Free simulators with exact math, checked against the Wizard's figures: open source, feedback welcome

I've built a set of free casino game simulators and trainers at casinogamessimulator . com, and I'd value this forum's scrutiny of the math, since it's checked against Wizard of Odds' published figures throughout:

• Video poker: exact returns and hand frequencies by enumeration, plus Easy/Best strategy charts scored over all 2,598,960 deals. The returns match the published figures for 5 paytables to 6 decimals.
• Blackjack: finite-shoe basic strategy and house edge for 26 rule sets, within 0.005 points of the Wizard's calculator for 22 of them.
• Baccarat and 3-card vs. dealer: exact enumeration, with outcome counts matching the published tables exactly.
• Hold'em vs. dealer: exact enumeration, 2.185% of the ante with optimal play.
• Craps: each bet solved as a Markov chain.
• Pai gow: exact hand frequencies; house edge from a 400M-round seeded simulation: 2.72% (95% CI 2.713–2.728%) vs. the published 2.7212%.

One place we differ: for the simple strategy in the game sold as Ultimate Texas Hold'em, enumerating it gives us 2.40% vs. the 2.43% quoted. I'd be curious whether anyone knows how that figure was computed.

The Methodology page on the site lists what's exact, what's simulated and what's approximated. The engines are open source under MIT on GitHub (search for casinogamessimulator/engines), so anyone can check them.

Everything is free, with virtual chips only, no ads and no affiliate links. There's a daily challenge and weekly leaderboards by trainer rating, if anyone here wants to put a name at the top of the video poker one.

Corrections very welcome.

great website, playing rn

Quote: alejandrobg

I've built a set of free casino game simulators and trainers at casinogamessimulator . com, and I'd value this forum's scrutiny of the math, since it's checked against Wizard of Odds' published figures throughout:

• Video poker: exact returns and hand frequencies by enumeration, plus Easy/Best strategy charts scored over all 2,598,960 deals. The returns match the published figures for 5 paytables to 6 decimals.
• Blackjack: finite-shoe basic strategy and house edge for 26 rule sets, within 0.005 points of the Wizard's calculator for 22 of them.
• Baccarat and 3-card vs. dealer: exact enumeration, with outcome counts matching the published tables exactly.
• Hold'em vs. dealer: exact enumeration, 2.185% of the ante with optimal play.
• Craps: each bet solved as a Markov chain.
• Pai gow: exact hand frequencies; house edge from a 400M-round seeded simulation: 2.72% (95% CI 2.713–2.728%) vs. the published 2.7212%.

One place we differ: for the simple strategy in the game sold as Ultimate Texas Hold'em, enumerating it gives us 2.40% vs. the 2.43% quoted. I'd be curious whether anyone knows how that figure was computed.

The Methodology page on the site lists what's exact, what's simulated and what's approximated. The engines are open source under MIT on GitHub (search for casinogamessimulator/engines), so anyone can check them.

Everything is free, with virtual chips only, no ads and no affiliate links. There's a daily challenge and weekly leaderboards by trainer rating, if anyone here wants to put a name at the top of the video poker one.

Corrections very welcome.
link to original post

Welcome to the forum. It seems like you put A LOT of thought and work into this! If I may ask, do you plan on monetizing this in the future? It seems like you should!

Thanks, harris! If anything looks off, whether a number, a hand that played strangely or a confusing UI, post it here. That’s exactly the feedback I’m after.

Quote: SOOPOO

Quote: alejandrobg

I've built a set of free casino game simulators and trainers at casinogamessimulator . com, and I'd value this forum's scrutiny of the math, since it's checked against Wizard of Odds' published figures throughout:

• Video poker: exact returns and hand frequencies by enumeration, plus Easy/Best strategy charts scored over all 2,598,960 deals. The returns match the published figures for 5 paytables to 6 decimals.
• Blackjack: finite-shoe basic strategy and house edge for 26 rule sets, within 0.005 points of the Wizard's calculator for 22 of them.
• Baccarat and 3-card vs. dealer: exact enumeration, with outcome counts matching the published tables exactly.
• Hold'em vs. dealer: exact enumeration, 2.185% of the ante with optimal play.
• Craps: each bet solved as a Markov chain.
• Pai gow: exact hand frequencies; house edge from a 400M-round seeded simulation: 2.72% (95% CI 2.713–2.728%) vs. the published 2.7212%.

