BROWN="\033[0;33m"
MAGENTA="\e[0;35m"
CYAN="\e[0;36m"

NC="\033[0m" # No Color

for i in grep awk printf blackjack; do
 if [ -z "$(which $i)" ]; then
  echo "error: $i not installed"
  exit 1
 fi
done

debug=0

declare -A strategy
declare -A ev

declare -A min
min["hard"]=4   # from 20 to 4 in hards
min["soft"]=12  # from 20 to 12 in softs

rm -f table.md hard.html soft.html pair.html

# start with standing
cp hard-stand.txt hard.txt
cp soft-stand.txt soft.txt

cat << EOF >> table.md
|  Hand  |  \$n\$  |  Stand [%]  |  Double [%]  |  Hit [%] |   Play    |
EOF


for type in hard soft; do
 for hand in $(seq 20 -1 ${min[${type}]}); do
 
  # choose two random cards that make up the player's assumed total
  if [ ${type} = "hard" ]; then
   t="h"
   card1=11
   card2=11
   while test $card1 -gt 10 -o $card2 -gt 10; do
    card1=$((${RANDOM} % (${hand}-3) + 2))
    card2=$((${hand} - ${card1}))
   done
  elif [ ${type} = "soft" ]; then
   t="s"
   # one card is an ace
   card1=1
   card2=$((${hand} - 10 - ${card1}))
  fi

  cat << EOF >> ${type}.html
 <tr>
  <td>${t}${hand}</td>
  <td>
   <div class="text-right">s<span class="d-none d-lg-inline">tand</span></div>
   <div class="text-right">h<span class="d-none d-lg-inline">it</span></div>
   <div class="text-right">d<span class="d-none d-lg-inline">ouble</span></div>
  </td>
EOF
  
  for upcard in $(seq 2 9) T A; do
  
   if [ "x$upcard" = "xT" ]; then
     upcard_n=10
   elif [ "x$upcard" = "xA" ]; then
     upcard_n=1
   else
     upcard_n=$(($upcard))
   fi
 
   n=${n0}   # start with n0 hands
   best="x"  # x means don't know what to so, so play
   
   while [ "${best}" = "x" ]; do
    # tell the user which combination we are trying and how many we will play
    echo -ne "${t}${hand}-${upcard} ($card1 $card2)\t"$(printf %.1e ${n})
    for play in s d h; do
     
     # start with options.conf as a template and add some custom stuff
     cp options.conf blackjack.conf
     cat << EOF >> blackjack.conf
hands = ${n}
player = internal
arranged_cards = ${card1}, $((${upcard_n} + 13)), $((${card2} + 26))
report = ${t}${hand}-${upcard}-${play}.yaml
#log = ${t}${hand}-${upcard}-${play}.log
EOF
 
     # read the current strategy
     while read w p2 p3 p4 p5 p6 p7 p8 p9 pT pA; do
      # w already has the "h" or the "s"
      strategy[${w},2]=$p2   
      strategy[${w},3]=$p3
      strategy[${w},4]=$p4    
      strategy[${w},5]=$p5    
      strategy[${w},6]=$p6    
      strategy[${w},7]=$p7    
      strategy[${w},8]=$p8    
      strategy[${w},9]=$p9    
      strategy[${w},T]=$pT    
      strategy[${w},A]=$pA    
     done < ${type}.txt
     
     # override the read strategy with the explicit play: s, d or h
     strategy[${t}${hand},${upcard}]=${play}
     
     # save the new (temporary) strategy
     rm -f ${type}.txt
     for h in $(seq 20 -1 ${min[${type}]}); do
      echo -n "${t}${h}  " >> ${type}.txt
      
      # extra space if h < 10
      if [ ${h} -lt 10 ]; then
       echo -n " " >> ${type}.txt
      fi 
      
      for u in $(seq 2 9) T A; do
       echo -n "${strategy[${t}${h},${u}]}  " >> ${type}.txt
      done
      echo >> ${type}.txt
     done

     # debug, comment for production
     if [ "${debug}" != "0" ]; then
      cp ${type}.txt ${t}${hand}-${upcard}-${play}.str
     fi
    
     # ensamble the full bs.txt with no pairing
     cat hard.txt soft.txt pair-no.txt > bs.txt
     
     # play!
     blackjack
    
     # evaluate the results
     ev[${t}${hand},${upcard},${play}]=$(grep mean ${t}${hand}-${upcard}-${play}.yaml | awk '{printf("%g", $2)}')
     error[${t}${hand},${upcard},${play}]=$(grep error ${t}${hand}-${upcard}-${play}.yaml | awk '{printf("%g", $2)}')
     
    done
   
    # choose the best one
    ev_s=$(echo ${ev[${t}${hand},${upcard},s]} | awk '{printf("%+.2f", 100*$1)}')
    ev_d=$(echo ${ev[${t}${hand},${upcard},d]} | awk '{printf("%+.2f", 100*$1)}')
    ev_h=$(echo ${ev[${t}${hand},${upcard},h]} | awk '{printf("%+.2f", 100*$1)}')
   
