EOR and EV of each card in dealer’s hand for UTH?
I’m looking for data on the effect of single card removal in UTH, and also the EV to all players for each card appearing in the dealer’s hand.
CAA has EV for each card in player’s hand and in the community cards, Jacobson has (slightly different) figures for EV if each card in players hand, and I’ve not been able to find anything anywhere else.
Does anyone have this data or would be willing to calculate it for me?
Many thanks
Enigma
That requires both knowledge (including unerring math crunching ability) and a perfect memory, to do it to the utmost.
No, I’m not trying to cheat.
As above, other authors in the AP world have published data on the EV for each of the 13 potential values of a single card in the player’s hand and the EV to each player for each of the 13 potential values of a single card in the community cards.
I’m looking for two things: firstly the EV to each player for each of the 13 potential values of a single card in the dealer’s hand and also the EOR of each of the 13 potential values of a single card to all players. (In other words, what is the effect to the house edge on removing a deuce from play in the next round versus a 9, and what is the effect of that deuce or nine being in the dealer’s hand instead of being removed from play, as an example).
I’d appreciate it if you could put me in touch with your friend, thank you.
Mental is the WOV Forum name of an individual; you can send him a personal message. He has not been dropping into the forum lately.
There are many of us who can do the EOR calculation, but it's complicated and requires a lot of work - 9 cards modeled per hand and 3 decision points. Most of the gaming analysts are serious people who aren't interested in putting in the work to do such a calculation because the pre-deal EOR is not of great interest. People are more interested in the EORs for specific post flop decisions.
- a collusion strategy for UTH that uses cards seen by the player to optimize decisions is published on the Wizard of Odds website as well as on Steve How's website called Discount Gambling.
Regarding pre-deal EORs for UTH:
Obviously, the presence of A, K, Q in the deck are important to the player because they create the opportunity for player to Bet 4X on various pre-flop hands in which the player has a positive EV.
The rank with the largest positive EOR on pre-deal EV will be when a deuce is removed from the deck. The removal of the deuce increases the odds of making a 4X wager preflop or a 2x wager post flop.
But beyond that there's not much of interest in a pre-deal EOR calculation for UTH.
I understand that you don’t see the value of EOR in UTH.
That’s fine, and in that case this question isn’t for you as it’s pretty obvious you aren’t able to help here. Have a nice day and all the best.
Good skills and positive variance
E
I’m not talking about HCing, I’m well aware of the value of HCing UTH as I own and have read Grosjean’s CAA.
I’m talking about the EV to the players from the simple fact that a dealer has a 2-9 in their hand.
No, Jacobson hasn’t published that data in his book, or on his site. The closest is a table publishing the EV to the player for each of the 13 card values being held in the player’s hand (not the dealer’s). Grosjean has also published this table in CAA (albeit with slightly different values) and also a table of the EV to all players for each of the 13 card values being on the flop.
Common sense dictates that a 2-9 appearing in the dealer’s hand will help the player, I’m trying to quantify that.
I’m aware of Stephen’s site and have been for some time. I’ve also contacted him via the site with the same request.
I’m also trying to establish if the house edge can be overcome by removing a single card (most likely a deuce, trey, etc but hopefully anything up to a 9) from the next potential round of UTH.
I appreciate that you think my question is pointless. TBH, I’m kind of glad that you do… the more people that think like you, the better.
I’ll reach out to your friend and see if they will help me. Obviously I’m happy to pay for their work.
Quote: EnigmaAPHi everyone,
I’m looking for data on the effect of single card removal in UTH, and also the EV to all players for each card appearing in the dealer’s hand.
CAA has EV for each card in player’s hand and in the community cards, Jacobson has (slightly different) figures for EV if each card in players hand, and I’ve not been able to find anything anywhere else.
Does anyone have this data or would be willing to calculate it for me?
Many thanks
Enigma
link to original post
https://wizardofvegas.com/forum/gaming-business/game-inventors/24259-ultimate-texas-hold-em-face-up-cards/
Look at my comments on this thread and you will see what my UTH program is able to do. The more cards are known, the faster the calculation is.