One place we differ: for the simple strategy in the game sold as Ultimate Texas Hold'em, enumerating it gives us 2.40% vs. the 2.43% quoted. I'd be curious whether anyone knows how that figure was computed.

The Methodology page on the site lists what's exact, what's simulated and what's approximated. The engines are open source under MIT on GitHub (search for casinogamessimulator/engines), so anyone can check them.

Everything is free, with virtual chips only, no ads and no affiliate links. There's a daily challenge and weekly leaderboards by trainer rating, if anyone here wants to put a name at the top of the video poker one.

Corrections very welcome.
link to original post

Welcome to the forum. It seems like you put A LOT of thought and work into this! If I may ask, do you plan on monetizing this in the future? It seems like you should!
link to original post

A bit of background: I was looking for a place to play and train properly, with real strategy feedback and the exact math behind it. I couldn’t find one that did both, so I built it.

Full transparency: I might add premium features someday to cover costs. Free play will always stay, and so will the trainers and the published math. I’d only charge for extras on top.

So I’m curious: what would actually be worth paying for? Some ideas I’ve had:

Detailed session history and error tracking (which hands you keep getting wrong)
Custom rule sets and paytables
Advanced drills, like a quiz on just your weak spots
Strategy charts you can export or print
Or nothing at all, and it should just stay free. Honest answers welcome.

I’ve tried these simulators and fount them beautiful. In Pai Gow poker, there are no $5 and $25 chips there. This is the only problem I could find there.

Quote: alejandrobg

I've built a set of free casino game simulators and trainers at casinogamessimulator . com, and I'd value this forum's scrutiny of the math, since it's checked against Wizard of Odds' published figures throughout:

• Video poker: exact returns and hand frequencies by enumeration, plus Easy/Best strategy charts scored over all 2,598,960 deals. The returns match the published figures for 5 paytables to 6 decimals.
• Blackjack: finite-shoe basic strategy and house edge for 26 rule sets, within 0.005 points of the Wizard's calculator for 22 of them.
• Baccarat and 3-card vs. dealer: exact enumeration, with outcome counts matching the published tables exactly.
• Hold'em vs. dealer: exact enumeration, 2.185% of the ante with optimal play.
• Craps: each bet solved as a Markov chain.
• Pai gow: exact hand frequencies; house edge from a 400M-round seeded simulation: 2.72% (95% CI 2.713–2.728%) vs. the published 2.7212%.

One place we differ: for the simple strategy in the game sold as Ultimate Texas Hold'em, enumerating it gives us 2.40% vs. the 2.43% quoted. I'd be curious whether anyone knows how that figure was computed.

The Methodology page on the site lists what's exact, what's simulated and what's approximated. The engines are open source under MIT on GitHub (search for casinogamessimulator/engines), so anyone can check them.

Everything is free, with virtual chips only, no ads and no affiliate links. There's a daily challenge and weekly leaderboards by trainer rating, if anyone here wants to put a name at the top of the video poker one.

Corrections very welcome.
link to original post

Your description is a bit confusing. What you initially call "Hold'em vs. dealer:" seems to match the EV of UTH. From https://wizardofodds.com/games/ultimate-texas-hold-em/
Total Combinations: 27,813,810,024,000 Return: -0.021850

I have done the combinatorial analysis (what you call 'exact enumeration') and agree with the Wizard and other authors.

Quote:

One place we differ: for the simple strategy in the game sold as Ultimate Texas Hold'em, enumerating it gives us 2.40% vs. the 2.43% quoted. I'd be curious whether anyone knows how that figure was computed.

