Double Draw Poker
Photo from G2E:
Rack Card:
Bonus Bet pay table.
| Hand | Pays |
|---|---|
| Five of a Kind | 500 |
| Natural Royal Flush | 100 |
| Wild Royal Flush | 50 |
| Straight Flush | 50 |
| Four of a Kind | 20 |
| Full House | 5 |
| Flush | 3 |
| Straight | 2 |
| Three of a kind | 1 |
| Two pair | Push |
| Pair | Loss |
| Trash | Loss |
I have just started my analysis. To give you a preview, here is my count of poker hands on the deal. Remember that the two jokers are treated the same was as in pai gow poker.
| Hand | Count |
|---|---|
| Five of a Kind | 6 |
| Natural Royal Flush | 4 |
| Wild Royal Flush | 80 |
| Straight Flush | 540 |
| Four of a Kind | 1,320 |
| Full House | 5,328 |
| Flush | 11,388 |
| Straight | 34,704 |
| Three of a kind | 75,048 |
| Two pair | 156,816 |
| Pair | 1,341,888 |
| Trash | 1,535,388 |
| Total | 3,162,510 |
Quote: tringlomaneSo the bonus bet at the Cromwell pays even money for three of a kind? Unfortunately reports say casinos in Detroit treat three of a kind as a push for the bonus bet, which definitely makes the game terrible.
Yup. ShuffleMaster, I mean Ballys, was kind enough to give me the math report by GLI on the game. The report indicates the two pay tables mentioned. They report a house edge of 1.7479% for the Cromwell pay table and 3.8503% for the Detroit pay table.
The number of combinations in this game is 28,587,727,358,190. To put that in perspective there are 3,986,646,103,440 in video poker. So, this game as 7.17 times as many. Without any short cuts, this game would take months, perhaps years, of computer time to cycle through. With enough short cuts, I could get it down to days, but it is time consuming for me to put in short cuts.
I hate to do this, but I think I'm going to trust the GLI number until the game gets enough placements for me to warrant the time to analyze.
Quote: WizardYup. ShuffleMaster, I mean Ballys, was kind enough to give me the math report by GLI on the game. The report indicates the two pay tables mentioned. They report a house edge of 1.7479% for the Cromwell pay table and 3.8503% for the Detroit pay table.
The number of combinations in this game is 28,587,727,358,190. To put that in perspective there are 3,986,646,103,440 in video poker. So, this game as 7.17 times as many. Without any short cuts, this game would take months, perhaps years, of computer time to cycle through. With enough short cuts, I could get it down to days, but it is time consuming for me to put in short cuts.
I hate to do this, but I think I'm going to trust the GLI number until the game gets enough placements for me to warrant the time to analyze.
Didn't you already do an analysis of generic double draw VP? Isn't this just a modified version with two bugs and restricted drawing rules?
Quote: MathExtremistDidn't you already do an analysis of generic double draw VP? Isn't this just a modified version with two bugs and restricted drawing rules?
He did do double draw VP, but double draw VP didn't allow you to have a double draw for every hand either...
https://wizardofodds.com/games/video-poker/tables/double-draw/
Quote: IbeatyouracesI would change five of a kind to five aces. Some people might get confused as to how the jokers work even though it's mentioned on the rack sheet.
I agree with this.
Quote: WizardYup. ShuffleMaster, I mean Ballys, was kind enough to give me the math report by GLI on the game. The report indicates the two pay tables mentioned. They report a house edge of 1.7479% for the Cromwell pay table and 3.8503% for the Detroit pay table.
The number of combinations in this game is 28,587,727,358,190. To put that in perspective there are 3,986,646,103,440 in video poker. So, this game as 7.17 times as many. Without any short cuts, this game would take months, perhaps years, of computer time to cycle through. With enough short cuts, I could get it down to days, but it is time consuming for me to put in short cuts.
I hate to do this, but I think I'm going to trust the GLI number until the game gets enough placements for me to warrant the time to analyze.
I analyzed this game a while back and my numbers differ from these published results. I get a HE of 3.915% for the Cromwell paytable. I did it pretty quickly, so there might have been an error, but double checking my code nothing stands out. It is a big enough difference, though, that something must be wrong.