The article is still presenting a very poor estimate of the collusion advantage based on a sample of only 500 hands. This number of hands is totally inadequate. If Eliot would have done statistical analysis of the spread of EVs within his sample, he would probably realize this himself. To be fair, he write: "Here are the (very approximate) results for six-player collusion at UTH:", but he never estimates the uncertainty in his 'approximate' answer.
Quote: MentalEliot has an updated article date Nov 19 2025: /blog/novelty-games/ultimate-texas-holdem-collusion
The article is still presenting a very poor estimate of the collusion advantage based on a sample of only 500 hands. This number of hands is totally inadequate. If Eliot would have done statistical analysis of the spread of EVs within his sample, he would probably realize this himself. To be fair, he write: "Here are the (very approximate) results for six-player collusion at UTH:", but he never estimates the uncertainty in his 'approximate' answer.
500 is only useful for determining a ballpark figure.
For any kind of exactitude you should basically throw the data out.
Quote: acesideActually, I myself is able to find all these numbers from Eliot’s work. It only takes some patience and statistics skills; however, I leave this as a homework to others.
link to original post
Are you able to download the XLSX files linked in Eliot's article? https://www.888casino.com/blog/novelty-games/ultimate-texas-holdem-hole-card-play-one-dealer-card
When I try to open these files, I get a message saying the files are corrupted:
https://www.888casino.com/blog/sites/newblog.888casino.com/files/inline-files/uth_dhc_computer_perfect_strategy.xlsx
https://www.888casino.com/blog/sites/newblog.888casino.com/files/inline-files/uth_dhc_grosjean_strategy1.xlsx
If you have any of these detailed results, I would love to get a copy so I can compare my calculations in detail.
Quote: Mental
Are you able to download the XLSX files linked in Eliot's article? https://www.888casino.com/blog/novelty-games/ultimate-texas-holdem-hole-card-play-one-dealer-card
When I try to open these files, I get a message saying the files are corrupted:
https://www.888casino.com/blog/sites/newblog.888casino.com/files/inline-files/uth_dhc_computer_perfect_strategy.xlsx
https://www.888casino.com/blog/sites/newblog.888casino.com/files/inline-files/uth_dhc_grosjean_strategy1.xlsx
If you have any of these detailed results, I would love to get a copy so I can compare my calculations in detail.
link to original post
Eliot sent me a link to a page with all the XLSX files: https://advancedadvantageplay.com/downloads/
He knew that 888 had somehow corrupted the files linked from their site. The direct links work fine for me.
Quote: acesideActually, I myself is able to find all these numbers from Eliot’s work. It only takes some patience and statistics skills; however, I leave this as a homework to others.
link to original post
Now that I have the files, I can confirm this. I just sorted Eliot's whole UTH_DHC_Computer_Perfect_Strategy.xlsx file by the column that has the dealer hole card, then split it into thirteen sub-lists for each dealer rank. Then, I use three simple formulas to sum the permutations column and weighted 'Optimal EV' column and divided the two to get the EV for computer-perfect play when you know one dealer hole card. This took me five minutes.
I had a minor bug in my new program which I found and fixed by comparing my results to the XLSX file. Now, my results are agreeing with Eliot's to seven significant digits that I use in my output. These are the EVs for dealer deuce through ace:
0.6242647
0.5266145
0.4321708
0.3381928
0.2860220
0.2284905
0.1615280
0.0856705
-0.0124859
-0.0817021
-0.1625157
-0.2582048
-0.4010560
Quote: MichaelLandonThere are some serious methodological problems with Jacobson's work on this. You need a more reliable source on this to test your numbers against.
link to original post
I disagree. We are both calculating computer perfect play, and we should both agree on the exact number. Our calculation do agree, now that I debugged my new code.
This is not an EV that is attainable by players just using their grey matter at a table. Nobody could possibly remember of all the deviations from basic strategy when the dealer is known to have an six and you have a marginal post-flop decision.