You don't say which simple strategy you follow for UTH, but the you quote the same 2.43% that is on the reference link above, so I assume that is your comparison.
"Following my Wizard strategy will result in a house edge of 2.43% and an Element of Risk of 0.58%."

To calculate the house edge of a strategy, you need to do the same calculation as the optimal house edge, but substitute the rule-based decisions for the optimal ones. Assuming the basic evaluator is correct in both cases, it seems that one of you is interpreting the simple rule set wrong or coding it up wrong.

I recall that JB did this work, but Wizard can confirm.

Quote: Mental

Quote: alejandrobg

I've built a set of free casino game simulators and trainers at casinogamessimulator . com, and I'd value this forum's scrutiny of the math, since it's checked against Wizard of Odds' published figures throughout:

• Video poker: exact returns and hand frequencies by enumeration, plus Easy/Best strategy charts scored over all 2,598,960 deals. The returns match the published figures for 5 paytables to 6 decimals.
• Blackjack: finite-shoe basic strategy and house edge for 26 rule sets, within 0.005 points of the Wizard's calculator for 22 of them.
• Baccarat and 3-card vs. dealer: exact enumeration, with outcome counts matching the published tables exactly.
• Hold'em vs. dealer: exact enumeration, 2.185% of the ante with optimal play.
• Craps: each bet solved as a Markov chain.
• Pai gow: exact hand frequencies; house edge from a 400M-round seeded simulation: 2.72% (95% CI 2.713–2.728%) vs. the published 2.7212%.

One place we differ: for the simple strategy in the game sold as Ultimate Texas Hold'em, enumerating it gives us 2.40% vs. the 2.43% quoted. I'd be curious whether anyone knows how that figure was computed.

The Methodology page on the site lists what's exact, what's simulated and what's approximated. The engines are open source under MIT on GitHub (search for casinogamessimulator/engines), so anyone can check them.

Everything is free, with virtual chips only, no ads and no affiliate links. There's a daily challenge and weekly leaderboards by trainer rating, if anyone here wants to put a name at the top of the video poker one.

Corrections very welcome.
link to original post

Your description is a bit confusing. What you initially call "Hold'em vs. dealer:" seems to match the EV of UTH. From /games/ultimate-texas-hold-em/
Total Combinations: 27,813,810,024,000 Return: -0.021850

I have done the combinatorial analysis (what you call 'exact enumeration') and agree with the Wizard and other authors.

Quote:

One place we differ: for the simple strategy in the game sold as Ultimate Texas Hold'em, enumerating it gives us 2.40% vs. the 2.43% quoted. I'd be curious whether anyone knows how that figure was computed.

You don't say which simple strategy you follow for UTH, but the you quote the same 2.43% that is on the reference link above, so I assume that is your comparison.
"Following my Wizard strategy will result in a house edge of 2.43% and an Element of Risk of 0.58%."

To calculate the house edge of a strategy, you need to do the same calculation as the optimal house edge, but substitute the rule-based decisions for the optimal ones. Assuming the basic evaluator is correct in both cases, it seems that one of you is interpreting the simple rule set wrong or coding it up wrong.

I recall that JB did this work, but Wizard can confirm.
link to original post

Mystery solved, I think. You were right: same evaluator, I was just reading two of the rules differently.

Flop “two pair or better”: I required the hole cards to help, so I checked with trips on the flop alone. Taken literally, you bet those.
River “hidden pair or better”: I counted any hole card matching the board, even when it doesn’t play (e.g. pocket 3s on A-A-K-K). Counting it only when your hole cards actually improve on the board fixes that.
With both changes the enumeration gives a 2.433% house edge and 0.584% element of risk, which is the published 2.43% / 0.58%. Thanks for the push!

