## Internal player If `blackjack` is called with the `-i` option, it uses an _internal_ player to play against itself. By default it plays basic strategy. Run ```terminal blackjack -i ``` and you will get the following report with the results of playing one million hands with basic strategy. ```yaml 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 ``` ## Always stand To play Blackjack as an “always-stander” run the following command: ```terminal yes stand | blackjack -n1e5 --flat_bet=true --no_insurance=true > /dev/null ``` The UNIX command `yes stand` writes the string “stand” repeteadly to the standard output, which is piped to the executable `blackjack` (assumed to be installed system-wide). The arguments tell Libre Blackjack to play one hundred thousand hands (`-n1e5`) using a flat bet (`flat_bet`, it defaults to a unit bet in each hand) and without asking for insurance if the dealer shows an ace (`no_insurance`). As there is no `blackjack.conf` file, the rules are---as expected---the default ones (see the documentation for details). The `/dev/null` part is important, otherwise Libre Blackjack will think that there is a human at the other side of the table and will 1. run slower (it will add explicit time delays to mimic an actual human dealer), and 2. give all the details of the dealt hands in the terminal as ASCII (actually UTF-8) art 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 `/dev/null` to save execution time. The results are written as a [YAML](http://yaml.org/)-formatted data to `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.). ```yaml 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 ``` > **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... ## 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): ```terminal mkfifo fifo ``` Then we execute `blackjack`, piping its output to the player (say `no-bust.pl`) and reading the standard input from `fifo`, whilst at the same time we redirect the player's standard output to `fifo`: ```terminal rm -f fifo; mkfifo fifo blackjack -n1e5 < fifo | ./no-bust.pl > fifo ``` As this time the player is coded in an interpreted langauge, it is far smarter than the previous `yes`-based player. So the player can handle bets and insurances, and there is not need to pass the options `--flat_bet` nor `--no_insurance` (though they can be passed anyway). Let us take a look at the Perl implementation: ```perl ##!/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($command = ); if ($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"; } } } ``` The very same player may be implemented as a shell script: ```bash ##!/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 ``` 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: ```terminal 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 ``` 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. > **Exercise:** modify the players so they always insure aces and see if it improves or degrades the result. ## 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 `blackjack.conf` is used. If a file with this name exists in the directory where `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: `player2dealer` and `dealer2player`. The first one sets the communication mechanism from the player to the dealer (by default is `blackjack`’s standard input), and the second one sets the mechanism from the dealer to the player (by default `blackjack`’s standard output). In this case, the configuration file reads: ```ini h17 = true ``` This means that two FIFOs (a.k.a. named pipes) are to be used for communication, `player2dealer` from the player to the dealer and `dealer2player` for the dealer to the player. If these FIFOs do not exist, they are created by `blackjack` upon execution. The player this time is implemented as an awk script, whose input should be read from `dealer2player` and whose output should be written to `player2dealer`. To run the game, execute `blackjack` in one terminal making sure the current directory is where the `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: ```terminal $ blackjack ``` In another terminal run the player ```terminal $ ./mimic-the-dealer.awk < dealer2player > player2dealer ``` Both dealer and player may be run in the same terminal putting the first one on the background: ```terminal rm -f d2p p2d; mkfifo d2p p2d gawk -f mimic-the-dealer.awk < d2p > p2d & blackjack -n1e5 > d2p < p2d ``` To understand the decisions taken by the player, we have to remember that when Libre Blackjack receives the command `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 `fflush()` are needed in order to avoid deadlocks on the named pipes: ```awk ##!/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; } ``` ```yaml 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 ``` > **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) ## Derivation of the basic strategy ### Quick run Execute the `run.sh` script. It should take a few minutes: ```terminal $ ./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 ``` A new text file called `bs.txt` with the strategy should be created from scratch: ``` ## 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 ``` ### Full table with results The script computes the expected value of each combination 1. Player’s hand (hard, soft and pair) 2. Dealer upcard 3. Hit, double or stand (for hard and soft hands) and splitting or not (for pairs) The results are given as the expected value in percentage with the uncertainty (one standard deviation) in the last significant digit. ```{=html}

Hand		2	3	4	5	6	7	8	9	T	A
