Casino Hold'em Poker revisited
If anyone else is trying to do this, I would point out that in the COMBINATIONS column, the number of Ties does not include ties where the dealer does not qualify. These are counted as losses, since they are paid as if the dealer lost. That makes perfect sense, but it took me a while to figure this out and get my numbers to match on every row.
Quote: WizardThe above analysis was performed using a brute force combinatorial program that analyzed all 27,813,810,024,000 possible outcomes, and played every hand according to optimal player strategy. It took my computers about a month to crank though the 27.8 trillion hands.
I was able to calculate all four pay tables in 220 minutes on a Macbook (using a single threaded process, but running four processes in parallel). I can do a single pay table in 158 minutes. I use equivalence classes to reduce the number of actual hands evaluated to 1,443,279,637,800. I am using a perfect hash for evaluation of the 133,784,560 7-card poker hands.
Quote: Wizard
The only pay tables fully analyzed are 3 and 4, because it takes my computer about a month to cycle through all possible combinations. However, assuming the player was following optimal strategy under pay table 3 but playing under pay table 1 or 2, the house edge under pay table 1 would be 2.40, and under pay table 2 would be 1.96%. The actual house edge under pay tables 1 and 2 would be slightly less or equal to these figures.
I don't care at all about the EV as expressed in initial bet Units. I really want the EV expressed as a percentage of coin in. The Wizard does not provide the latter, but you can simply derive it from the information in his tables.
The Wizard follows the abhorrent (to me) convention of specifying the house edge of table games as percentage of the initial bet unit. This might be helpful if you are figuring out how many initial bet units you are expected to lose per game. But, when I am playing online for cash back at a certain percentage of coin in, then I want to know the house edge of the game on the basis of coin in, not the number of initial bets. I earn points and cash back on all bets, even returned bets and pushes.
In this table, I give the house edge both as a percentage of the initial bet unit and as a percentage of all action.
| PayTable | EV (% Unit) | EV (% Action) |
|---|---|---|
| #1 | -2.4039% | -0.9106% |
| #2 | -0.44436% | -0.227223% |
| #3 | -2.1576% | -0.8173% |
| #4 | -0.3511% | -0.1329% |
The first thing that I will note is that the Wizard's approximation is pretty close for pay table #1 which he did not analyze in full. We have perfect agreement for #3 and #4. However, we are miles apart for pay table #2. Is there a typo in the #2 pay table? I have only seen PT #3 out in the wild.
Second, the house edge (the way I prefer to think about it) is much lower than Mike implies. This is why the longstanding conventions that players use to state house edge for table games seem so stupid to me. The rules of Playtech and NETENT Casino Hold'em both give the optimal RTP as 99.18% versus my 99.1827% with PT #3. The game developer quotes RTP on the Help page. Online RTPs are always based on total rated action, even for BJ.
Ironically, the Wizard is all over this matter of figuring house edge based on initial bet *these days* ... this would seem to mean the analysis above is older. In a new analysis, he is likely going to include what he calls the 'element of risk'. I won't look for such comments, but he has indicated, I think, that casino personnel expect the HE to be always on initial bet, and it throws them to have some other way to do it. Actually, I think this is a good thing, and Ultimate Texas Holdem is a good example. If the element of risk was presented to them, they might reject it out of hand as too low in HE ... for UTH it is 0.526%, he saysQuote: MentalI have been playing Casino Hold'em online the last few weeks. On The Wizard of Odds Casino Holdem page (https://wizardofodds.com/games/casino-hold-em/), Mike posts the expected values and a table of combinations. I wrote my own Casino Hold'em analyzer and I was able to reproduce Mike's entire 'Return Table — Pay Table 3'.
If anyone else is trying to do this, I would point out that in the COMBINATIONS column, the number of Ties does not include ties where the dealer does not qualify. These are counted as losses, since they are paid as if the dealer lost. That makes perfect sense, but it took me a while to figure this out and get my numbers to match on every row.Quote: WizardThe above analysis was performed using a brute force combinatorial program that analyzed all 27,813,810,024,000 possible outcomes, and played every hand according to optimal player strategy. It took my computers about a month to crank though the 27.8 trillion hands.
I was able to calculate all four pay tables in 220 minutes on a Macbook (using a single threaded process, but running four processes in parallel). I can do a single pay table in 158 minutes. I use equivalence classes to reduce the number of actual hands evaluated to 1,443,279,637,800. I am using a perfect hash for evaluation of the 133,784,560 7-card poker hands.Quote: Wizard
The only pay tables fully analyzed are 3 and 4, because it takes my computer about a month to cycle through all possible combinations. However, assuming the player was following optimal strategy under pay table 3 but playing under pay table 1 or 2, the house edge under pay table 1 would be 2.40, and under pay table 2 would be 1.96%. The actual house edge under pay tables 1 and 2 would be slightly less or equal to these figures.