The pre-game EV advantage that he calculates knowing that the player will have a 6c in the hole is essentially the real-world EV the player accrues, because he simply needs to play basic strategy well to achieve it,
What specific methodological problems do you see?
Quote: MentalQuote: MichaelLandonThere are some serious methodological problems with Jacobson's work on this. You need a more reliable source on this to test your numbers against.
link to original post
I disagree. We are both calculating computer perfect play, and we should both agree on the exact number. Our calculation do agree, now that I debugged my new code.
You will end up with the same numbers if you replicate the same flaws.
Remember Jacobson is a casino consultant. I don't want to encourage casinos to find any errors in his work, or encourage them to stop paying for faulty and exploitable analysis. Sorry I can't go into this further for obvious reasons.
Firstly, I am grateful to you all for contributing to this, and for the time and effort that you have spent in both calculating and posting.
This information (the EV to the player from the dealer having each one of the 13 values, and the playing altering their play accordingly) is very useful (although I would be keen to discuss further with Michael regarding the accuracy concerns).
However, I am still looking for the two sets of data in the original request to this post:
1) The EOR for each of the 13 card values to the player’s EV
(in other words, if a player had the ability to prevent a 6 from being dealt in the next round, what would be the EV to all players, even if they just played basic? What would it be if they removed an A instead?)
2) The EV to each player for each of the 13 card values in the dealer’s hand.
(In other words, if a player were to be able to somehow control one of the dealer’s two cards, what is the EV to all players for each of the 13 cards merely being in the dealer’s hand, even if they just played basic because they might not know any better?)
As posted above, Grosjean provides the EV for each value appearing both in the player’s hand and on the flop, and Jacobson provides the EV for each value in the player’s hand. I’m simply looking for the obvious omission.
I’d be very grateful if someone is willing to assist with calculating these sets of data and am willing to pay for the information or hire them to calculate it.
(Aceside - I do not believe this data is obtainable from Jacobson’s excel sheet. Feel free to prove me wrong, however…)
Thanks again
Enigma
For your question #2, I kinda understand it. So this is hole carding. Mental has listed the EV of each of the thirty ranks above. The problem is players cannot change their ante+blind bet amount after seeing the dealer hole cards, so the player EV gain is still very limited. I haven’t found any opportunities for hole carding in any casinos around the world.
Quote: EnigmaAPHi Aceside, Michael and Mental.
Firstly, I am grateful to you all for contributing to this, and for the time and effort that you have spent in both calculating and posting.
This information (the EV to the player from the dealer having each one of the 13 values, and the playing altering their play accordingly) is very useful (although I would be keen to discuss further with Michael regarding the accuracy concerns).
However, I am still looking for the two sets of data in the original request to this post:
1) The EOR for each of the 13 card values to the player’s EV
(in other words, if a player had the ability to prevent a 6 from being dealt in the next round, what would be the EV to all players, even if they just played basic? What would it be if they removed an A instead?)
2) The EV to each player for each of the 13 card values in the dealer’s hand.
(In other words, if a player were to be able to somehow control one of the dealer’s two cards, what is the EV to all players for each of the 13 cards merely being in the dealer’s hand, even if they just played basic because they might not know any better?)
As posted above, Grosjean provides the EV for each value appearing both in the player’s hand and on the flop, and Jacobson provides the EV for each value in the player’s hand. I’m simply looking for the obvious omission.
I’d be very grateful if someone is willing to assist with calculating these sets of data and am willing to pay for the information or hire them to calculate it.
(Aceside - I do not believe this data is obtainable from Jacobson’s excel sheet. Feel free to prove me wrong, however…)
Thanks again
Enigma
link to original post
I pointed out the earlier thread where I did a boatload of calculations on exposed cards and collusion in UTH. The clear result of that thread is that there is very little EV change for a single card removed from the deck EVEN WHEN YOU PLAY OPTIMALLY. If you play basic strategy when you know a deuce is removed from the deck, the EV change will be even less. My program plays optimally. I suppose I could do a bit of work to have the program calculate the basic strategy for every hand and then recalculate the EV with a card removed from the deck. This means doing 22 trillion hands twice for each card removed from the deck, so 44 x 13 = 572T hands. I doubt it is worth the effort, but I might think about how I could code this without too much effort. I am cruising in the Mediterranean, so maybe it is something I can do to to amuse myself on a sea day.
