This project started with a few of my own curiosities:
- "Advantage Players" in casino Blackjack know how to deviate from a basic strategy based on the frequency of remaining cards in the deck. If a computer were to perfectly analyze the remaining deck composition in a fair game of Blackjack, how frequently would its playing choices deviate from basic strategy?
- How much of an advantage would that computer have compared to playing perfect basic strategy even if it never changed its bets like an Advantage Player would?
- What's the big deal with Rust?
To answer these, I built a program that simulates as many games of Blackjack as my 24-thread CPU could handle. For every decision the player makes:
- Calculate the optimal decision to hit, stand, double, or split. Since every decision comes with a unique set of cards in play and remaining cards in the deck, I brute-force the exact odds of every outcome to be as precise as possible.
- Compare the expected value of that choice with the expected value of the action suggested by a basic strategy chart.
Rust was the perfect choice for the job since this brute-force simulation is CPU-intensive, and I get plenty of manual memory management at my day job.
Image courtesy of ChatGPT 4.
The casino's edge in Blackjack is already very slim with standard rules. A proper game will carry an edge of around 0.5%, which means that for every $100 wagered on the game, the player can expect to lose around 50 cents in the long run if played according to basic strategy.
I chose rules for my simulation based on the most common rules I've seen for single-deck games: 1 deck, shuffled after half of the cards have been played, that pays 3:2 on a player Blackjack. The dealer hits a soft 17. The player can split up to 4 hands, but Aces can only be split once and can only take one additional card. Most restrictively, the player can only double down a hard 10 or 11, and cannot double down after a split. These rules can all be tweaked in the simulation.
Notably, I only chose to simulate single-deck. The goal is to determine an upper bound of "how much better can a computer play", and even two decks severely dilute the amount of information available to the computer on any given hand.
I let the simulation run overnight a few times. Full records are logged in record.txt. Here's an example:
Played 144180610 hands (27931824 shoes) and had total of -70705.5 returned. Edge = -0.04903953451161013%,
2261 hands/sec total (2238 hands/433 shoes in last second), 11846522/197323361 deviant actions
0.44857870762458846% average +EV/hand. 872283/2368688/11058450 insurances won/taken/offered (+0.08587506541799021 EV).
Click for full deviations breakdown by hand.
Left axis is the player's hand. Top axis is the dealer's up-card. In each cell, the left number is the number of times the computer chose to deviate from basic strategy in that situation. The right number is the total number of times that situation came up.
Hard 2 3 4 5 6 7 8 9 10 A
4 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 4
5 0/111312 0/114802 0/159739 0/160048 0/160198 0/156548 0/145393 0/141915 0/518413 0/95510 5
6 0/108750 0/150011 0/113471 0/152190 0/152695 0/148971 0/144431 0/143489 0/525938 0/96827 6
7 0/258803 0/266588 0/273562 0/273436 0/312062 0/306646 0/298868 0/295116 0/1080338 0/198721 7
8 0/272629 0/281258 0/326026 0/290986 0/287577 0/322272 0/312109 0/308114 0/1126698 0/206845 8
9 0/444534 0/453161 0/463356 0/471506 0/468020 0/456335 0/482023 0/474659 0/1730881 0/319016 9
10 5259/474108 4033/482118 2571/494588 1342/532661 941/499300 17345/490565 35270/475757 85410/506620 330501/1847260 74663/338871 10