Quote: alejandrobg

Quote: Mental

Quote: alejandrobg

I've built a set of free casino game simulators and trainers at casinogamessimulator . com, and I'd value this forum's scrutiny of the math, since it's checked against Wizard of Odds' published figures throughout:

• Video poker: exact returns and hand frequencies by enumeration, plus Easy/Best strategy charts scored over all 2,598,960 deals. The returns match the published figures for 5 paytables to 6 decimals.
• Blackjack: finite-shoe basic strategy and house edge for 26 rule sets, within 0.005 points of the Wizard's calculator for 22 of them.
• Baccarat and 3-card vs. dealer: exact enumeration, with outcome counts matching the published tables exactly.
• Hold'em vs. dealer: exact enumeration, 2.185% of the ante with optimal play.
• Craps: each bet solved as a Markov chain.
• Pai gow: exact hand frequencies; house edge from a 400M-round seeded simulation: 2.72% (95% CI 2.713–2.728%) vs. the published 2.7212%.

One place we differ: for the simple strategy in the game sold as Ultimate Texas Hold'em, enumerating it gives us 2.40% vs. the 2.43% quoted. I'd be curious whether anyone knows how that figure was computed.

The Methodology page on the site lists what's exact, what's simulated and what's approximated. The engines are open source under MIT on GitHub (search for casinogamessimulator/engines), so anyone can check them.

Everything is free, with virtual chips only, no ads and no affiliate links. There's a daily challenge and weekly leaderboards by trainer rating, if anyone here wants to put a name at the top of the video poker one.

Corrections very welcome.
link to original post

Your description is a bit confusing. What you initially call "Hold'em vs. dealer:" seems to match the EV of UTH. From /games/ultimate-texas-hold-em/
Total Combinations: 27,813,810,024,000 Return: -0.021850

I have done the combinatorial analysis (what you call 'exact enumeration') and agree with the Wizard and other authors.

Quote:

One place we differ: for the simple strategy in the game sold as Ultimate Texas Hold'em, enumerating it gives us 2.40% vs. the 2.43% quoted. I'd be curious whether anyone knows how that figure was computed.

You don't say which simple strategy you follow for UTH, but the you quote the same 2.43% that is on the reference link above, so I assume that is your comparison.
"Following my Wizard strategy will result in a house edge of 2.43% and an Element of Risk of 0.58%."

To calculate the house edge of a strategy, you need to do the same calculation as the optimal house edge, but substitute the rule-based decisions for the optimal ones. Assuming the basic evaluator is correct in both cases, it seems that one of you is interpreting the simple rule set wrong or coding it up wrong.

I recall that JB did this work, but Wizard can confirm.
link to original post

Mystery solved, I think. You were right: same evaluator, I was just reading two of the rules differently.

Flop “two pair or better”: I required the hole cards to help, so I checked with trips on the flop alone. Taken literally, you bet those.
River “hidden pair or better”: I counted any hole card matching the board, even when it doesn’t play (e.g. pocket 3s on A-A-K-K). Counting it only when your hole cards actually improve on the board fixes that.
With both changes the enumeration gives a 2.433% house edge and 0.584% element of risk, which is the published 2.43% / 0.58%. Thanks for the push!
link to original post

Great! I was assuming it would take you a long time to recheck the house edge for the simple strategy.

I suppose analyzing the rules is much faster than playing out the 27,813,810,024,000 combinations to get optimal strategy from scratch. Could you please give us estimates of the time it takes to run the check of the 'simple strategy' versus grinding out the optimal strategy?

Quote: Mental

Quote: alejandrobg

Quote: Mental

Quote: alejandrobg

I've built a set of free casino game simulators and trainers at casinogamessimulator . com, and I'd value this forum's scrutiny of the math, since it's checked against Wizard of Odds' published figures throughout:

• Video poker: exact returns and hand frequencies by enumeration, plus Easy/Best strategy charts scored over all 2,598,960 deals. The returns match the published figures for 5 paytables to 6 decimals.
• Blackjack: finite-shoe basic strategy and house edge for 26 rule sets, within 0.005 points of the Wizard's calculator for 22 of them.
• Baccarat and 3-card vs. dealer: exact enumeration, with outcome counts matching the published tables exactly.
• Hold'em vs. dealer: exact enumeration, 2.185% of the ante with optimal play.
• Craps: each bet solved as a Markov chain.
• Pai gow: exact hand frequencies; house edge from a 400M-round seeded simulation: 2.72% (95% CI 2.713–2.728%) vs. the published 2.7212%.