The way I did it was to reduce the number of starting hands from 3,162,510 to 226,408. I did this by building a map of hand rank and suit patterns. The nice thing is this map reduction is useful for other games, it isn't specific to Double Draw. It works like this:
Each card rank has a prime number value. Multiply them together to get a unique total for any given hand. This is the technique used by Cactus Kev's hand evaluator.
I then also make a suit pattern, which is a 5 char string that works something like this:
if the hand is AsKd3s4c5h the suit map would be: "01023". This shows that the first and third card are the same suit, the others are different suits. This works because from an EV perspective, AsKd3s4c5h and AdKs3d4c5h are equivalent.
I then calculate the frequency for each rank and suit pattern. For instance:
Example Hand: JsQhKhAsAd
Rank Value: 55915103
Suit Pattern: 01102
# of occurrences: 12
I took the 206,408 hands and split them in to 100 hand files. I wrote code to read the file, loop over each hand and evaluate the EV. I spun up a bunch of Amazon EC2 instances to churn through the files.
A few hours later I used another program to consolidate the results and multiply each hand EV by the number of occurrences for that hand. Then I found the average EV.
Anybody see an issue with this approach?
Quote: jopkeAnybody see an issue with this approach?
Huh, can it be that easy? I recently tried to write a similar bucketing algorithm for the standard 52c deck (and thus avoid manually copying the suit patterns by hand from the wizard's vp programming tip page).
Are you sorting beforehand? By value and suit? 2s3s3c6d7h == 2d3c3d6s7h, but my algorithm misses a couple of cases like this, where the suits in pairs are reversed, and I end up with approx 136k buckets instead of 134k.
How many buckets do you get for the standard 52c deck, w/o jokers (~2.6M hands)?
Quote: socksHuh, can it be that easy? I recently tried to write a similar bucketing algorithm for the standard 52c deck (and thus avoid manually copying the suit patterns by hand from the wizard's vp programming tip page).
Are you sorting beforehand? By value and suit? 2s3s3c6d7h == 2d3c3d6s7h, but my algorithm misses a couple of cases like this, where the suits in pairs are reversed, and I end up with approx 136k buckets instead of 134k.
How many buckets do you get for the standard 52c deck, w/o jokers (~2.6M hands)?
Standard deck reduces to 204087
The code grabs all 5 card combinations, sorted by value from low to high. It gets the rank product and the suit pattern and stores them in a map, keeping track of how many of each exists. Here is a sample from the file generated.
It is example hand | rank product | suit | frequency
2h6sThQsQh 486266 01010 6
2s6hTsQhQd 486266 01012 12
2h6dThQsQh 486266 01020 12
2s6dTsQhQd 486266 01021 12
2s6hTsQdQc 486266 01023 12
2s6hThQsQh 486266 01101 6
2s6hThQsQd 486266 01102 12
2h6sTsQsQh 486266 01110 6
2s6hThQhQd 486266 01112 12
2h6dTdQsQh 486266 01120 12
2s6dTdQhQd 486266 01121 12
2s6hThQdQc 486266 01123 12
2s6hTdQsQh 486266 01201 12
2s6hTdQsQd 486266 01202 12
2s6hTdQsQc 486266 01203 12
2h6sTdQsQh 486266 01210 12
2s6hTdQhQd 486266 01212 12
2s6hTdQhQc 486266 01213 12
2h6dTsQsQh 486266 01220 12
2s6dThQhQd 486266 01221 12
2s6hTdQdQc 486266 01223 12
2h6dTcQsQh 486266 01230 12
2s6dTcQhQd 486266 01231 12
2s6hTcQdQc 486266 01232 12
3s7s7hQsQh 487227 00101 6
3s7s7hQsQd 487227 00102 8
3s7s7hQhQd 487227 00112 4
3h7h7dQsQh 487227 00120 4
3s7s7dQhQd 487227 00121 8
Quote: jopkeStandard deck reduces to 204087
The code grabs all 5 card combinations, sorted by value from low to high. It gets the rank product and the suit pattern and stores them in a map, keeping track of how many of each exists. Here is a sample from the file generated.