I don't care at all about the EV as expressed in initial bet Units. I really want the EV expressed as a percentage of coin in. The Wizard does not provide the latter, but you can simply derive it from the information in his tables.
The Wizard follows the abhorrent (to me) convention of specifying the house edge of table games as percentage of the initial bet unit. This might be helpful if you are figuring out how many initial bet units you are expected to lose per game. But, when I am playing online for cash back at a certain percentage of coin in, then I want to know the house edge of the game on the basis of coin in, not the number of initial bets. I earn points and cash back on all bets, even returned bets and pushes.
In this table, I give the house edge both as a percentage of the initial bet unit and as a percentage of all action.
PayTable EV (% Unit) EV (% Action) #1 -2.4039% -0.9106% #2 -0.44436% -0.227223% #3 -2.1576% -0.8173% #4 -0.3511% -0.1329%
The first thing that I will note is that the Wizard's approximation is pretty close for pay table #1 which he did not analyze in full. We have perfect agreement for #3 and #4. However, we are miles apart for pay table #2. Is there a typo in the #2 pay table? I have only seen PT #3 out in the wild.
Second, the house edge (the way I prefer to think about it) is much lower than Mike implies. This is why the longstanding conventions that players use to state house edge for table games seem so stupid to me. The rules of Playtech and NETENT Casino Hold'em both give the optimal RTP as 99.18% versus my 99.1827% with PT #3. The game developer quotes RTP on the Help page. Online RTPs are always based on total rated action, even for BJ.
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https://wizardofodds.com/games/ultimate-texas-hold-em/ .
https://wizardofodds.com/gambling/glossary/#element_of_risk .Quote:Element of Risk
The ratio of the expected player loss to the total amount bet. This is a good measure of comparing the value of one bet against another. For games in which there is no raising the element of risk will be the same as the house edge. However for games in which the player may increase his bet the element of risk will be lower. For example in Caribbean Stud Poker the house edge is 5.22%. However the average amount bet per hand is 2.0445 times the initial bet, so the element of risk would be 0.0522/2.0445 = 2.55%.
B&M casinos rate players based on the number of initial bets. It makes sense for pit bosses to want to calculate the players theoretical loss based on fractions of an initial bet, not total action. I just object to calling this a 0.5% house edge rather than a 0.005 unit edge.
Online casinos track every penny of action effortlessly.
In Casino Hold'em Poker, there is no strategy provided. However, the Playtech version will often stop you when you want to fold a hand. To proceed, you have to click "FOLD" on a pop up screen that says "ARE YOU SURE YOU WANT TO FOLD?".
I was a bit puzzled by this because sometimes the pop up comes when the decision is a clear fold, at other times, it would allow you to fold a callable hand with no pop up.
I quickly realized that I got this pop up whenever I had a pair on the flop, even a board pair. This is totally demented and ass backwards. In Casino Hold'em Poker, board pairs above the rank of three actually hurt you when you have a lousy hand. The dealer needs a pair of fours or higher to qualify. If you have a lousy hand, you can still win if the dealer does not qualify. Once there is a board pair 44+ out there, the dealer can beat your rags simply by out-kicking you. If neither you or the dealer improve, you are most likely going to lose.
Let us say your hole cards are 2h5d-unsuited. The flop comes 4c4s9c, giving you a pair. You need to fold because the EV of this hand is a horrible -1.73 units. The pop up will stop you from folding this garbage until you confirm you really want to fold ASAP.
Instead, let us say your hole cards are 2h5d-unsuited. The flop comes 3c4s9c, giving you a 9-high hand, so far. You need to call because the EV of this hand is -0.45 units, and folding has an EV of -1.0. The pop up will not stop you from folding this callable hand!
Quote: Mentalremember the auto-hold feature on the old Bally's video poker game that would suggest the right hold most of the time, but would suggest holds in some instances that were pretty far from optimal. I found the same thing to be true for the strategy provided for DK Bingo Blackjack in the Help screen.
Yes, some of the autoholds were terrible. With the technology now in the machines they could easily incorporate a 100% correct hold.
There is no excuse for the new DK Bingo BJ game to include such an awful strategy table. I suspect malice. It is not like it is a simplified strategy -- it is as complicated as a perfect strategy and gives the wrong strategy for dozens of cases.Quote: DRichQuote: Mentalremember the auto-hold feature on the old Bally's video poker game that would suggest the right hold most of the time, but would suggest holds in some instances that were pretty far from optimal. I found the same thing to be true for the strategy provided for DK Bingo Blackjack in the Help screen.
Yes, some of the autoholds were terrible. With the technology now in the machines they could easily incorporate a 100% correct hold.
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Many online BJ games have a safety-net feature that you can turn on or off. They will ask for confirmation on some boneheaded plays. I have not studied them in detail, but they seem to save me from some accidental clicks like when I drop my mouse on the floor.