One place we differ: for the simple strategy in the game sold as Ultimate Texas Hold'em, enumerating it gives us 2.40% vs. the 2.43% quoted. I'd be curious whether anyone knows how that figure was computed.

The Methodology page on the site lists what's exact, what's simulated and what's approximated. The engines are open source under MIT on GitHub (search for casinogamessimulator/engines), so anyone can check them.

Everything is free, with virtual chips only, no ads and no affiliate links. There's a daily challenge and weekly leaderboards by trainer rating, if anyone here wants to put a name at the top of the video poker one.

Corrections very welcome.
link to original post

Your description is a bit confusing. What you initially call "Hold'em vs. dealer:" seems to match the EV of UTH. From /games/ultimate-texas-hold-em/
Total Combinations: 27,813,810,024,000 Return: -0.021850

I have done the combinatorial analysis (what you call 'exact enumeration') and agree with the Wizard and other authors.

Quote:

One place we differ: for the simple strategy in the game sold as Ultimate Texas Hold'em, enumerating it gives us 2.40% vs. the 2.43% quoted. I'd be curious whether anyone knows how that figure was computed.

You don't say which simple strategy you follow for UTH, but the you quote the same 2.43% that is on the reference link above, so I assume that is your comparison.
"Following my Wizard strategy will result in a house edge of 2.43% and an Element of Risk of 0.58%."

To calculate the house edge of a strategy, you need to do the same calculation as the optimal house edge, but substitute the rule-based decisions for the optimal ones. Assuming the basic evaluator is correct in both cases, it seems that one of you is interpreting the simple rule set wrong or coding it up wrong.

I recall that JB did this work, but Wizard can confirm.
link to original post

Mystery solved, I think. You were right: same evaluator, I was just reading two of the rules differently.

Flop “two pair or better”: I required the hole cards to help, so I checked with trips on the flop alone. Taken literally, you bet those.
River “hidden pair or better”: I counted any hole card matching the board, even when it doesn’t play (e.g. pocket 3s on A-A-K-K). Counting it only when your hole cards actually improve on the board fixes that.
With both changes the enumeration gives a 2.433% house edge and 0.584% element of risk, which is the published 2.43% / 0.58%. Thanks for the push!
link to original post

Great! I was assuming it would take you a long time to recheck the house edge for the simple strategy.

I suppose analyzing the rules is much faster than playing out the 27,813,810,024,000 combinations to get optimal strategy from scratch. Could you please give us estimates of the time it takes to run the check of the 'simple strategy' versus grinding out the optimal strategy?
link to original post

Ha, it’s actually the same cost: one run computes both. The full enumeration of the game takes about 90 seconds on a modest 4-core / 8-thread machine, and that gives the optimal strategy and the simple strategy’s house edge together. To track down the 2.40/2.43 difference, I scored 32 different readings of the rules in a single pass, which took about 2 minutes.

It’s fast because it never actually plays out the 27.8 trillion combinations one by one:

Suit symmetry: the 22,100 flops collapse to 1,755 distinct ones, each weighted by how often it occurs.
Sort instead of compare: for each flop and each of the 1,081 turn/river pairs, every two-card hand from the remaining 47 cards is evaluated once and sorted. A player hand’s dealer opponents are all the hands that share no card with it, so its wins, ties and losses against all 990 dealer hands come from running counts in that sorted list (inclusion–exclusion on the two shared cards). That’s one evaluation per hand instead of 990.
Decisions are cheap once you have the EVs: with every outcome counted, the optimal play at each decision point is just “take the better EV”. The simple strategy uses the exact same numbers but takes the rule-based decision instead. Scoring a strategy costs almost nothing on top of the enumeration, which is why trying 32 readings barely added time.