11 4932/655005 4200/666773 2716/676796 1502/682471 1261/682496 22736/670014 43681/662261 67863/653544 330955/2494574 90922/462466 11
12 292039/1062909 468358/1187625 553490/1211680 305156/1222185 247231/1267248 0/1189847 1312/1171718 19231/1183683 81524/4408908 7761/828961 12
13 433669/1298407 309049/1167930 207115/1288401 138814/1282682 100939/1271007 4740/1317249 1835/1302444 22143/1310392 277934/4871807 32977/896795 13
14 211004/1160388 137785/1153072 79196/1009844 54722/1098861 38157/1098202 46688/1259945 8843/1242948 25714/1262201 466692/4700130 60843/867841 14
15 90245/1209187 64299/1191650 39358/1187633 25971/1039177 19660/1127515 86657/1363450 67550/1364435 148459/1382570 913009/5095014 115137/947851 15
16 44614/1081997 33311/1075119 20580/1041977 14431/1031039 20028/874675 185822/1300625 163655/1348995 300101/1316832 1970236/4835503 185790/909275 16
17 783/1107663 652/1095327 390/1074329 1155/1054797 870/1046895 3671/1321973 156571/1421617 49108/1411174 42610/5109052 142943/980830 17
18 0/892363 0/874875 0/858693 0/831315 0/819997 0/1248231 0/1119866 32/1305915 0/4549721 20/905471 18
19 0/888345 0/872126 0/855775 0/836141 0/825143 0/1196847 0/1280428 0/1099982 0/4259114 0/877481 19
20 0/177800 0/159819 0/134663 0/109254 0/100981 0/509054 0/536790 0/551233 0/2203217 0/444481 20
21 0/180406 0/158261 0/129857 0/101154 0/90553 0/524128 0/547780 0/533006 0/1930677 0/403401 21
Soft 2 3 4 5 6 7 8 9 10 A
12 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 12
13 0/108144 0/148718 0/149261 0/149415 0/149991 0/148248 0/141168 0/140452 0/526750 0/70948 13
14 0/150806 0/117137 0/158669 0/158571 0/159112 0/156340 0/151799 0/149247 0/557602 0/73670 14
15 0/156012 0/158654 0/124827 0/160184 0/160643 0/159079 0/161043 0/160657 0/605673 0/80080 15
16 0/177720 0/178166 0/186526 0/153277 0/188107 0/186364 0/185564 0/184086 0/690880 0/91242 16
17 119/197827 7/202232 111/198647 200/205603 289/167928 4991/202125 1/200223 1/199702 13/750097 0/98920 17
18 9427/226134 6930/228194 6331/229245 5376/230297 18939/237131 68/198223 15644/229929 8052/226189 105882/847710 12703/111502 18
19 0/233244 5/235065 9/238315 0/237037 0/236188 0/244240 0/203931 1/253445 6163/945014 2/123649 19
20 0/248661 0/245984 0/249644 0/251073 0/251990 0/243218 0/256014 0/232761 0/977960 0/133037 20
21 0/73659 0/79246 0/85257 0/96930 0/96988 0/75022 0/75116 0/97424 0/439669 0/57439 21
Pair 2 3 4 5 6 7 8 9 10 A
2 8482/27496 16317/57088 9956/57723 4013/57829 3102/58088 11376/56557 7658/53753 906/53414 1008/193890 66/36202 2
3 14619/54921 14296/27436 4339/57630 2103/57572 2883/57485 10561/56416 17390/54352 4910/53164 6324/194906 808/35781 3
4 151/53825 556/53348 1081/26648 8356/53693 12211/54015 248/52568 316/52940 114/52809 180/191925 9/35808 4
5 562/53341 370/52971 223/52998 35/26538 61/52566 1558/52842 3804/52542 8686/52293 42275/192506 9598/35282 5
6 15764/56446 9086/56337 3528/56603 1397/56919 506/27315 23708/54218 9588/53095 2926/52505 6050/191664 1126/35152 6
7 1675/57156 1054/56995 539/56630 218/56960 80/56618 4/27369 19807/52907 6075/52315 79407/190222 7639/35301 7
8 1/56951 0/57105 0/56736 6/56596 20/56881 0/56430 0/26942 7532/55953 36658/201911 10361/36986 8
9 12077/55969 10889/55601 7275/55837 3469/55988 3057/56314 5548/52289 2350/56078 1146/26808 1841/188999 13995/35843 9
10 17100/1051650 40591/1064379 81490/1086505 133077/1111364 138603/1116528 6297/1035039 89/1029499 0/1027439 0/3293996 23/737914 0
A 459/52455 346/51981 247/51868 191/52411 169/52007 939/52015 1743/51748 2073/51918 11555/199072 1620/17522 1
Let's break it down:
- The computer deviated from the basic strategy play on 11.8M out of 197M decisions. That works out to almost exactly 6% of decisions.