+48(2) +8(1)	+52(2) +10(1)	+57(2) +12(1)	+61(2) +16(1)	+66(2) +20(2)	+47(2) +16(1)	+35(2) +10(1)	+23(2) -0(1)	+9(2) -14(1)	-24(2) -35.4(9)	-4(4) +63.3(7)	+5(4) +64.3(7)	+15(4) +65.6(7)	+28(4) +66.8(7)	+38(4) +67.9(7)	+9(3) +77.5(6)	-23(2) +79.0(6)	-61(3) +76.2(6)	-76(3) +43.6(7)	-86(2) +11(1)	+20(2) +11(1)	+26(2) +14(1)	+33(2) +16(1)	+39(2) +19(1)	+47(2) +23(1)	+36.6(9) +40.0(4)	+24(2) +10.1(8)	-8(2) -18(1)	-37(2) -24(1)	-48.4(8) -46.8(4)	+8(2) -29(1)	+15(2) -24(1)	+21(2) -21(1)	+30(2) -16(1)	+39(3) -12(1)	+32(2) -41.8(9)	-3(2) -45.6(9)	-39(2) -50.7(9)	-51(2) -57.3(8)	-66.5(8) -68.4(4)	-13(3) -28(1)	-6(3) -24(1)	+7(3) -21(1)	+16(3) -16(1)	+25(3) -12(1)	-4(2) -32(1)	-39(1) -36.9(5)	-57(2) -43.2(9)	-68(2) -51.0(9)	-82(2) -63.0(8)	-20(2) -25(1)	-13(3) -23(1)	-1(3) -21(1)	+9(3) -16(1)	+18(3) -12(1)	-26(2) -21(1)	-42(2) -27(1)	-61(2) -34(1)	-74(2) -43.0(9)	-81(2) -57.5(8)	-29(2) +36(2)	-21(2) +41(2)	-14(3) +46(2)	-2(3) +51(2)	+8(3) +55(2)	-30(2) +39(2)	-46(2) +30(2)	-63(2) +15(2)	-76(2) -5(1)	-83(1) -29(1)	-18(2) -2(1)	-10(3) -0(1)	-2(3) +4(1)	+8.3(7) +6.9(3)	+19(3) +9(1)	-17(2) +8(1)	-32(2) -6(1)	-52(2) -21(1)	-65(2) -31(1)	-76(2) -48.9(9)	-13.55(1) -13.84(1)	-5(1) -10.6(5)	+4(3) -8(1)	+13(3) -3(1)	+22(3) +0(1)	-6(2) -15(1)	-23(1) -21.6(5)	-43(2) -30(1)	-59(2) -38.9(9)	-70(2) -54.1(9)	-8(2) -12(1)	-2(2) -8(1)	+5(3) -4(1)	+17(3) -1(1)	+26(3) +3(1)	+1(2) -9(1)	-17.8(5) -16.0(2)	-40(2) -24(1)	-53(2) -34.2(9)	-67(2) -51.1(9)
Hand		2	3	4	5	6	7	8	9	T	A
s20	stand hit double	+63(2) +18(1) +36(2)	+65(2) +20(1) +42(2)	+65(2) +23(1) +46(2)	+67(2) +26(1) +51(2)	+68(2) +28(1) +57(2)	+77(2) +26(1) +40(2)	+79(2) +19(1) +28(2)	+76(2) +12(1) +16(2)	+44(2) -5(1) -8(2)	+11(2) -28(1) -35(2)
s19	stand hit double	+38(2) +12(1) +23(2)	+39(2) +14(1) +29(2)	+42(2) +18(1) +34(2)	+44(1) +20.2(5) +40(1)	+45.3(2) +23.1(1) +46.2(2)	+62(2) +23(1) +32(2)	+60(2) +15(1) +20(2)	+29(2) +1(1) -7(2)	-2(2) -15(1) -29(2)	-18(2) -34.4(9) -47(2)
s18	stand hit double	+11.01(1) +6.00(1) +11.43(1)	+14(1) +8.8(5) +18(1)	+16(2) +12(1) +22(2)	+20(2) +14(1) +29(2)	+22(2) +16(1) +36(2)	+40(2) +17(1) +22(2)	+11(2) +4(1) -2(2)	-18(2) -10(1) -28(2)	-24(2) -20(1) -39(2)	-46(2) -41.6(9) -60(2)
s17	stand hit double	-15.67(1) -0.59(1) -0.88(1)	-12(1) +2.4(5) +6(1)	-8(2) +5(1) +12(2)	-4(2) +9(1) +19(2)	-0(2) +10(1) +26(2)	-11(2) +6(1) -2(2)	-38(2) -8(1) -25(2)	-43(2) -14(1) -39(2)	-46(2) -26(1) -50(2)	-67(2) -46.6(9) -69(2)
s16	stand hit double	-29(2) -2(1) -8(2)	-25(1) +0.3(5) -1(1)	-21(1) +3.6(5) +6(1)	-16(2) +7(1) +14(2)	-13(2) +8(1) +20(2)	-48(2) -0(1) -18(2)	-51(2) -7(1) -31(2)	-54(2) -15(1) -45(2)	-58(2) -27(1) -56(2)	-73(2) -44.4(9) -72(2)
s15	stand hit double	-28(2) -1(1) -7(2)	-25(2) +3(1) -0(2)	-20.6(3) +5.6(1) +6.2(3)	-16(2) +9(1) +14(2)	-13(2) +10(1) +20(2)	-48(2) +3(1) -18(2)	-51(2) -3(1) -32(2)	-55(2) -11(1) -46(2)	-57(2) -24(1) -55(2)	-72(2) -42.4(9) -72(2)
s14	stand hit double	-28(2) +2(1) -6(2)	-24(2) +4(1) -1(2)	-20(2) +8(1) +6(2)	-16(1) +10.8(5) +12(1)	-12(2) +12(1) +20(2)	-48(2) +8(1) -19(2)	-51(2) +2(1) -32(2)	-54(2) -8(1) -45(2)	-57(2) -20(1) -56(2)	-72(2) -39.8(9) -71(2)
s13	stand hit double	-29(2) +4(1) -7(2)	-24(2) +7(1) -0(2)	-20(2) +10(1) +5(2)	-16.46(1) +12.88(1) +12.73(1)	-13(2) +14(1) +19(2)	-48(2) +12(1) -19(2)	-51(2) +6(1) -33(2)	-54(2) -3(1) -46(2)	-57(2) -17(1) -55(2)	-72(2) -37.2(9) -72(2)
s12	stand hit double	-29(2) +7(1) -6(2)	-25(2) +10(1) -0(2)	-20(2) +12(1) +6(2)	-16(2) +15(1) +13(2)	-12(2) +15(1) +21(2)	-47(2) +17(1) -18(2)	-51(2) +9(1) -31(2)	-54(2) +0(1) -45(2)	-58(2) -15(1) -57(2)	-72(2) -35.2(9) -72(2)
Hand		2	3	4	5	6	7	8	9	T	A
h20	stand hit double	+64(1) -85.6(5) -171(1)	+65(1) -85.6(5) -171(1)	+65(1) -85.7(5) -171(1)	+67(1) -85.5(5) -171(1)	+68(1) -85.2(5) -171(1)	+77(1) -85.1(6) -170(1)	+79(1) -85.0(6) -170(1)	+76(1) -85.0(6) -170(1)	+44(1) -86.1(5) -164(1)	+11(1) -89.6(5) -149(1)
h19	stand hit double	+37(1) -73.1(7) -146(1)	+40(1) -73.0(7) -145(1)	+41(1) -72.8(7) -146(1)	+44(1) -72.8(7) -144(1)	+46(1) -72.6(7) -145(1)	+61(2) -71.5(7) -143(2)	+60(2) -71.4(7) -143(2)	+28(2) -71.6(7) -143(2)	-2(1) -74.9(7) -143(1)	-18(1) -81.4(6) -131(1)
h18	stand hit double	+11(2) -62.2(8) -124(2)	+13(2) -61.5(8) -124(2)	+17(2) -61.8(8) -124(2)	+19(2) -61.4(8) -124(2)	+22(2) -61.4(8) -122(2)	+39(2) -59.4(8) -118(2)	+10(2) -58.4(8) -117(2)	-19(2) -61.5(8) -123(2)	-24(2) -67.5(8) -128(2)	-47(1) -74.6(7) -119(1)