The code’s open source (github / casinogamessimulator/engines) if anyone wants to check it or reuse it.

Quote: alejandrobg

Ha, it’s actually the same cost: one run computes both. The full enumeration of the game takes about 90 seconds on a modest 4-core / 8-thread machine, and that gives the optimal strategy and the simple strategy’s house edge together. To track down the 2.40/2.43 difference, I scored 32 different readings of the rules in a single pass, which took about 2 minutes.

It’s fast because it never actually plays out the 27.8 trillion combinations one by one:

Suit symmetry: the 22,100 flops collapse to 1,755 distinct ones, each weighted by how often it occurs.
Sort instead of compare: for each flop and each of the 1,081 turn/river pairs, every two-card hand from the remaining 47 cards is evaluated once and sorted. A player hand’s dealer opponents are all the hands that share no card with it, so its wins, ties and losses against all 990 dealer hands come from running counts in that sorted list (inclusion–exclusion on the two shared cards). That’s one evaluation per hand instead of 990.
Decisions are cheap once you have the EVs: with every outcome counted, the optimal play at each decision point is just “take the better EV”. The simple strategy uses the exact same numbers but takes the rule-based decision instead. Scoring a strategy costs almost nothing on top of the enumeration, which is why trying 32 readings barely added time.

The code’s open source (github / casinogamessimulator/engines) if anyone wants to check it or reuse it.
link to original post

Thanks for the quick explanation. The symmetry argument makes sense and I think the sorting idea is possibly a very useful idea to understand for these sorts of games with a dealer and player hand and common board.

I found the github page. It appears that this code has been around for 6 years and I was unaware of it. Thanks for the link. Do you have any relationship to the folks who did this project?

Quote: Mental

Quote: alejandrobg

Ha, it’s actually the same cost: one run computes both. The full enumeration of the game takes about 90 seconds on a modest 4-core / 8-thread machine, and that gives the optimal strategy and the simple strategy’s house edge together. To track down the 2.40/2.43 difference, I scored 32 different readings of the rules in a single pass, which took about 2 minutes.

It’s fast because it never actually plays out the 27.8 trillion combinations one by one:

Suit symmetry: the 22,100 flops collapse to 1,755 distinct ones, each weighted by how often it occurs.
Sort instead of compare: for each flop and each of the 1,081 turn/river pairs, every two-card hand from the remaining 47 cards is evaluated once and sorted. A player hand’s dealer opponents are all the hands that share no card with it, so its wins, ties and losses against all 990 dealer hands come from running counts in that sorted list (inclusion–exclusion on the two shared cards). That’s one evaluation per hand instead of 990.
Decisions are cheap once you have the EVs: with every outcome counted, the optimal play at each decision point is just “take the better EV”. The simple strategy uses the exact same numbers but takes the rule-based decision instead. Scoring a strategy costs almost nothing on top of the enumeration, which is why trying 32 readings barely added time.

The code’s open source (github / casinogamessimulator/engines) if anyone wants to check it or reuse it.
link to original post

Thanks for the quick explanation. The symmetry argument makes sense and I think the sorting idea is possibly a very useful idea to understand for these sorts of games with a dealer and player hand and common board.

I found the github page. It appears that this code has been around for 6 years and I was unaware of it. Thanks for the link. Do you have any relationship to the folks who did this project?
link to original post

It might be the wrong github repo (I can't post links). github / casinogamessimulator/engines was done by me (mostly coded with AI) it was published yesterday

Quote: alejandrobg

It might be the wrong github repo (I can't post links). github / casinogamessimulator/engines was done by me (mostly coded with AI) it was published yesterday
link to original post

Yes, my original search took me to a different engine repository. I did a new search and found the repository that you posted yesterday. https://github.com/casinogamessimulator/engines I will download it and take a look.

Thanks for sharing.