It is example hand | rank product | suit | frequency
2h6sThQsQh 486266 01010 6
...
Do you have any 24 buckets? I think there should be.
Take a look at the pairs section on https://wizardofodds.com/games/video-poker/methodology/
I might be missing something, but there are no 6's, and the majority are 24's. Similarly, the 2 pairs section is half 12's and half 24's.
Quote: jopkeStandard deck reduces to 204087
2s6hTdQhQd 486266 01212 12
2s6dThQhQd 486266 01221 12
There should be 134459 buckets. The two lines I copied should be in the same bucket.
I forget the paytable at the Hard Rock (Hollywood, FL) but I did enjoy playing the game despite the initial confusion when I heard the dealer mention something about five aces and thought it was a joke.
Quote: MathExtremistThere should be 134459 buckets. The two lines I copied should be in the same bucket.
Good point, that would cut down on the number of hands needed to be calculated, but won't affect the overall outcome.
The reduced set I have is pretty quick when spread over EC2 instances, but reducing further would cut the costs a bit, so I'll likely look into it. Thanks for pointing it out.
Quote: jopkeI analyzed this game a while back and my numbers differ from these published results. I get a HE of 3.915% for the Cromwell paytable. I did it pretty quickly, so there might have been an error, but double checking my code nothing stands out. It is a big enough difference, though, that something must be wrong.
What also concerns me is that the house edge number submitted to the Washington gaming commission doesn't match either of these numbers...
http://www.wsgc.wa.gov/activities/game-rules/double-draw-poker.pdf
Obviously the math is very difficult for this game.
Quote: tringlomaneWhat also concerns me is that the house edge number submitted to the Washington gaming commission doesn't match either of these numbers...
http://www.wsgc.wa.gov/activities/game-rules/double-draw-poker.pdf
Obviously the math is very difficult for this game.
Very strange, those numbers are very different. Maybe there is some "House Edge" vs "Element of Risk" confusion in the Bally literature...
If anyone wants to compare notes with individual hands, I'd be happy to. I played around with a basic strategy but didn't get very far. It is complicated for sure and I doubt any reasonable strategy will be close to the theoretical house edge. I haven't looked at it for a while, I might dig back into it with a clear head.
Quote: tringlomane...Obviously the math is very difficult for this game.
This makes me laugh a little. I've put an embarrassing # of hours into trying to get any correct result for various games and I've come up pretty empty. As far as I can tell all games are absurdly hard to analyze. With Video poker, my result was so close, the error disappeared if I forced it into a float, but there had been no division to that point, so it couldn't be a rounding error. The smallest additional error I could purposefully introduce was 4x as large.
Quote: MathExtremistThere should be 134459 buckets. The two lines I copied should be in the same bucket.
I tweaked my code, it isn't pretty, but it is now reduced to 134459 hands. With 2 jokers the total is 152646.
Thanks guys for the input. Now if we could figure out why the DoubleDraw HE numbers are so inconsistent...
Quote: socksThis makes me laugh a little. I've put an embarrassing # of hours into trying to get any correct result for various games and I've come up pretty empty. As far as I can tell all games are absurdly hard to analyze. With Video poker, my result was so close, the error disappeared if I forced it into a float, but there had been no division to that point, so it couldn't be a rounding error. The smallest additional error I could purposefully introduce was 4x as large.
IMO, probably the easiest poker based table game to analyze is Mississippi Stud. There are no draws, it is just a straight 5 card hand. I started with that one and it is quick enough no reductions are needed.
Quote: jopkeIMO, probably the easiest poker based table game to analyze is Mississippi Stud. There are no draws, it is just a straight 5 card hand. I started with that one and it is quick enough no reductions are needed.
A good suggestion. Maybe I'll take a second look. I actually noticed MS a couple of months back when I was working on video poker. Like you say, it's small enough that a straightforward approach can work. At the time, I didn't want to get too far off track of what I was doing, but I took a couple of days to look at it anyway. The initial results were pretty close, but a run with my naive approach went overnight, and I went back to VP before figuring out exactly what went wrong.