A basic strategy would contain every optimal decision for every possible player hand, flop, and river. (52 choose 2)*(50 choose 3)*(47 choose 2) = 28,094,757,600 ways. If you specify a basic strategy for UTH without compacting it, it will not fit in RAM. A program to calculate UTH EVs will typically calculate the strategy on the fly and then discard it to avoid consuming memory. If you have a compact basic strategy in mind, the programmer could calculate the EV for that strategy. I think I could also calculate the basic strategy in smaller chunks and then apply it to the deck missing one specific card.
1) I disagree.
2) No, it’s not holecarding.
Incidentally though, not being able to increase your bet after seeing the HC is irrelevant in HCing anyway…that’s not how an AP obtains an edge on a HC play.
As per my previous post, it’s clear you aren’t able to help with this query.
I don’t expect you to be able to answer my questions, and wasn’t placing you under any obligation to offer your opinions on them either.
Please be assured that there is no need to trouble yourself by continuing to reply to the thread with your valuable contributions.
All the best and I wish you well
This question isn’t meant for someone who doesn’t understand it.
Especially when that person feels they can offer their unsolicited opinion that the OP was invalid and questionable, and when that person tries to suggest that the data will be used for cheating or deception in a home poker game, or when they resort to trying to taunt the OP with the childish claim that they easily calculated the data required themselves but aren’t willing provide it.
These are just a few examples, and the reader will find plenty more above. Despite all of the provocations from you, I’ve been nothing but polite, courteous and respectful towards you in my replies, every single time.
I don’t know if you are intentionally trolling or sealioning here, or if you are genuinely curious about trying to figure out an Advanced play that you aren’t aware of and trying to pick away until I tell you, but neither of those things are helpful or conducive to this thread.
As before, I wish you all the best, despite your behaviour towards me. But that’s it, no more engagement now, I’m not prepared to waste any more time and energy with you.
I’m obviously happy to engage with people who are able to help me and even if they aren’t, people who are prepared to discuss this seriously.
I am going to block you now, to prevent me from seeing your inevitable reply which will no doubt be attempt to troll / sea-lion further. Most likely ‘I was only asking the question…’, I would imagine.
Condensing both your posts into one reply for thread tidiness:
1) I’ve read through that thread and whilst it is very detailed on collusion play and the minimal impact of increased EV obtained through that additional information, I don’t see any data on the impact to EV from the fact that a card cannot appear in your hand at all.
To phrase it another way, my understanding of that thread is that you have calculated there isn’t much EV gain from noting that one of the outs linked to one of your cards is live or dead.
What it doesn’t cover is the EV gain from the fact a specific card cannot appear in your hand in the first place.
Correct me if I’m wrong, but I cannot see any data on this in that thread
2) Thank you, I’d be grateful if you could calculate that, although I understand you are on a cruise at the moment.
Incidentally, when I use the phrase ‘basic strategy’ for UTH, I am using it in a loose sense and am referring to either the Wizard’s version or Grosjean’s (both of which are very similar and provide an edge very close to computer perfect). I appreciate that true computer perfect BS is impossible for a player to achieve, and never meant that.
Quote: EnigmaAPHi Mental,
Condensing both your posts into one reply for thread tidiness:
1) I’ve read through that thread and whilst it is very detailed on collusion play and the minimal impact of increased EV obtained through that additional information, I don’t see any data on the impact to EV from the fact that a card cannot appear in your hand at all.
To phrase it another way, my understanding of that thread is that you have calculated there isn’t much EV gain from noting that one of the outs linked to one of your cards is live or dead.
What it doesn’t cover is the EV gain from the fact a specific card cannot appear in your hand in the first place.
Correct me if I’m wrong, but I cannot see any data on this in that thread
2) Thank you, I’d be grateful if you could calculate that, although I understand you are on a cruise at the moment.