    
    if [ ${n} -le ${n_max} ]; then 
     # if we still have room, take into account errors
     error_s=$(echo ${error[${t}${hand},${upcard},s]} | awk '{printf("%.1f", 100*$1)}')
     error_d=$(echo ${error[${t}${hand},${upcard},d]} | awk '{printf("%.1f", 100*$1)}')
     error_h=$(echo ${error[${t}${hand},${upcard},h]} | awk '{printf("%.1f", 100*$1)}')
    else
     # instead of running infinite hands, above a threshold asume errors are zero
     error_s=0
     error_d=0
     error_h=0
    fi  
 
    echo -ne "\t${ev_s}\t(${error_s})"
    echo -ne "\t${ev_d}\t(${error_d})"
    echo -ne "\t${ev_h}\t(${error_h})"
   
    if   (( $(echo ${ev_s} ${error_s} ${ev_d} ${error_d} | awk '{print (($1-$2) > ($3+$4))}') )) &&
         (( $(echo ${ev_s} ${error_s} ${ev_h} ${error_h} | awk '{print (($1-$2) > ($3+$4))}') )); then
         
     best="s"
     color=${BROWN}
     best_string="stand"
     
    elif (( $(echo ${ev_d} ${error_d} ${ev_s} ${error_s} | awk '{print (($1-$2) > ($3+$4))}') )) &&
         (( $(echo ${ev_d} ${error_d} ${ev_h} ${error_h} | awk '{print (($1-$2) > ($3+$4))}') )); then
         
     best="d"
     color=${CYAN}
     best_string="double"
     
    elif (( $(echo ${ev_h}-${error_h} ${ev_s} ${error_s} | awk '{print (($1-$2) > ($3+$4))}') )) &&
         (( $(echo ${ev_h}-${error_h} ${ev_d} ${error_d} | awk '{print (($1-$2) > ($3+$4))}') )); then
         
     best="h"
     color=${MAGENTA}
     best_string="hit"
         
    else
    
     best="x"
     color=${NC}
     best_string="uncertain"
     
     n=$((${n} * 4))
     
    fi
    
    echo -e ${color}"\t"${best_string}${NC}
    
   done

   strategy[${t}${hand},${upcard}]=${best}
   
   
   
   echo "| ${t}${hand}-${upcard} | $(printf %.1e ${n}) | ${ev_s} (${error_s}) | ${ev_h} (${error_h}) | ${ev_d} (${error_d}) | ${best_string} | " >> table.md
    
   echo " <!-- ${upcard} -->" >> ${type}.html
   echo " <td>" >> ${type}.html
   echo ${ev_s} ${error_s} | awk -f html_cell.awk >> ${type}.html
   echo ${ev_h} ${error_h} | awk -f html_cell.awk >> ${type}.html
   echo ${ev_d} ${error_d} | awk -f html_cell.awk >> ${type}.html
   echo " </td>" >> ${type}.html
   
   
   # save the strategy again with the best strategy
   rm -f ${type}.txt
   for h in $(seq 20 -1 ${min[${type}]}); do
    echo -n "${t}${h}  " >> ${type}.txt
    
    # extra space if h < 10
    if [ ${h} -lt 10 ]; then
     echo -n " " >> ${type}.txt
    fi 
    
    for u in $(seq 2 9) T A; do
     echo -n "${strategy[${t}${h},${u}]}  " >> ${type}.txt
    done
    
    echo >> ${type}.txt
    
   done
  done
  
#   echo "</tr>" >> ${type}.html
  
 done
done


cat << EOF >> table.md


|  Hand  |  \$n\$  |   Yes [%]  |   No [%]   |
EOF

# pairs
type="pair"
t="p"
cp pair-no.txt pair.txt

for hand in A T $(seq 9 -1 2); do
 if [ "${hand}" = "A" ]; then
  pair=1
 elif [ "${hand}" = "T" ]; then
  pair=10
 else
  pair=$((${hand}))
 fi
  
#  cat << EOF >> ${type}.html
#  <tr>
#   <td>${t}${hand}</td>
#   <td>
#    <div class="text-right">y<span class="d-none d-lg-inline">es</span></div>
#    <div class="text-right">n<span class="d-none d-lg-inline">o</span></div>
#   </td>
# EOF
  
 for upcard in $(seq 2 9) T A; do
  if [ "$upcard" = "T" ]; then
    upcard_n=10
  elif [ "$upcard" = "A" ]; then
    upcard_n=1
  else
    upcard_n=$(($upcard))
  fi
 
  n=${n0}   # start with n0 hands
  best="x"  # x means don't know what to so, so play
   
  while [ "${best}" = "x" ]; do
   # tell the user which combination we are trying and how many we will play
   echo -ne "${t}${hand}-${upcard}\t\t$(printf %.0e ${n})"
   
   for play in y n; do
    
    # start with options.conf as a template and add some custom stuff
    cp options.conf blackjack.conf
    cat << EOF >> blackjack.conf
hands = ${n}
player = internal
arranged_cards = ${pair}, $((${upcard_n} + 13)), $((${pair} + 26))
report = ${t}${hand}-${upcard}-${play}.yaml
# log = ${t}${hand}-${upcard}-${play}.log
EOF
 