Quote: socksA good suggestion. Maybe I'll take a second look. I actually noticed MS a couple of months back when I was working on video poker. Like you say, it's small enough that a straightforward approach can work. At the time, I didn't want to get too far off track of what I was doing, but I took a couple of days to look at it anyway. The initial results were pretty close, but a run with my naive approach went overnight, and I went back to VP before figuring out exactly what went wrong.
As a baseline for you to measure against, my code runs through the full combinatorial analysis of Mississippi Stud in about 23 seconds. My results match Wizards exactly.
I don't write my code to focus on performance, I'm more concerned with maintainability and development time than with execution time, so it would certainly be possible to get it quite a bit faster.
First step is to get a quick hand evaluator, which it sounds like by now you have.
Quote: jopkeAs a baseline for you to measure against, my code runs through the full combinatorial analysis of Mississippi Stud in about 23 seconds. My results match Wizards exactly.
Thanks for the reference time. I also focus on development time and expressiveness. When I made that half hearted attempt at MS, I used a mutual recursion technique I'd worked on for BJ. Outside of the unoptimized eval, my code was a couple of dozen lines. Mostly I needed a break from VP, But I think I will drop in a better eval in and try to get some results. It'd be nice to post a win. Thanks.
JB has to get the glory on this game. As I mentioned before, there are an ungodly number of combinations to this game. However, JB is a genius at time-saving short-cuts and multi-core processing. His program takes only several hours to crunch the numbers. So, a well-deserved BRAVO to JB on this one. Check out his strategy as well.
I welcome all comments, questions, and especially corrections on the page.
Quote: WizardYup. ShuffleMaster, I mean Ballys, was kind enough to give me the math report by GLI on the game. The report indicates the two pay tables mentioned. They report a house edge of 1.7479% for the Cromwell pay table and 3.8503% for the Detroit pay table.
I'm annoyed that they consider the "element of risk" as the "house edge". A game with a complex strategy, where the element of risk is 1.75% and the cheapest available average bet is $19.29? And 5 aces are worth 5.92% of an ante?? I'm shocked this game is a success. People have to play this game pretty badly. Straight flush draws are super important. And the Detroit casinos (Motorcity was mentioned in an older thread) should definitely be ashamed of themselves for offering the weaker paytable. And it was reported that AC also offers this crappier paytable too??? :(
Also, even though the second draw strategy is probably obvious...
My guess is...any chance for a paying hand is a call I suppose, otherwise fold? If that's the general case, it's probably worth adding that in the main page just for completeness.
Also this is only 7 times more hands to cycle through than standard VP? I would expect an answer way higher than that. Am I interpreting the statement incorrectly?
"An analysis of this game requires looping through 28,587,727,358,190 different hands. To put that in comparison, that is seven times as many as video poker."
As for typos/grammar suggestions, looks pretty good to me.
Only noticed these:
"The problem isn't writing a program to do it, but making it finish withing within months, or years."
"Where I threw my hands up in the air and said "forget it," my sidekick JB wasn't discouraged."
I would probably use "when" instead of "where".
And last but absolutely not least, BRAVO JB!!!
Quote: tringlomaneI'm annoyed that they consider the "element of risk" as the "house edge".
I was able to determine that Washington's definition (at least for this game) of the house edge was the average loss relative to the initial wager (the ante and the bonus). GLI's definition of the house edge (at least for this game) was the average loss relative to the average total wager.
Quote: tringlomaneAlso this is only 7 times more hands to cycle through than standard VP? I would expect an answer way higher than that. Am I interpreting the statement incorrectly?
"An analysis of this game requires looping through 28,587,727,358,190 different hands. To put that in comparison, that is seven times as many as video poker."
It's difficult to say how many combinations there actually are because it depends on how many cards get discarded at each decision. What I did was equalize the possibilities with weight factors.
Quote: socksNice work JB. At some point, the man hours going in (and the tradeoff w/computer hours) becomes just as interesting. Care to comment on whether this was hours of computer time for man-week of human time, or was it more like a day?