h17	stand hit double	-15(2) -53.9(9) -109(2)	-12(2) -53.6(9) -106(2)	-8(2) -52.9(9) -106(2)	-5(2) -51.7(9) -105(2)	-1(2) -51.9(9) -104(2)	-11(2) -48.1(9) -96(2)	-39(2) -50.4(9) -101(2)	-43(2) -55.4(8) -111(2)	-46(2) -61.9(8) -116(2)	-66(1) -70.8(7) -111(1)
h16	stand hit double	-29(2) -47.5(9) -95(2)	-25(2) -46.5(9) -94(2)	-21(2) -45.5(9) -92(2)	-16(2) -45.2(9) -90(2)	-12(2) -44.4(9) -88(2)	-47(2) -41.3(9) -84(2)	-51(2) -46.0(9) -92(2)	-55(2) -50.7(9) -103(2)	-57.57(1) -57.53(1) -107.34(1)	-72(1) -68.2(7) -105(1)
h15	stand hit double	-28(2) -41.5(9) -84(2)	-25(2) -40.7(9) -81(2)	-21(2) -39.4(9) -79(2)	-17(2) -38(1) -78(2)	-12(2) -38(1) -75(2)	-48(2) -37.0(9) -75(2)	-51(2) -41.5(9) -84(2)	-54(2) -47.3(9) -95(2)	-58(2) -54.6(9) -100(2)	-72(2) -65.8(8) -101(2)
h14	stand hit double	-29(2) -37(1) -72(2)	-24(2) -35(1) -71(2)	-20(2) -34(1) -69(2)	-17(2) -32(1) -64(2)	-12(2) -31(1) -61(2)	-48(2) -32(1) -66(2)	-51(2) -37.6(9) -77(2)	-54(2) -43.3(9) -88(2)	-58(2) -50.7(9) -95(2)	-72(2) -62.8(8) -98(2)
h13	stand hit double	-29(1) -31.0(5) -62(1)	-25(2) -29(1) -59(2)	-21(2) -27(1) -55(2)	-17(2) -26(1) -50(2)	-12(2) -24(1) -48(2)	-48(2) -27(1) -58(2)	-51(2) -32(1) -70(2)	-54(2) -38.6(9) -81(2)	-57(2) -46.5(9) -87(2)	-72(2) -60.3(8) -92(2)
h12	stand hit double	-29(2) -25(1) -50(2)	-25(1) -23.2(5) -47(1)	-20.5(3) -21.3(1) -42.8(3)	-17(1) -19.5(5) -39(1)	-12(2) -17(1) -35(2)	-47(2) -21(1) -50(2)	-51(2) -27(1) -62(2)	-54(2) -34(1) -74(2)	-58(2) -43.0(9) -82(2)	-73(2) -57.4(8) -89(2)
h11	stand hit double	-28(2) +24(1) +47(2)	-25(2) +26(1) +51(2)	-20(2) +29(1) +56(2)	-16(2) +31(1) +61(2)	-12(2) +33(1) +67(2)	-47(2) +29(1) +46(2)	-51(2) +22(1) +35(2)	-54(2) +16(1) +24(2)	-58(2) +3(1) +10(2)	-72.22(1) -23.67(1) -23.56(1)
h10	stand hit double	-28(2) +18(1) +36(2)	-24(2) +21(1) +41(2)	-21(2) +23(1) +47(2)	-16(2) +26(1) +52(2)	-12(2) +29(1) +56(2)	-47(2) +25(1) +38(2)	-51(2) +20(1) +29(2)	-54(1) +11.8(5) +15(1)	-58(2) -6(1) -8(2)	-72(2) -29(1) -34(2)
h9	stand hit double	-29(1) +7.6(5) +6(1)	-25(1) +9.9(5) +12(1)	-21(2) +12(1) +18(2)	-16(2) +16(1) +24(2)	-12(2) +20(1) +31(2)	-48(2) +17(1) +10(2)	-51(2) +10(1) -2(2)	-54(2) -5(1) -30(2)	-58(2) -22(1) -52(2)	-72(2) -38.8(9) -61(2)
h8	stand hit double	-29(2) -2(1) -19(2)	-26(2) +0(1) -14(2)	-21(2) +4(1) -8(2)	-16(2) +8(1) +0(2)	-12(1) +10.5(5) +7(1)	-47(2) +9(1) -17(2)	-52(2) -6(1) -46(2)	-54(2) -22(1) -72(2)	-57(2) -31(1) -76(2)	-72(2) -49.0(9) -88(2)
h7	stand hit double	-29(2) -11(1) -43(2)	-24(2) -8(1) -35(2)	-19(2) -4(1) -28(2)	-17(2) -1(1) -21(2)	-12(2) +3(1) -11(2)	-47(2) -7(1) -60(2)	-51(2) -20(1) -85(2)	-54(2) -29(1) -96(2)	-58(2) -37.1(9) -96(2)	-72(1) -55.5(8) -105(1)
h6	stand hit double	-28(2) -16(1) -56(2)	-24(2) -12(1) -47(2)	-21(2) -8(1) -39(2)	-17(2) -5(1) -31(2)	-12(2) -1(1) -22(2)	-48(2) -16(1) -90(2)	-51(2) -24(1) -99(2)	-54(2) -31(1) -107(2)	-58(2) -40.0(9) -105(2)	-72(1) -56.3(8) -112(1)
h5	stand hit double	-28(2) -14(1) -56(2)	-25(2) -12(1) -49(2)	-21(2) -8(1) -43(2)	-16(2) -4(1) -33(2)	-12(2) -0(1) -24(2)	-47(2) -16(1) -94(2)	-51(2) -22(1) -102(2)	-54(2) -30(1) -108(2)	-57(2) -38.8(9) -107(2)	-72(1) -54.7(8) -114(1)
h4	stand hit double	-29(2) -13(1) -57(2)	-25(2) -10(1) -50(2)	-21(2) -7(1) -41(2)	-17(2) -3(1) -33(2)	-12(2) +0(1) -24(2)	-48(2) -13(1) -95(2)	-51(2) -20(1) -103(2)	-54(2) -28(1) -109(2)	-57(2) -36.6(9) -107(2)	-72(1) -53.8(9) -113(1)

``` | Hand | Hands needed | Stand [%] | Double [%] | Hit [%] | Play | |:------:|:-----:|:-----------:|:------------:|:--------:|:---------:| | h20-2 | 8.0e+04 | +63.55 (1.1) | -85.62 (0.5) | -171.16 (1.1) | stand | | h20-3 | 8.0e+04 | +64.58 (1.1) | -85.56 (0.5) | -170.56 (1.1) | stand | | h20-4 | 8.0e+04 | +65.30 (1.1) | -85.70 (0.5) | -171.10 (1.1) | stand | | h20-5 | 8.0e+04 | +67.04 (1.1) | -85.52 (0.5) | -171.34 (1.1) | stand | | h20-6 | 8.0e+04 | +67.81 (1.1) | -85.17 (0.5) | -170.61 (1.1) | stand | | h20-7 | 8.0e+04 | +77.12 (1.1) | -85.13 (0.6) | -170.32 (1.1) | stand | | h20-8 | 8.0e+04 | +79.14 (1.1) | -84.99 (0.6) | -170.46 (1.1) | stand | | h20-9 | 8.0e+04 | +75.94 (1.1) | -85.00 (0.6) | -169.53 (1.1) | stand | | h20-T | 8.0e+04 | +43.51 (1.1) | -86.06 (0.5) | -164.02 (1.1) | stand | | h20-A | 8.0e+04 | +10.86 (1.0) | -89.62 (0.5) | -149.06 (1.0) | stand | | h19-2 | 8.0e+04 | +37.41 (1.4) | -73.09 (0.7) | -146.25 (1.4) | stand | | h19-3 | 8.0e+04 | +40.09 (1.4) | -72.96 (0.7) | -145.29 (1.4) | stand | | h19-4 | 8.0e+04 | +41.38 (1.4) | -72.80 (0.7) | -146.07 (1.4) | stand | | h19-5 | 8.0e+04 | +43.76 (1.4) | -72.76 (0.7) | -144.19 (1.4) | stand | | h19-6 | 8.0e+04 | +45.80 (1.4) | -72.62 (0.7) | -145.14 (1.4) | stand | | h19-7 | 8.0e+04 | +61.39 (1.5) | -71.50 (0.7) | -143.39 (1.5) | stand | | h19-8 | 8.0e+04 | +59.58 (1.5) | -71.36 (0.7) | -143.03 (1.5) | stand | | h19-9 | 8.0e+04 | +28.15 (1.5) | -71.59 (0.7) | -143.42 (1.5) | stand | | h19-T | 8.0e+04 | -2.00 (1.3) | -74.91 (0.7) | -142.88 (1.3) | stand | | h19-A | 8.0e+04 | -18.04 (1.2) | -81.43 (0.6) | -131.46 (1.2) | stand | | h18-2 | 8.0e+04 | +11.02 (1.6) | -62.22 (0.8) | -123.71 (1.6) | stand | | h18-3 | 8.0e+04 | +13.20 (1.6) | -61.53 (0.8) | -124.35 (1.6) | stand | | h18-4 | 8.0e+04 | +16.92 (1.6) | -61.80 (0.8) | -123.86 (1.6) | stand | | h18-5 | 8.0e+04 | +19.35 (1.6) | -61.38 (0.8) | -123.62 (1.6) | stand | | h18-6 | 8.0e+04 | +22.36 (1.6) | -61.35 (0.8) | -121.85 (1.6) | stand | | h18-7 | 8.0e+04 | +39.35 (1.7) | -59.36 (0.8) | -118.48 (1.7) | stand | | h18-8 | 8.0e+04 | +10.34 (1.7) | -58.35 (0.8) | -117.43 (1.7) | stand | | h18-9 | 8.0e+04 | -18.74 (1.6) | -61.52 (0.8) | -123.34 (1.6) | stand | | h18-T | 8.0e+04 | -23.84 (1.5) | -67.55 (0.8) | -127.96 (1.5) | stand | | h18-A | 8.0e+04 | -46.97 (1.3) | -74.64 (0.7) | -118.63 (1.3) | stand | | h17-2 | 8.0e+04 | -15.44 (1.7) | -53.93 (0.9) | -108.61 (1.7) | stand | | h17-3 | 8.0e+04 | -12.18 (1.8) | -53.57 (0.9) | -105.83 (1.8) | stand | | h17-4 | 8.0e+04 | -8.33 (1.8) | -52.86 (0.9) | -105.69 (1.8) | stand | | h17-5 | 8.0e+04 | -4.55 (1.8) | -51.66 (0.9) | -105.34 (1.8) | stand | | h17-6 | 8.0e+04 | -1.15 (1.8) | -51.94 (0.9) | -104.07 (1.8) | stand | | h17-7 | 8.0e+04 | -10.79 (1.8) | -48.10 (0.9) | -96.43 (1.8) | stand | | h17-8 | 8.0e+04 | -38.74 (1.8) | -50.44 (0.9) | -101.03 (1.8) | stand | | h17-9 | 8.0e+04 | -42.54 (1.7) | -55.42 (0.8) | -111.46 (1.7) | stand | | h17-T | 8.0e+04 | -46.12 (1.6) | -61.91 (0.8) | -115.53 (1.6) | stand | | h17-A | 8.0e+04 | -66.10 (1.4) | -70.80 (0.7) | -110.62 (1.4) | stand | | h16-2 | 8.0e+04 | -28.62 (1.8) | -47.46 (0.9) | -95.12 (1.8) | stand | | h16-3 | 8.0e+04 | -24.81 (1.8) | -46.52 (0.9) | -93.95 (1.8) | stand | | h16-4 | 8.0e+04 | -20.67 (1.8) | -45.46 (0.9) | -91.92 (1.8) | stand | | h16-5 | 8.0e+04 | -16.48 (1.8) | -45.18 (0.9) | -89.56 (1.8) | stand | | h16-6 | 8.0e+04 | -11.55 (1.9) | -44.39 (0.9) | -87.64 (1.9) | stand | | h16-7 | 8.0e+04 | -47.14 (1.9) | -41.27 (0.9) | -84.33 (1.9) | hit | | h16-8 | 8.0e+04 | -50.94 (1.8) | -45.95 (0.9) | -92.12 (1.8) | hit | | h16-9 | 8.0e+04 | -54.50 (1.7) | -50.71 (0.9) | -102.58 (1.7) | hit | | h16-T | 2.0e+07 | -57.57 (0) | -57.53 (0) | -107.34 (0) | hit | | h16-A | 8.0e+04 | -72.39 (1.4) | -68.19 (0.7) | -105.24 (1.4) | hit | | h15-2 | 8.0e+04 | -28.36 (1.9) | -41.53 (0.9) | -83.65 (1.9) | stand | | h15-3 | 8.0e+04 | -25.21 (1.9) | -40.72 (0.9) | -81.48 (1.9) | stand | | h15-4 | 8.0e+04 | -20.66 (1.9) | -39.38 (0.9) | -78.86 (1.9) | stand | | h15-5 | 8.0e+04 | -16.55 (1.9) | -38.38 (1.0) | -77.57 (1.9) | stand | | h15-6 | 8.0e+04 | -11.88 (1.9) | -37.86 (1.0) | -75.22 (1.9) | stand | | h15-7 | 8.0e+04 | -48.09 (1.9) | -37.04 (0.9) | -75.35 (1.9) | hit | | h15-8 | 8.0e+04 | -50.84 (1.9) | -41.53 (0.9) | -83.94 (1.9) | hit | | h15-9 | 8.0e+04 | -54.38 (1.8) | -47.30 (0.9) | -95.06 (1.8) | hit | | h15-T | 8.0e+04 | -57.73 (1.7) | -54.62 (0.9) | -99.81 (1.7) | hit | | h15-A | 8.0e+04 | -72.00 (1.5) | -65.80 (0.8) | -101.33 (1.5) | hit | | h14-2 | 8.0e+04 | -28.56 (1.9) | -36.62 (1.0) | -72.17 (1.9) | stand | | h14-3 | 8.0e+04 | -24.12 (1.9) | -34.63 (1.0) | -70.90 (1.9) | stand | | h14-4 | 8.0e+04 | -20.48 (1.9) | -33.73 (1.0) | -68.97 (1.9) | stand | | h14-5 | 8.0e+04 | -16.54 (2.0) | -31.93 (1.0) | -64.39 (2.0) | stand | | h14-6 | 8.0e+04 | -12.19 (2.0) | -30.64 (1.0) | -60.75 (2.0) | stand | | h14-7 | 8.0e+04 | -47.85 (1.9) | -32.25 (1.0) | -66.44 (1.9) | hit | | h14-8 | 8.0e+04 | -51.09 (1.9) | -37.59 (0.9) | -76.59 (1.9) | hit | | h14-9 | 8.0e+04 | -54.41 (1.8) | -43.27 (0.9) | -87.80 (1.8) | hit | | h14-T | 8.0e+04 | -57.74 (1.7) | -50.71 (0.9) | -94.55 (1.7) | hit | | h14-A | 8.0e+04 | -72.11 (1.5) | -62.81 (0.8) | -97.57 (1.5) | hit | | h13-2 | 3.2e+05 | -28.58 (1.0) | -30.97 (0.5) | -61.58 (1.0) | stand | | h13-3 | 8.0e+04 | -24.61 (2.0) | -29.17 (1.0) | -58.99 (2.0) | stand | | h13-4 | 8.0e+04 | -21.15 (2.0) | -27.05 (1.0) | -55.31 (2.0) | stand | | h13-5 | 8.0e+04 | -16.88 (2.0) | -26.15 (1.0) | -50.31 (2.0) | stand | | h13-6 | 8.0e+04 | -12.35 (2.0) | -24.12 (1.0) | -47.91 (2.0) | stand | | h13-7 | 8.0e+04 | -47.50 (2.0) | -26.69 (1.0) | -58.47 (2.0) | hit | | h13-8 | 8.0e+04 | -51.33 (1.9) | -32.22 (1.0) | -70.19 (1.9) | hit | | h13-9 | 8.0e+04 | -54.31 (1.9) | -38.57 (0.9) | -80.97 (1.9) | hit | | h13-T | 8.0e+04 | -56.99 (1.8) | -46.49 (0.9) | -87.36 (1.8) | hit | | h13-A | 8.0e+04 | -72.39 (1.5) | -60.33 (0.8) | -92.36 (1.5) | hit | | h12-2 | 8.0e+04 | -28.53 (2.0) | -25.26 (1.0) | -50.10 (2.0) | hit | | h12-3 | 3.2e+05 | -24.74 (1.0) | -23.20 (0.5) | -47.17 (1.0) | hit | | h12-4 | 5.1e+06 | -20.50 (0.3) | -21.34 (0.1) | -42.80 (0.3) | stand | | h12-5 | 3.2e+05 | -16.67 (1.0) | -19.49 (0.5) | -38.59 (1.0) | stand | | h12-6 | 8.0e+04 | -11.57 (2.0) | -17.45 (1.0) | -34.91 (2.0) | stand | | h12-7 | 8.0e+04 | -47.16 (2.0) | -21.22 (1.0) | -49.56 (2.0) | hit | | h12-8 | 8.0e+04 | -51.23 (2.0) | -26.82 (1.0) | -61.96 (2.0) | hit | | h12-9 | 8.0e+04 | -54.02 (1.9) | -33.93 (1.0) | -74.10 (1.9) | hit | | h12-T | 8.0e+04 | -58.14 (1.8) | -43.03 (0.9) | -82.21 (1.8) | hit | | h12-A | 8.0e+04 | -72.60 (1.5) | -57.38 (0.8) | -88.89 (1.5) | hit | | h11-2 | 8.0e+04 | -27.73 (2.0) | +23.74 (1.0) | +46.68 (2.0) | double | | h11-3 | 8.0e+04 | -24.81 (2.0) | +26.30 (1.0) | +50.86 (2.0) | double | | h11-4 | 8.0e+04 | -20.28 (2.0) | +28.91 (1.0) | +56.46 (2.0) | double | | h11-5 | 8.0e+04 | -16.10 (1.9) | +31.08 (1.0) | +60.58 (1.9) | double | | h11-6 | 8.0e+04 | -12.18 (1.9) | +33.16 (1.0) | +67.19 (1.9) | double | | h11-7 | 8.0e+04 | -47.49 (2.0) | +29.49 (1.0) | +46.09 (2.0) | double | | h11-8 | 8.0e+04 | -51.29 (2.0) | +22.39 (1.0) | +34.94 (2.0) | double | | h11-9 | 8.0e+04 | -54.41 (2.0) | +15.62 (1.0) | +23.72 (2.0) | double | | h11-T | 8.0e+04 | -57.70 (2.0) | +3.21 (1.0) | +9.84 (2.0) | double | | h11-A | 2.0e+07 | -72.22 (0) | -23.67 (0) | -23.56 (0) | double | | h10-2 | 8.0e+04 | -28.45 (2.0) | +18.25 (1.0) | +35.73 (2.0) | double | | h10-3 | 8.0e+04 | -23.99 (2.0) | +20.70 (1.0) | +40.59 (2.0) | double | | h10-4 | 8.0e+04 | -21.31 (2.0) | +23.14 (1.0) | +46.54 (2.0) | double | | h10-5 | 8.0e+04 | -16.23 (2.0) | +25.77 (1.0) | +51.66 (2.0) | double | | h10-6 | 8.0e+04 | -12.01 (2.0) | +28.63 (1.0) | +56.36 (2.0) | double | | h10-7 | 8.0e+04 | -47.40 (2.0) | +25.23 (1.0) | +38.38 (2.0) | double | | h10-8 | 8.0e+04 | -51.35 (2.0) | +20.43 (1.0) | +28.63 (2.0) | double | | h10-9 | 3.2e+05 | -54.13 (1.0) | +11.81 (0.5) | +15.23 (1.0) | double | | h10-T | 8.0e+04 | -58.04 (1.9) | -5.84 (1.0) | -7.94 (1.9) | hit | | h10-A | 8.0e+04 | -72.46 (1.7) | -28.90 (1.0) | -33.54 (1.7) | hit | | h9-2 | 3.2e+05 | -28.82 (1.0) | +7.62 (0.5) | +5.95 (1.0) | hit | | h9-3 | 3.2e+05 | -24.59 (1.0) | +9.93 (0.5) | +11.94 (1.0) | double | | h9-4 | 8.0e+04 | -20.79 (2.0) | +12.15 (1.0) | +17.76 (2.0) | double | | h9-5 | 8.0e+04 | -16.50 (2.0) | +15.92 (1.0) | +23.51 (2.0) | double | | h9-6 | 8.0e+04 | -12.35 (2.0) | +19.57 (1.0) | +30.65 (2.0) | double | | h9-7 | 8.0e+04 | -47.62 (2.0) | +17.20 (1.0) | +10.00 (2.0) | hit | | h9-8 | 8.0e+04 | -50.97 (2.0) | +10.15 (1.0) | -2.38 (2.0) | hit | | h9-9 | 8.0e+04 | -53.95 (1.9) | -4.61 (1.0) | -29.84 (1.9) | hit | | h9-T | 8.0e+04 | -57.78 (1.9) | -21.96 (1.0) | -52.36 (1.9) | hit | | h9-A | 8.0e+04 | -72.33 (1.7) | -38.77 (0.9) | -61.26 (1.7) | hit | | h8-2 | 8.0e+04 | -29.03 (2.0) | -1.76 (1.0) | -19.15 (2.0) | hit | | h8-3 | 8.0e+04 | -25.61 (2.1) | +0.33 (1.0) | -13.68 (2.1) | hit | | h8-4 | 8.0e+04 | -20.69 (2.1) | +3.82 (1.0) | -7.66 (2.1) | hit | | h8-5 | 8.0e+04 | -16.06 (2.1) | +7.53 (1.0) | +0.32 (2.1) | hit | | h8-6 | 3.2e+05 | -12.16 (1.0) | +10.49 (0.5) | +7.43 (1.0) | hit | | h8-7 | 8.0e+04 | -47.34 (2.0) | +8.56 (1.0) | -16.64 (2.0) | hit | | h8-8 | 8.0e+04 | -51.56 (1.9) | -6.18 (1.0) | -46.07 (1.9) | hit | | h8-9 | 8.0e+04 | -54.44 (1.9) | -21.65 (1.0) | -71.64 (1.9) | hit | | h8-T | 8.0e+04 | -57.15 (1.8) | -30.78 (1.0) | -76.35 (1.8) | hit | | h8-A | 8.0e+04 | -72.01 (1.5) | -49.02 (0.9) | -88.47 (1.5) | hit | | h7-2 | 8.0e+04 | -29.15 (2.0) | -11.08 (1.0) | -42.53 (2.0) | hit | | h7-3 | 8.0e+04 | -24.37 (2.0) | -7.51 (1.0) | -34.68 (2.0) | hit | | h7-4 | 8.0e+04 | -19.40 (2.1) | -3.90 (1.0) | -27.57 (2.1) | hit | | h7-5 | 8.0e+04 | -16.84 (2.1) | -0.53 (1.0) | -20.59 (2.1) | hit | | h7-6 | 8.0e+04 | -11.93 (2.1) | +2.53 (1.0) | -10.66 (2.1) | hit | | h7-7 | 8.0e+04 | -47.22 (1.9) | -7.33 (1.0) | -59.71 (1.9) | hit | | h7-8 | 8.0e+04 | -50.71 (1.8) | -20.35 (1.0) | -85.02 (1.8) | hit | | h7-9 | 8.0e+04 | -54.12 (1.8) | -29.11 (1.0) | -96.16 (1.8) | hit | | h7-T | 8.0e+04 | -57.52 (1.7) | -37.06 (0.9) | -95.95 (1.7) | hit | | h7-A | 8.0e+04 | -72.23 (1.4) | -55.54 (0.8) | -104.99 (1.4) | hit | | h6-2 | 8.0e+04 | -28.38 (2.0) | -15.59 (1.0) | -56.44 (2.0) | hit | | h6-3 | 8.0e+04 | -24.34 (2.1) | -11.56 (1.0) | -46.62 (2.1) | hit | | h6-4 | 8.0e+04 | -21.46 (2.1) | -8.03 (1.0) | -39.24 (2.1) | hit | | h6-5 | 8.0e+04 | -16.68 (2.1) | -4.95 (1.0) | -31.10 (2.1) | hit | | h6-6 | 8.0e+04 | -12.40 (2.1) | -0.76 (1.0) | -22.49 (2.1) | hit | | h6-7 | 8.0e+04 | -47.58 (1.9) | -16.43 (1.0) | -89.56 (1.9) | hit | | h6-8 | 8.0e+04 | -50.72 (1.8) | -24.13 (1.0) | -98.62 (1.8) | hit | | h6-9 | 8.0e+04 | -54.32 (1.8) | -31.34 (1.0) | -107.41 (1.8) | hit | | h6-T | 8.0e+04 | -57.56 (1.7) | -40.02 (0.9) | -105.28 (1.7) | hit | | h6-A | 8.0e+04 | -72.04 (1.4) | -56.31 (0.8) | -112.37 (1.4) | hit | | h5-2 | 8.0e+04 | -28.46 (2.0) | -14.09 (1.0) | -56.05 (2.0) | hit | | h5-3 | 8.0e+04 | -24.73 (2.1) | -11.80 (1.0) | -49.48 (2.1) | hit | | h5-4 | 8.0e+04 | -20.62 (2.1) | -8.01 (1.0) | -42.69 (2.1) | hit | | h5-5 | 8.0e+04 | -16.47 (2.1) | -4.45 (1.0) | -32.88 (2.1) | hit | | h5-6 | 8.0e+04 | -12.29 (2.1) | -0.24 (1.0) | -24.41 (2.1) | hit | | h5-7 | 8.0e+04 | -47.07 (1.9) | -16.19 (1.0) | -94.28 (1.9) | hit | | h5-8 | 8.0e+04 | -51.00 (1.8) | -22.25 (1.0) | -101.66 (1.8) | hit | | h5-9 | 8.0e+04 | -54.20 (1.8) | -29.74 (1.0) | -107.96 (1.8) | hit | | h5-T | 8.0e+04 | -57.42 (1.7) | -38.76 (0.9) | -107.09 (1.7) | hit | | h5-A | 8.0e+04 | -72.39 (1.4) | -54.71 (0.8) | -114.03 (1.4) | hit | | h4-2 | 8.0e+04 | -28.76 (2.0) | -12.93 (1.0) | -57.47 (2.0) | hit | | h4-3 | 8.0e+04 | -24.68 (2.1) | -10.48 (1.0) | -49.92 (2.1) | hit | | h4-4 | 8.0e+04 | -20.67 (2.1) | -7.36 (1.0) | -41.23 (2.1) | hit | | h4-5 | 8.0e+04 | -16.91 (2.1) | -3.43 (1.0) | -32.59 (2.1) | hit | | h4-6 | 8.0e+04 | -12.25 (2.1) | +0.40 (1.0) | -23.57 (2.1) | hit | | h4-7 | 8.0e+04 | -47.75 (1.9) | -12.72 (1.0) | -94.73 (1.9) | hit | | h4-8 | 8.0e+04 | -50.76 (1.8) | -19.63 (1.0) | -103.23 (1.8) | hit | | h4-9 | 8.0e+04 | -54.06 (1.8) | -28.13 (1.0) | -108.57 (1.8) | hit | | h4-T | 8.0e+04 | -56.86 (1.7) | -36.62 (0.9) | -107.18 (1.7) | hit | | h4-A | 8.0e+04 | -72.38 (1.4) | -53.76 (0.9) | -113.22 (1.4) | hit | | s20-2 | 8.0e+04 | +63.48 (2.0) | +18.05 (1.0) | +36.30 (2.0) | stand | | s20-3 | 8.0e+04 | +64.51 (2.0) | +19.96 (1.0) | +41.60 (2.0) | stand | | s20-4 | 8.0e+04 | +65.47 (2.0) | +22.51 (1.0) | +45.96 (2.0) | stand | | s20-5 | 8.0e+04 | +66.59 (2.0) | +26.19 (1.0) | +50.69 (2.0) | stand | | s20-6 | 8.0e+04 | +68.22 (2.0) | +28.20 (1.0) | +56.57 (2.0) | stand | | s20-7 | 8.0e+04 | +77.45 (2.0) | +26.05 (1.0) | +40.16 (2.0) | stand | | s20-8 | 8.0e+04 | +79.03 (2.0) | +19.32 (1.0) | +28.44 (2.0) | stand | | s20-9 | 8.0e+04 | +76.09 (2.0) | +11.81 (1.0) | +15.62 (2.0) | stand | | s20-T | 8.0e+04 | +43.83 (1.9) | -5.43 (1.0) | -8.42 (1.9) | stand | | s20-A | 8.0e+04 | +10.71 (1.7) | -28.46 (1.0) | -34.57 (1.7) | stand | | s19-2 | 8.0e+04 | +37.55 (2.0) | +12.16 (1.0) | +23.28 (2.0) | stand | | s19-3 | 8.0e+04 | +39.32 (2.0) | +14.45 (1.0) | +28.76 (2.0) | stand | | s19-4 | 8.0e+04 | +41.67 (2.0) | +17.97 (1.0) | +34.49 (2.0) | stand | | s19-5 | 3.2e+05 | +43.87 (1.0) | +20.25 (0.5) | +39.76 (1.0) | stand | | s19-6 | 5.1e+06 | +45.30 (0.2) | +23.06 (0.1) | +46.24 (0.2) | double | | s19-7 | 8.0e+04 | +62.00 (2.0) | +22.80 (1.0) | +32.42 (2.0) | stand | | s19-8 | 8.0e+04 | +59.68 (2.0) | +15.07 (1.0) | +20.09 (2.0) | stand | | s19-9 | 8.0e+04 | +28.83 (2.0) | +1.29 (1.0) | -7.06 (2.0) | stand | | s19-T | 8.0e+04 | -1.88 (1.9) | -15.44 (1.0) | -29.40 (1.9) | stand | | s19-A | 8.0e+04 | -17.63 (1.7) | -34.41 (0.9) | -46.94 (1.7) | stand | | s18-2 | 2.0e+07 | +11.01 (0) | +6.00 (0) | +11.43 (0) | double | | s18-3 | 3.2e+05 | +13.74 (1.0) | +8.81 (0.5) | +17.70 (1.0) | double | | s18-4 | 8.0e+04 | +16.38 (2.0) | +12.04 (1.0) | +22.38 (2.0) | double | | s18-5 | 8.0e+04 | +19.67 (2.0) | +14.12 (1.0) | +29.11 (2.0) | double | | s18-6 | 8.0e+04 | +22.24 (2.0) | +15.83 (1.0) | +35.89 (2.0) | double | | s18-7 | 8.0e+04 | +39.95 (2.0) | +17.00 (1.0) | +21.63 (2.0) | stand | | s18-8 | 8.0e+04 | +10.59 (2.0) | +3.79 (1.0) | -2.20 (2.0) | stand | | s18-9 | 8.0e+04 | -18.25 (2.0) | -10.19 (1.0) | -27.83 (2.0) | hit | | s18-T | 8.0e+04 | -23.79 (1.9) | -20.45 (1.0) | -38.84 (1.9) | hit | | s18-A | 8.0e+04 | -46.31 (1.7) | -41.64 (0.9) | -59.63 (1.7) | hit | | s17-2 | 2.0e+07 | -15.67 (0) | -0.59 (0) | -0.88 (0) | hit | | s17-3 | 3.2e+05 | -12.35 (1.0) | +2.42 (0.5) | +5.90 (1.0) | double | | s17-4 | 8.0e+04 | -8.13 (2.0) | +5.14 (1.0) | +11.54 (2.0) | double | | s17-5 | 8.0e+04 | -4.49 (2.0) | +9.23 (1.0) | +18.59 (2.0) | double | | s17-6 | 8.0e+04 | -0.39 (2.0) | +10.43 (1.0) | +25.67 (2.0) | double | | s17-7 | 8.0e+04 | -10.70 (2.0) | +5.76 (1.0) | -2.08 (2.0) | hit | | s17-8 | 8.0e+04 | -37.98 (2.0) | -7.55 (1.0) | -24.83 (2.0) | hit | | s17-9 | 8.0e+04 | -42.52 (2.0) | -14.14 (1.0) | -39.49 (2.0) | hit | | s17-T | 8.0e+04 | -46.49 (1.9) | -25.88 (1.0) | -50.28 (1.9) | hit | | s17-A | 8.0e+04 | -66.85 (1.6) | -46.56 (0.9) | -68.92 (1.6) | hit | | s16-2 | 8.0e+04 | -28.87 (2.1) | -1.93 (1.0) | -7.79 (2.1) | hit | | s16-3 | 3.2e+05 | -24.68 (1.0) | +0.27 (0.5) | -1.15 (1.0) | hit | | s16-4 | 3.2e+05 | -20.67 (1.0) | +3.56 (0.5) | +5.67 (1.0) | double | | s16-5 | 8.0e+04 | -16.41 (2.1) | +6.62 (1.0) | +13.59 (2.1) | double | | s16-6 | 8.0e+04 | -12.54 (2.1) | +8.47 (1.0) | +19.52 (2.1) | double | | s16-7 | 8.0e+04 | -47.66 (2.1) | -0.28 (1.0) | -17.54 (2.1) | hit | | s16-8 | 8.0e+04 | -51.11 (2.0) | -6.82 (1.0) | -31.42 (2.0) | hit | | s16-9 | 8.0e+04 | -54.37 (2.0) | -14.60 (1.0) | -45.25 (2.0) | hit | | s16-T | 8.0e+04 | -57.80 (1.9) | -26.81 (1.0) | -55.92 (1.9) | hit | | s16-A | 8.0e+04 | -72.59 (1.6) | -44.41 (0.9) | -71.66 (1.6) | hit | | s15-2 | 8.0e+04 | -28.34 (2.1) | -0.67 (1.0) | -7.00 (2.1) | hit | | s15-3 | 8.0e+04 | -24.95 (2.1) | +2.82 (1.0) | -0.19 (2.1) | hit | | s15-4 | 5.1e+06 | -20.55 (0.3) | +5.57 (0.1) | +6.16 (0.3) | double | | s15-5 | 8.0e+04 | -16.27 (2.1) | +8.83 (1.0) | +13.72 (2.1) | double | | s15-6 | 8.0e+04 | -12.97 (2.1) | +9.96 (1.0) | +20.24 (2.1) | double | | s15-7 | 8.0e+04 | -47.65 (2.1) | +3.09 (1.0) | -18.41 (2.1) | hit | | s15-8 | 8.0e+04 | -51.20 (2.0) | -2.86 (1.0) | -31.79 (2.0) | hit | | s15-9 | 8.0e+04 | -55.02 (2.0) | -11.42 (1.0) | -46.42 (2.0) | hit | | s15-T | 8.0e+04 | -57.45 (1.9) | -23.91 (1.0) | -54.95 (1.9) | hit | | s15-A | 8.0e+04 | -72.00 (1.6) | -42.42 (0.9) | -72.09 (1.6) | hit | | s14-2 | 8.0e+04 | -28.06 (2.1) | +1.78 (1.0) | -6.21 (2.1) | hit | | s14-3 | 8.0e+04 | -24.41 (2.1) | +4.42 (1.0) | -0.62 (2.1) | hit | | s14-4 | 8.0e+04 | -20.41 (2.1) | +8.31 (1.0) | +5.82 (2.1) | hit | | s14-5 | 3.2e+05 | -16.31 (1.0) | +10.82 (0.5) | +12.34 (1.0) | double | | s14-6 | 8.0e+04 | -12.35 (2.1) | +11.90 (1.0) | +19.57 (2.1) | double | | s14-7 | 8.0e+04 | -47.52 (2.0) | +7.95 (1.0) | -18.64 (2.0) | hit | | s14-8 | 8.0e+04 | -51.41 (2.0) | +2.01 (1.0) | -32.23 (2.0) | hit | | s14-9 | 8.0e+04 | -54.26 (2.0) | -7.57 (1.0) | -45.45 (2.0) | hit | | s14-T | 8.0e+04 | -57.20 (1.9) | -20.14 (1.0) | -56.20 (1.9) | hit | | s14-A | 8.0e+04 | -72.41 (1.6) | -39.75 (0.9) | -70.94 (1.6) | hit | | s13-2 | 8.0e+04 | -28.79 (2.1) | +4.29 (1.0) | -7.07 (2.1) | hit | | s13-3 | 8.0e+04 | -24.39 (2.1) | +7.00 (1.0) | -0.33 (2.1) | hit | | s13-4 | 8.0e+04 | -20.39 (2.1) | +10.23 (1.0) | +5.20 (2.1) | hit | | s13-5 | 2.0e+07 | -16.46 (0) | +12.88 (0) | +12.73 (0) | hit | | s13-6 | 8.0e+04 | -12.67 (2.1) | +13.75 (1.0) | +18.93 (2.1) | double | | s13-7 | 8.0e+04 | -47.64 (2.1) | +12.41 (1.0) | -18.59 (2.1) | hit | | s13-8 | 8.0e+04 | -51.47 (2.0) | +5.50 (1.0) | -32.61 (2.0) | hit | | s13-9 | 8.0e+04 | -54.33 (2.0) | -3.30 (1.0) | -45.89 (2.0) | hit | | s13-T | 8.0e+04 | -57.17 (1.9) | -17.29 (1.0) | -55.28 (1.9) | hit | | s13-A | 8.0e+04 | -71.95 (1.6) | -37.21 (0.9) | -72.33 (1.6) | hit | | s12-2 | 8.0e+04 | -28.81 (2.1) | +7.24 (1.0) | -5.99 (2.1) | hit | | s12-3 | 8.0e+04 | -24.61 (2.1) | +9.89 (1.0) | -0.34 (2.1) | hit | | s12-4 | 8.0e+04 | -20.33 (2.1) | +12.28 (1.0) | +6.36 (2.1) | hit | | s12-5 | 8.0e+04 | -16.22 (2.1) | +15.13 (1.0) | +12.78 (2.1) | hit | | s12-6 | 8.0e+04 | -12.16 (2.1) | +15.30 (1.0) | +21.02 (2.1) | double | | s12-7 | 8.0e+04 | -46.97 (2.1) | +16.64 (1.0) | -18.22 (2.1) | hit | | s12-8 | 8.0e+04 | -50.98 (2.0) | +9.38 (1.0) | -31.23 (2.0) | hit | | s12-9 | 8.0e+04 | -54.01 (2.0) | +0.31 (1.0) | -45.30 (2.0) | hit | | s12-T | 8.0e+04 | -58.03 (1.9) | -14.72 (1.0) | -56.96 (1.9) | hit | | s12-A | 8.0e+04 | -72.15 (1.6) | -35.17 (0.9) | -72.39 (1.6) | hit | | Hand | Hands needed | Yes [%] | No [%] | |:------:|:-------:|:----------:|:----------:| | pA-2 | 8.0e+04 | +47.79 (1.7) | +7.70 (1.0) | yes | | pA-3 | 8.0e+04 | +51.89 (1.7) | +9.80 (1.0) | yes | | pA-4 | 8.0e+04 | +56.73 (1.7) | +11.81 (1.0) | yes | | pA-5 | 8.0e+04 | +61.10 (1.7) | +15.76 (1.0) | yes | | pA-6 | 8.0e+04 | +66.09 (1.7) | +20.40 (2.1) | yes | | pA-7 | 8.0e+04 | +46.61 (1.6) | +15.82 (1.0) | yes | | pA-8 | 8.0e+04 | +35.34 (1.6) | +9.87 (1.0) | yes | | pA-9 | 8.0e+04 | +23.05 (1.6) | -0.13 (1.0) | yes | | pA-T | 8.0e+04 | +8.79 (1.6) | -14.01 (1.0) | yes | | pA-A | 8.0e+04 | -24.16 (1.5) | -35.35 (0.9) | yes | | pT-2 | 8.0e+04 | -4.29 (3.7) | +63.29 (0.7) | no | | pT-3 | 8.0e+04 | +4.76 (3.8) | +64.28 (0.7) | no | | pT-4 | 8.0e+04 | +15.42 (4.0) | +65.64 (0.7) | no | | pT-5 | 8.0e+04 | +27.51 (4.1) | +66.77 (0.7) | no | | pT-6 | 8.0e+04 | +37.88 (4.1) | +67.89 (0.7) | no | | pT-7 | 8.0e+04 | +8.98 (2.6) | +77.52 (0.6) | no | | pT-8 | 8.0e+04 | -22.83 (2.5) | +79.04 (0.6) | no | | pT-9 | 8.0e+04 | -61.33 (2.6) | +76.23 (0.6) | no | | pT-T | 8.0e+04 | -76.47 (2.7) | +43.58 (0.7) | no | | pT-A | 8.0e+04 | -85.74 (2.2) | +11.11 (1.0) | no | | p9-2 | 8.0e+04 | +20.25 (2.2) | +10.96 (1.0) | yes | | p9-3 | 8.0e+04 | +25.62 (2.2) | +13.70 (1.0) | yes | | p9-4 | 8.0e+04 | +32.53 (2.2) | +16.16 (1.0) | yes | | p9-5 | 8.0e+04 | +39.22 (2.2) | +19.08 (1.0) | yes | | p9-6 | 8.0e+04 | +47.19 (2.2) | +22.75 (1.0) | yes | | p9-7 | 3.2e+05 | +36.59 (0.9) | +40.03 (0.4) | no | | p9-8 | 8.0e+04 | +23.50 (1.9) | +10.05 (0.8) | yes | | p9-9 | 8.0e+04 | -8.46 (1.8) | -18.37 (1.0) | yes | | p9-T | 8.0e+04 | -36.82 (1.9) | -23.97 (1.0) | no | | p9-A | 3.2e+05 | -48.36 (0.8) | -46.75 (0.4) | no | | p8-2 | 8.0e+04 | +7.89 (2.4) | -29.06 (1.0) | yes | | p8-3 | 8.0e+04 | +14.84 (2.4) | -23.82 (1.0) | yes | | p8-4 | 8.0e+04 | +21.02 (2.5) | -20.67 (1.0) | yes | | p8-5 | 8.0e+04 | +29.77 (2.5) | -16.36 (1.0) | yes | | p8-6 | 8.0e+04 | +39.44 (2.7) | -11.93 (1.1) | yes | | p8-7 | 8.0e+04 | +31.80 (2.1) | -41.83 (0.9) | yes | | p8-8 | 8.0e+04 | -2.54 (2.1) | -45.57 (0.9) | yes | | p8-9 | 8.0e+04 | -39.27 (2.2) | -50.67 (0.9) | yes | | p8-T | 8.0e+04 | -51.15 (1.9) | -57.31 (0.8) | yes | | p8-A | 3.2e+05 | -66.55 (0.8) | -68.39 (0.4) | yes | | p7-2 | 8.0e+04 | -12.93 (2.6) | -28.38 (1.0) | yes | | p7-3 | 