Incidentally, when I use the phrase ‘basic strategy’ for UTH, I am using it in a loose sense and am referring to either the Wizard’s version or Grosjean’s (both of which are very similar and provide an edge very close to computer perfect). I appreciate that true computer perfect BS is impossible for a player to achieve, and never meant that.
link to original post
I agree that the collusion thread is addressing a slightly different question. Whether you know about a specific dead card before or after you size your bet is different for practical purposes. Knowing you will become aware of random dead cards after the deal is different than knowing a specific card is out of play before the deal. It is also more valuable. I think I see a way of calculating what you want efficiently. We try to sightsee all day and dance all night when we are cruising. I have a sea day coming up on the 7th and I will try to do some work on this.
BTW, we have cruised to 110 different ports and will add 17 more ports this year.
Enigma
Quote: gordonm888I also have figured out your reason for asking this question. It would not be appropriate for me to mention it publicly. May I send you a PM?
link to original post
Please do! And THANK YOU for your discretion. It’s a pleasure talking with people that realise the way this kind of thing needs to be handled.
Enigma
Quote: gordonm888I also have figured out your reason for asking this question. It would not be appropriate for me to mention it publicly. May I send you a PM?
link to original post
I actually haven’t understood what OP is asking. Please PM me too to let me know, if you can.
(52 choose 2) * (50 choose 3) * (47 choose 2) * (45 choose 2) = 27.81381e+12
By using suit equivalences, this is reduced to:
169 * (50 choose 3) * (47 choose 2) * (45 choose 2) = 3.5448974e+12
If we choose one card to be in the dealer's hand, the number of equivalence classes increases to 391, but fixing one card makes the problem smaller:
391 * (49 choose 3) * (46 choose 2) * (45 choose 1) = 0.335516e+12
If you just remove a card from the deck, it becomes a bigger calculation again:
391 * (49 choose 3) * (46 choose 2) * (44 choose 2) = 7.053297e+12
This is why the card removed from play took me three days to calculate. Here are the results in CSV form. There are kinks if you plot the EVs by rank. This is probably due to weird straight interactions with the removed card.
2,0,-0.013437
3,1,-0.013320
4,2,-0.013760
5,3,-0.014134
6,4,-0.013922
7,5,-0.015412
8,6,-0.016965
9,7,-0.019548
T,8,-0.026877
J,9,-0.026982
Q,10,-0.029061
K,11,-0.032753
A,12,-0.035955
I believe I can efficiently recalculate the EVs using basic strategy, but it is not something I want to code up while I am vacationing in Europe. The pre-deal EV when you know the dealer has a deuce of clubs as a hole card will be much smaller if you play basic strategy instead of optimal strategy, but I still expect the EV to be significantly positive. I will update if I make progress on this problem.
Quote: MentalThe unique ways to deal a UTH hand are:
(52 choose 2) * (50 choose 3) * (47 choose 2) * (45 choose 2) = 27.81381e+12
By using suit equivalences, this is reduced to:
169 * (50 choose 3) * (47 choose 2) * (45 choose 2) = 3.5448974e+12
If we choose one card to be in the dealer's hand, the number of equivalence classes increases to 391, but fixing one card makes the problem smaller:
391 * (49 choose 3) * (46 choose 2) * (45 choose 1) = 0.335516e+12
If you just remove a card from the deck, it becomes a bigger calculation again:
391 * (49 choose 3) * (46 choose 2) * (44 choose 2) = 7.053297e+12
This is why the card removed from play took me three days to calculate. Here are the results in CSV form. There are kinks if you plot the EVs by rank. This is probably due to weird straight interactions with the removed card.
2,0,-0.013437
3,1,-0.013320
4,2,-0.013760
5,3,-0.014134
6,4,-0.013922
7,5,-0.015412
8,6,-0.016965
9,7,-0.019548
T,8,-0.026877
J,9,-0.026982
Q,10,-0.029061
K,11,-0.032753
A,12,-0.035955
I believe I can efficiently recalculate the EVs using basic strategy, but it is not something I want to code up while I am vacationing in Europe. The pre-deal EV when you know the dealer has a deuce of clubs as a hole card will be much smaller if you play basic strategy instead of optimal strategy, but I still expect the EV to be significantly positive. I will update if I make progress on this problem.
link to original post
Just to be clear, the above analyses are EOR, optimal strategy?