I had to run my analyzer several times to get it right. After the first run, I realized that I neglected to allow for folding at the second decision after having drawn at the first decision. I fixed that and ran it again, but the totals didn't add up. I changed what I thought was the problem and ran it again, but the totals were still off. Then I found the real problem, fixed that and ran it again, and all was well. I don't quite know how much man-time I spent on it since I had worked on it off and on for the last couple of weeks, but I can safely say that my computer worked a lot harder on it than I did. After the Vegas paytable was successfully completed, I ran it again for the stingy paytable. And then when that finished, I ran it again to have it save strategy information. The strategy didn't take too long to put in writing; I probably spent more time writing the code which identified the strategy.
Quote: tringlomaneI'm annoyed that they consider the "element of risk" as the "house edge".
I agree 100%.
I saw this game at Monte Carlo at the end of March.
Being suspicious of new games that I don't know the strategy for, I opted not to play and just casually observe. It did not seem like a fun game. Everyone who played it seemed depressed.
Quote: Ibeatyouraces14.77% here in Detroit. Absolutely disgusting!!
And that's if you play your cards right. I can't imagine what it is with how bad many people do play.
It seems like that doesn't bode well for the game, right? If the sheep are getting slaughtered, so to speak, the game probably isn't going to generate that much action.
I can see the immediate appeal of the game, though. Very similar to the for-fun version of poker I played with my friends as kids, which was basically 5 card draw. The second draw is interesting. But with those odds, I'm not really interested in playing it at all!
But with a HE of 6,7%, not even with Collusion is beatable, I think
Quote: tringlomaneI'm annoyed that they consider the "element of risk" as the "house edge".
When I have my reports done by them I have them calculate both the HE and the EOR... However, the way they word them is the House Edge per ante wager and House Edge per average bet... The average bet for my game according to them is 1.67 units...
Although, my games the max bet including the side wager is 3 units if the player chooses to play the 2.68% HA, 28.2% hit rate side wager... (who wouldn't place that!)
Quote: jopkeStandard deck reduces to 204087
The standard deck has 134,459 unique hands. This is the number of distinct 5-card hands in this game without jokers.
Next, add the hands with one joker:
4 of a kind (13 of them)
3 of a kind and one card whose suit is in the 3 (13 x 12 = 156 distinct hands; there are 4 suits that can be missing from the 3, and 3 suits for the 1, so each distinct hand has 12 actual hands)
3 of a kind and one card whose suit is not in the 3 (13 x 12 = 156 distinct hands; there are 4 suits that can be missing from the 3, and 1 suit for the 1, so each distinct hand has 4 actual hands)
2 pair, with the suits matching (e.g. As Ah Ks Kh)
2 pair, with one card from each matching suit (As Ah Ks Kc)
2 pair, with no suits matching (As Ah Kc Kd)
2-1-1, with the 1s in the same suit that is also in the pair (As Ah Ks Qs)
2-1-1, with the 1s in the same suit that is not in the pair (As Ah Kc Qc)
2-1-1, with the 1s in different suits, both in the pair (As Ah Ks Qh)
2-1-1, with the 1s in different suits, the higher of which is in the pair (As Ah Ks Qc)
2-1-1, with the 1s in different suits, the lower of which is in the pair (As Ah Kc Qs)
2-1-1, with the 1s in different suits, neither of which is in the pair (As Ah Kc Qd)
No pair - these are broken down by suits:
4 (As Ks Qs Js)
3-1 (As Ks Qs Jh) (the "off" card can be in any of the 4 positions)
2-2 (As Ks Qh Jh) (there are three ways - AABB, ABBA, ans ABAB)
2-1-1 (As Ks Qh Jc) (there are 4 places for the higher single card, and 3 for the lower)
1-1-1-1 (As Kh Qc Jd)
Finally, add the hands with two jokers:
3 of a kind (13 distinct hands; each has 4 possibilities for the missing suit) (As Ah Ac)
Pair and one, with the one's suit in the pair (As Ah Ks)
Pair and one, with the one's suit not in the pair (As Ah Kc)
No pair, suited (As Ks Qs)
No pair, two suited (there are three places for the unsuited card - As Ks Qh, As Kh Qs, Ah Ks Qs)
No pair, three different suits (As Kh Qc)
The hard part is, for each initial deal, you have to take into account each draw; there's no easy way to describe a "perfect" strategy. You might be able to "fudge" a strategy by assuming discarded cards are put back into the deck; you then calculate the result with optimum play for the second draw (0 or 1 card) for each possible hand, then use those numbers to calculate optimum play (0-3 cards) for each of the original hands.