The Playtech system is too simple. It only looks at whether the player is paired without regard to whether the pair is on the board.
Quote: MentalIf the Wizard is just being discrete in only quoting the absolute house edge per game, then I salute him for his discretion. I have seen him use 'element of risk', but RTP seems to be the disclosure term that is required by most online jurisdictions for the equivalent number.
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I prefer to quote the house edge because it is the more standard statistic and gives the user an easy estimate of losses over time. Most players are not going to know the ratio of the total amount bet to average amount bet. I do agree that the Element of Risk is the better measurement of the value of a game, which I usually site as well in games were there is subsequent wagering.
Grosjean speaks at length to this topic in his latest book. He favors the house edge statistic as well. In Ultimate Texas Hold 'Em, it raises the question if the house edge should be based on the Ante only or combined Ante and Blind. Both of us base it on the Ante only, as that is what the table minimum/maximum refer to.
I am definitely not blaming the Wizard for this messy nomenclature. He inherited the conventions from a quaint bygone era. But do we have to put a percent sign after the house edge? When I am installed as benevolent overlord of the planet, I will decree that a special unit be used in place of the percent sign when specifying house edge in the form of initial bet units.Quote: WizardQuote: MentalIf the Wizard is just being discrete in only quoting the absolute house edge per game, then I salute him for his discretion. I have seen him use 'element of risk', but RTP seems to be the disclosure term that is required by most online jurisdictions for the equivalent number.
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I prefer to quote the house edge because it is the more standard statistic and gives the user an easy estimate of losses over time. Most players are not going to know the ratio of the total amount bet to average amount bet. I do agree that the Element of Risk is the better measurement of the value of a game, which I usually site as well in games were there is subsequent wagering.
Grosjean speaks at length to this topic in his latest book. He favors the house edge statistic as well. In Ultimate Texas Hold 'Em, it raises the question if the house edge should be based on the Ante only or combined Ante and Blind. Both of us base it on the Ante only, as that is what the table minimum/maximum refer to.
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It is like the quaintness of specifying engine output by horsepower when kilowatts would be much more convenient to use in calculations.
You example from UTH is a perfect example of the arbitrariness of the house edge standard. What units are we supposed to use? Are we supposed to use the base bet for house edge in UX games or the 10-coin bet unit?
I have not even tried. Every over card, kicker, and board pair affects the EV and the cards interact with each other in diabolical ways.Quote: gordonm888Some years ago, I invested quite a lot of time into analyzing Casino Hold-em Poker and trying to work out a strategy. The strategy became hopelessly complex and I threw up my hands.
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I am playing online. I can consult software for any close calls. If I continue to have opportunities where CHE is the best game available, I will probably develop my own list of minimum criteria for hands that never fold and maximum criteria for always fold. This will reduce my dependence on the software crutch. Even though I am currently using my crutch too often, it does not slow my throughput much. I am able to get coin in at a rate of $57K per hour.
:https://wizardofodds.com/games/casino-hold-em/
Without a 4 of a kind or Straight Flush, this game is a total drain.
It's possible to have a run of a dozen winning hands, but losing at max bet several times is more likely.
Play MS Stud, not this game.
Is this advice intended for me? Your random feelings don't constitute a sound basis for advice.Quote: ChumpChangeMy notes about UTH (which may be different than the game described in the rules at the link)
:https://wizardofodds.com/games/casino-hold-em/
Without a 4 of a kind or Straight Flush, this game is a total drain.
It's possible to have a run of a dozen winning hands, but losing at max bet several times is more likely.
Play MS Stud, not this game.
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I have determined that Casino Holdem is the best game for me to play under certain promo rules for technical reasons. For this particular promo, MS Stud isn't even close, nor is BJ.
Without a 4 of a kind or Straight Flush, MS Stud is a total drain.
If you don't get BJs and don't win your doubles, BJ is a total drain.
The ratio of winning hands to losing hands isn't important to me except for loss rebates.
I must have copied the wrong data into the table for pay table #2. Mike's number was number 1.96%. My calculation was actually in agreement all along. This is the corrected table.Quote: Mental
In this table, I give the house edge both as a percentage of the initial bet unit and as a percentage of all action.
PayTable EV (% Unit) EV (% Action) #1 -2.4039% -0.9106% #2 -0.44436% -0.227223% #3 -2.1576% -0.8173% #4 -0.3511% -0.1329%
The first thing that I will note is that the Wizard's approximation is pretty close for pay table #1 which he did not analyze in full. We have perfect agreement for #3 and #4. However, we are miles apart for pay table #2. Is there a typo in the #2 pay table? I have only seen PT #3 out in the wild.
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| PayTable | EV (% Unit) | EV (% Action) |
|---|---|---|
| #1 | -2.4039% | -0.9106% |
| #2 | -1.9566% | -0.7411% |
| #3 | -2.1576% | -0.8173% |
| #4 | -0.3511% | -0.1329% |