As you remarked, there are discontinuities in the trend of 'EV vs rank of card that is removed.' These are caused by the discontinuities in strategy as a function of ranks. In basic strategy, for example, you bet 4X on the pairs AA - 33 but check a 22 pair. When a 3 is removed from the deck there is considerable value in that knowledge because you avoid betting 4x on a 33 pair. When a 2 is removed, there is less value in that knowledge because you would have checked a 22 pair (which is basic strategy) anyway without knowing that a 2 is removed. So the effect of removal is lower for removal of a 2 than for removal of a 3.
Quote: acesideI’d say this non-monotonically increasing trend is caused by the A-2-3-4-5 straights.
link to original post
I have not thought about what experiments could be done to pin down all the wiggles in the EV versus rank plot. You want the dealer to qualify when you can beat the dealer hand and to not qualify when you have bupkis/zippo. If the dealer qualifies with a straight, you usually lose, so killing straights has a positive effect on EV, I think. Gordon's point about deuces/treys is a good one. It will be interesting to see if the results using basic strategy look qualitatively different.
These results above are obtained using full knowledge of the card removed from the deck and playing optimally. I agree with OP that it will be hard to play anywhere near optimal strategy when you know the dealer/board/player will have a deuce of clubs. The EV using ordinary basic strategy should be a floor, and any improvements beyond that are likely to be limited (unless the player spends a lot of time memorizing deviations or uses computer assistance).
I hope I can do the ordinary BS calculations quickly when I get home next week.
I agree with ace that optimal BS is not a practical strategy, but it will have to serve as a stand-in.
What I am saying is, if we do not regard A-2-3-4-5 as a straight, the above trend will become monotonically decreasing. Just a thought.
Quote: acesideSubtracting the usual house edge of -0.02185, the resulting EV change list is: (0.008413, 0.008530, 0.008090, 0.007716, 0.007928, 0.006438, 0.004885, 0.002302, -0.005027, -0.005132, -0.007211, -0.010903, -0.014105).
What I am saying is, if we do not regard A-2-3-4-5 as a straight, the above trend will become monotonically decreasing. Just a thought.
link to original post
Ace, deuce, and king are involved in 2 straights. 6-T are involved in 5 straights. If you subtract some coefficient times the number of possible straights from the EV, you can make the chart straighter than it is. I can easily program the removal of all four cards of each rank to make this even more obvious. If remove one deuce, you deaden the remaining deuces (whether they are in player or dealer hand). If you remove all four deuces, you don't deaden any rank for making pairs, 2P, trips, FH, quads. But, you completely kill many straights.
This is easy to program and would take about 3-4 days to calculate. Not my highest priority, but I will try to do it when my computer isn't busy with some other calculation.
Hiking in the Pyrénées now, back to the US on Tuesday. I try not to think about how much EV I gave up by going on vacation for three weeks.
Quote: acesideSubtracting the usual house edge of -0.02185, the resulting EV change list is: (0.008413, 0.008530, 0.008090, 0.007716, 0.007928, 0.006438, 0.004885, 0.002302, -0.005027, -0.005132, -0.007211, -0.010903, -0.014105).
What I am saying is, if we do not regard A-2-3-4-5 as a straight, the above trend will become monotonically decreasing. Just a thought.
link to original post
The ability to form an A-2-3-4-5 straight benefits the dealer and player equally on the Ante and Pay bets; it is only that a straight is paid 1 to 1 on the bonus bet that makes straights beneficial to the player's equity. Elimination of the A-2-3-4-5 straight will negatively affect the EVs of those 5 cards (A,2,3,4,5) by an equal amount, and so the trend of ΔEV vs ranks will not be straightened out. I believe that he kink in the trend exists for the reason I gave earlier.