- On average, the deviant plays earned a theoretical average of 0.449% of expected value. Over 144M hands, the actual loss was only 0.049% - around 5 cents for every $100 bet.
- Compared to a basic-strategy house edge of around 0.5%, the simulation and calculated EV gain agreed with each other.
Brute-forcing predicted a gain of around 0.449%, and
0.5% - 0.449% ≈ 0.049%. - In this particular game, the computer improved on the long-run performance of a basic strategy player by a factor of ten!
- Taking the "Insurance" bet when the deck composition called for it accounted for about 0.08% of the EV gained, around 1/6 of the total EV gain from all deviations.
- Looking at per-hand deviations, some make sense. Many hands have zero deviations because there is no draw pile extreme enough to tilt the odds away from the basic decision, such as hitting a hard 20 or 21.
- Borderline plays are noticeable on the per-hand chart. These agree with common Advantage Play deviations based on a
simple running count of high and low cards left in the deck.
- A hard 12 vs a 4-up deviates over 45% of the time.
- A hard 16 vs a 10-up deviates around 40% of the time.
- Perhaps the most infamous "dubious" play, splitting tens, is the correct choice against a 6-up around 12% of the time.
- With slightly more favorable table rules, a computer could reasonably beat the house without ever adjusting its bet sizes. This specific simulation comes so close to a fair game that simply increasing deck penetration, allowing a few more doubles, or altering bets with a very small spread would be enough to overcome the house edge.
- Do NOT try this for real money. Card counting with your head is not illegal. Card counting with a program like this is very illegal.
I also wanted to get a benchmark of pure hand simulation that did not involve any brute-forcing. Rust did its job beautifully - the simulator is blazing fast when using a simple lookup table for its decision-making. These are the results of using 20 threads on a Ryzen 5900X for about a minute:
Started 2573248431 hands (499288429 shoes, 2804642332 units placed) and had total of -14783846 returned.
Edge = -0.5271205469346814% (units returned/placed), 38982908 hands/sec total (39494662 hands/7663049 shoes in last
second), 0/3508287419 deviant actions 0% average +EV/hand. 0/0/197576910 insurances won/taken/offered (+0 EV).
Around 2.5 billion hands per minute! I did not investigate any other Blackjack simulator programs out there, but I would be surprised if anything else not written in Rust or C(++) even comes close.
Of course, that's not the performance we get when we start brute-forcing perfect strategies. As shown by the above result that ran for several hours, it's much closer to 130,000 hands per minute. This makes sense - there is a massive tree of outcomes to analyze, compounded especially by the ability to split hands. I memoized specific scenarios since there is overlap in outcomes within a single running hand, but any further speedup would mean the computer's decision being less "perfect".
I did not write this program to have any kind of user interface. You will need to be familiar with compiling code to run it and editing code to make changes.
Requires Rust and Cargo.
git clone https://github.com/joshuaprince/blackjack_composition.git
cd blackjack_composition
cargo build --release
./target/release/blackjack_compositionIf you are programmatically inclined, here are some things to try tweaking that really ought to have a user interface:
- Simulate fast Basic Strategy instead of a comparison: In
main.rs::play_hands_compare_and_report: Change thePlayerDecisionMethod::BasicPerfectComparisontoPlayerDecisionMethod::BasicStrategy. - Change the number of threads to use:
main.rs::THREADSconstant. - Change the game rules:
rules.rs. The rules struct is selected as themain.rs::RULESconstant. WARNING: Not all combinations of rules will work properly.
This was created by Joshua Prince in 2023. It is licensed under the AGPL. Any changes to the code must also be open-source and licensed under the AGPL. I make no guarantees about the accuracy or functionality of any data or code in this repository.
Special thanks to The Wizard of Odds for developing calculators and tools that I used frequently as a "ground truth" while developing and debugging this simulator.