8.0e+04 | -5.76 (2.8) | -24.45 (1.0) | yes | | p7-4 | 8.0e+04 | +7.32 (2.9) | -21.06 (1.0) | yes | | p7-5 | 8.0e+04 | +16.33 (2.9) | -16.50 (1.0) | yes | | p7-6 | 8.0e+04 | +24.66 (2.9) | -12.31 (1.1) | yes | | p7-7 | 8.0e+04 | -4.49 (2.1) | -31.70 (1.0) | yes | | p7-8 | 3.2e+05 | -39.47 (1.0) | -36.94 (0.5) | no | | p7-9 | 8.0e+04 | -57.27 (2.1) | -43.25 (0.9) | no | | p7-T | 8.0e+04 | -68.16 (1.8) | -51.02 (0.9) | no | | p7-A | 8.0e+04 | -81.66 (1.6) | -62.96 (0.8) | no | | p6-2 | 8.0e+04 | -20.02 (2.5) | -24.94 (1.0) | yes | | p6-3 | 8.0e+04 | -13.43 (2.9) | -22.86 (1.0) | yes | | p6-4 | 8.0e+04 | -1.32 (2.9) | -20.97 (1.0) | yes | | p6-5 | 8.0e+04 | +8.77 (2.9) | -16.47 (1.0) | yes | | p6-6 | 8.0e+04 | +18.02 (2.9) | -11.89 (1.1) | yes | | p6-7 | 8.0e+04 | -25.77 (2.0) | -21.03 (1.0) | no | | p6-8 | 8.0e+04 | -42.37 (2.0) | -26.97 (1.0) | no | | p6-9 | 8.0e+04 | -61.32 (1.9) | -33.82 (1.0) | no | | p6-T | 8.0e+04 | -73.60 (1.7) | -42.98 (0.9) | no | | p6-A | 8.0e+04 | -81.35 (1.5) | -57.46 (0.8) | no | | p5-2 | 8.0e+04 | -29.29 (2.3) | +35.74 (2.0) | no | | p5-3 | 8.0e+04 | -21.10 (2.5) | +40.83 (2.0) | no | | p5-4 | 8.0e+04 | -14.02 (2.8) | +46.13 (2.0) | no | | p5-5 | 8.0e+04 | -2.07 (2.8) | +51.38 (2.0) | no | | p5-6 | 8.0e+04 | +7.69 (2.8) | +55.49 (2.0) | no | | p5-7 | 8.0e+04 | -29.78 (1.8) | +38.74 (2.0) | no | | p5-8 | 8.0e+04 | -45.59 (1.8) | +29.74 (2.0) | no | | p5-9 | 8.0e+04 | -63.38 (1.7) | +15.31 (2.0) | no | | p5-T | 8.0e+04 | -75.56 (1.7) | -5.45 (1.0) | no | | p5-A | 8.0e+04 | -83.48 (1.4) | -28.71 (1.0) | no | | p4-2 | 8.0e+04 | -18.34 (2.4) | -2.08 (1.0) | no | | p4-3 | 8.0e+04 | -10.15 (2.7) | -0.10 (1.0) | no | | p4-4 | 8.0e+04 | -1.53 (2.9) | +4.20 (1.0) | no | | p4-5 | 1.3e+06 | +8.32 (0.7) | +6.94 (0.3) | yes | | p4-6 | 8.0e+04 | +18.54 (2.9) | +9.10 (1.0) | yes | | p4-7 | 8.0e+04 | -17.13 (2.0) | +8.29 (1.0) | no | | p4-8 | 8.0e+04 | -32.41 (2.0) | -5.80 (1.0) | no | | p4-9 | 8.0e+04 | -52.03 (2.0) | -21.40 (1.0) | no | | p4-T | 8.0e+04 | -65.35 (1.7) | -30.80 (1.0) | no | | p4-A | 8.0e+04 | -75.64 (1.5) | -48.85 (0.9) | no | | p3-2 | 2.0e+07 | -13.55 (0) | -13.84 (0) | yes | | p3-3 | 3.2e+05 | -4.62 (1.3) | -10.55 (0.5) | yes | | p3-4 | 8.0e+04 | +4.29 (2.7) | -7.50 (1.0) | yes | | p3-5 | 8.0e+04 | +12.98 (2.9) | -2.79 (1.0) | yes | | p3-6 | 8.0e+04 | +22.13 (2.9) | +0.27 (1.0) | yes | | p3-7 | 8.0e+04 | -5.64 (2.0) | -14.75 (1.0) | yes | | p3-8 | 3.2e+05 | -23.16 (1.0) | -21.60 (0.5) | no | | p3-9 | 8.0e+04 | -43.38 (2.0) | -29.61 (1.0) | no | | p3-T | 8.0e+04 | -59.02 (1.7) | -38.86 (0.9) | no | | p3-A | 8.0e+04 | -70.21 (1.5) | -54.10 (0.9) | no | | p2-2 | 8.0e+04 | -7.83 (2.3) | -11.55 (1.0) | yes | | p2-3 | 8.0e+04 | -1.63 (2.5) | -8.20 (1.0) | yes | | p2-4 | 8.0e+04 | +5.17 (2.7) | -4.49 (1.0) | yes | | p2-5 | 8.0e+04 | +16.51 (2.7) | -1.33 (1.0) | yes | | p2-6 | 8.0e+04 | +25.65 (2.9) | +2.57 (1.0) | yes | | p2-7 | 8.0e+04 | +0.70 (2.0) | -8.63 (1.0) | yes | | p2-8 | 1.3e+06 | -17.84 (0.5) | -15.96 (0.2) | no | | p2-9 | 8.0e+04 | -40.15 (2.0) | -23.95 (1.0) | no | | p2-T | 8.0e+04 | -53.36 (1.8) | -34.25 (0.9) | no | | p2-A | 8.0e+04 | -67.42 (1.5) | -51.15 (0.9) | no | ### 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 _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. #### 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 1. always standing 2. always doubling 3. always hitting 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. #### 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. #### 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. #### Number of hands The output is the expected value\ $e$ of the bankroll, which is a random variable with an associated uncertainty\ $\Delta e$ (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 $e_h-\Delta e_h > e_s+\Delta e_s$ and $e_h-\Delta e_h > e_d+\Delta e_d$. 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. ### Implementation The steps above can be written in a [Bash](https://en.wikipedia.org/wiki/Bash_%28Unix_shell%29) script that * loops over hands and upcards, * creates a strategy file for each possible play hit, double or stand (or split or not), * runs [Libre Blackjack](https://www.seamplex.com/blackjack), * checks the results and picks the best play, * updates the strategy file ```bash ##!/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 ${t}${hand}

stand

hit

double

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 " " >> ${type}.html echo " " >> ${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 " " >> ${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 "" >> ${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 ## ## ${t}${hand} ## ##

yes

## ## 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 " " >> ${type}.html echo " " >> ${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 " " >> ${type}.html strategy[${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 ```