    # read the current strategy
    while read w p2 p3 p4 p5 p6 p7 p8 p9 pT pA; do
     # w already has the "p"
     strategy[${w},2]=$p2   
     strategy[${w},3]=$p3
     strategy[${w},4]=$p4    
     strategy[${w},5]=$p5    
     strategy[${w},6]=$p6    
     strategy[${w},7]=$p7    
     strategy[${w},8]=$p8    
     strategy[${w},9]=$p9    
     strategy[${w},T]=$pT    
     strategy[${w},A]=$pA    
    done < ${type}.txt
     
    # override the read strategy with the explicit play: y or n
    strategy[${t}${hand},${upcard}]=${play}
     
    # save the new (temporary) strategy
    rm -f ${type}.txt
    for h in A T $(seq 9 -1 2); do
     echo -n "${t}${h}   " >> ${type}.txt
     for u in $(seq 2 9) T A; do
      echo -n "${strategy[${t}${h},${u}]}  " >> ${type}.txt
     done
     echo >> ${type}.txt
    done
     
    if [ "${debug}" != "0" ]; then
     cp ${type}.txt ${t}${hand}-${upcard}-${play}.str
    fi  
    
    # ensamble the full bs.txt
    cat hard.txt soft.txt pair.txt > bs.txt

    # play!
    blackjack

    # evaluate the results
    ev[${t}${hand},${upcard},${play}]=$(grep mean ${t}${hand}-${upcard}-${play}.yaml | awk '{printf("%g", $2)}')
    error[${t}${hand},${upcard},${play}]=$(grep error ${t}${hand}-${upcard}-${play}.yaml | awk '{printf("%g", $2)}')
    
   done
   
   # choose the best one
   ev_y=$(echo ${ev[${t}${hand},${upcard},y]} | awk '{printf("%+.2f", 100*$1)}')
   ev_n=$(echo ${ev[${t}${hand},${upcard},n]} | awk '{printf("%+.2f", 100*$1)}')
   
   if [ $n -le ${n_max} ]; then 
    # if we still have room, take into account errors
    error_y=$(echo ${error[${t}${hand},${upcard},y]} | awk '{printf("%.1f", 100*$1)}')
    error_n=$(echo ${error[${t}${hand},${upcard},n]} | awk '{printf("%.1f", 100*$1)}')
   else
    # instead of running infinite hands, above a threshold asume errors are zero
    error_y=0
    error_n=0
   fi  
 
   echo -ne "\t${ev_y}\t(${error_y})"
   echo -ne "\t${ev_n}\t(${error_n})"
   
   if   (( $(echo ${ev_y} ${error_y} ${ev_n} ${error_n} | awk '{print (($1-$2) > ($3+$4))}') )); then
   
    best="y"
    color=${GREEN}
    best_string="yes"
    
   elif (( $(echo ${ev_n} ${error_n} ${ev_y} ${error_y} | awk '{print (($1-$2) > ($3+$4))}') )); then
   
    best="n"
    color=${RED}
    best_string="no"
   
   else
   
    best="x"
    color=${NC}
    best_string="uncertain"
    
    n=$((${n} * 4))
    
   fi
   
   echo -e ${color}"\t"${best_string}${NC}
  done

  echo "| ${t}${hand}-${upcard} | $(printf %.1e ${n}) | ${ev_y} (${error_y}) | ${ev_n} (${error_n}) | ${best_string} | " >> table.md
  
  echo " <!-- ${upcard} -->" >> ${type}.html
  echo " <td>" >> ${type}.html
  echo ${ev_y} ${error_y} | awk -f html_cell.awk >> ${type}.html
  echo ${ev_n} ${error_n} | awk -f html_cell.awk >> ${type}.html
  echo " </td>" >> ${type}.html
  
  
  strategy[${t}${hand},${upcard}]=${best}
   
  # save the strategy again with the best strategy
  rm -f ${type}.txt
  for h in A T $(seq 9 -1 2); do
   echo -n "${t}${h}   " >> ${type}.txt
   for u in $(seq 2 9) T A; do
    echo -n "${strategy[${t}${h},${u}]}  " >> ${type}.txt
   done
   echo >> ${type}.txt
  done
 done
done

 
cat header.txt hard.txt header.txt soft.txt header.txt pair.txt > bs.txt
rm -f hard.txt soft.txt pair.txt blackjack.conf
if [ "${debug}" == "0" ]; then
 rm -f *.yaml
 rm -f *.str
 rm -f *.log
fi
 
@end verbatim

@bye