Aceside, I will be willing to bet you $100 that Mental's analysis will show that elimination of the A-2-3-4-5 straight does NOT eliminate the kink in the trend of EV (or ΔEV) vs ranks.
2,0.017925
3,0.020279
4,0.021814
5,0.022412
6,0.026110
7,0.019294
8,0.014763
9,0.007374
T,-0.026775
J,-0.025586
Q,-0.021721
K,-0.012373
A,-0.003099
If you eliminate all rank T or higher, you kill all Royals. The lack of RF will only reduce EV by about 0.01, so I cannot easily explain the big jump between 9 and T. There are so many different factors contributing to EV for and against the player when you kill straights, SFs, and RFs. I would have to separately track the contributions to EV from Ante, Blind, and Play bets to get a real understanding of the EOR.
What I don't understand is why the focus on the A-2-3-4-5 wheel straight. Removing all 6s kills many straights, and this is good for the player. Removing high cards is bad for the player because the player has fewer favorable raise opportunities.
Single card EOR seems kind of meaningless here because it's not going to be additive (or even close to additive) the way it is for other games. For example, of you remove all cards of a certain rank, high-paint hands like quads and fill houses become much common. If you remove all the cards of many ranks they become MUCH more common. At the extreme, in a game with only 8 cards of two ranks, every hand is quads or a full house (it's not possible to make any other hand) so you get massive payouts on half the hands (the hands you win). So the EOR is highly dependent on which cards have already been removed. This is less true in other games (certainly not to anywhere near this extreme). In blackjack, removing low cards pretty much always helps the player and removing high cards pretty much always helps the house (as long as blackjacks are still possible)
Quote: SkinnyTonyWhen you say "optimal strategy", are you recalculating the optimal strategy for the short deck, or using the same strategy you would for a full deck?
Single card EOR seems kind of meaningless here because it's not going to be additive (or even close to additive) the way it is for other games. For example, of you remove all cards of a certain rank, high-paint hands like quads and fill houses become much common. If you remove all the cards of many ranks they become MUCH more common. At the extreme, in a game with only 8 cards of two ranks, every hand is quads or a full house (it's not possible to make any other hand) so you get massive payouts on half the hands (the hands you win). So the EOR is highly dependent on which cards have already been removed. This is less true in other games (certainly not to anywhere near this extreme). In blackjack, removing low cards pretty much always helps the player and removing high cards pretty much always helps the house (as long as blackjacks are still possible)
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I answered this question several times up-thread.
In the future, I will use CPOS for computer perfect optimal strategy to describe what I have done so far, which is to use all knowledge available to the player to make perfect strategy choices at every decision point. If the player somehow knows that a deuce of clubs will appear on the flop before the game is dealt, then I will make pre-flop decisions optimally exploiting that knowledge. I have only done CPOS calculations for this thread and the UTH collusion thread.
I wrote code to do the BS calculations, but I have not tested or verified the code yet. By BS, I mean that I will calculate the EV for a certain game constraint assuming the player plays perfect basic strategy according to the base game with no cards missing and a random deal. As aceside pointed out, no player can play BS or CPOS without a computer, so both CPOS and BS are theoretical concepts for a B&M game unless you are cheating.
Yes, removing an entire rank increases the frequency of big hands like FH and quads. That is only part of the reason why the player has an edge when low ranks are removed. There are a lot of moving parts, and you would need to do an analysis of the changes in all of the EV table. There are 52 lines in the EV table. (See the https://wizardofodds.com/games/ultimate-texas-hold-em/ EV table.) Removing a card or rank affects all of the frequencies in the EV table. In particular, it changes the probability for the dealer to qualify and it changes the probability of a 4x raise. I removed all cards of a rank just to accentuate the effects of removing one card from play and make it easier to differentiate dead card effects from the effects of killing straight/SF/RF hands.
I don't really play UTH. I am only doing these calculations because I was asked to do them by PM and I know how to do them very quickly. I am going to attempt to get started on this today, but I am heading to the Indy 500 and may not get the calculations set up before I leave.