Quote: ThatDonGuy
The hard part is, for each initial deal, you have to take into account each draw; there's no easy way to describe a "perfect" strategy. You might be able to "fudge" a strategy by assuming discarded cards are put back into the deck; you then calculate the result with optimum play for the second draw (0 or 1 card) for each possible hand, then use those numbers to calculate optimum play (0-3 cards) for each of the original hands.
I think JB did the first draw pretty well personally.
And the 2nd draw strategy is pretty much...do you have a chance to win? Call.
I literally don't think I ever saw one person playing this at Monte Carlo, compared to the last trip where there always seemed to be a large group of suckers trying it out.
This time it was always just a bored dealer standing watch over an empty $5 table.
In Double Draw, one of the Wild Royal/Natural Royal/Five Aces happens once every 1300 hands. Those three events contribute 9.5% to the RTP. The overall HE starts off at 6.7%, so you are looking at playing against a 16.2% HE until you hit one of those hands. My guess is the pace of this game with the two drawing opportunities for each player is about 35-40 hands per hour per player. Even if you use 40 HPH, it will take a player over 30 hours of play to ever see one of these three events. In the meantime, they will be getting whacked at a 16%+ HE.
The Monte Carlo is one install that represents a very small sample size. I think Babs had mentioned there were two tables at a property near her in FL that were getting lots of action. As with every new game concept, trial time and results will be determine the verdict. I do think a draw poker concept is a good one and will find a niche in the live table game pit at some point.
ps. I just found this forum and am very thankful i did. Great work JB
Quote: BobbyMacI noticed the strategy has "joker+Ace" ranked above "4 to a Flush having one joker and one Ace". If this is accurate then one would never hold "4 to a Flush having one joker and one Ace" for the first draw. Im also confused as to why TJQA is better than 7Ts but 36s is better than A235. I realize many of these hands run very close and change with straight/flush penalties. So given 36s,A25 what is the best hold? How about 47s,68Q or KJs,AT3? The expected value of first draw hands would be a great addition as well if you have them.
ps. I just found this forum and am very thankful i did. Great work JB
The strategy is somewhat inaccurate because it is based solely on the average return of each play in hands where that play was the best play; it doesn't take into account hands where it exists but was not the best play, because that would have been an enormous undertaking.
So for example, in all of the hands where Joker+Ace was the best play, the EV is 2.01265; in all of the hands where 4 to a Flush including a Joker+Ace was the best play, the EV is 1.98682 or lower. Since I sorted the strategy list by EV, this puts Joker+Ace higher than 4 to a Flush w/Joker+Ace, but in reality, Joker+Ace is the better play only if you don't also have 4 to a Flush.
With 36 suited vs. unsuited A235 in the same hand, such as 3♣ 6♣ 2♦ 5♥ A♠, holding A235 is the better play.
Quote: BobbyMacI noticed the strategy has "joker+Ace" ranked above "4 to a Flush having one joker and one Ace". If this is accurate then one would never hold "4 to a Flush having one joker and one Ace" for the first draw. Im also confused as to why TJQA is better than 7Ts but 36s is better than A235. I realize many of these hands run very close and change with straight/flush penalties. So given 36s,A25 what is the best hold? How about 47s,68Q or KJs,AT3? The expected value of first draw hands would be a great addition as well if you have them.
ps. I just found this forum and am very thankful i did. Great work JB
Welcome to the forum, BobbyMac! My understanding (having played this game, not as a mathematician) of the joker+A hold is (and the rest, though I think this is the best example), you have to take into account the paytables/return on the holds, not just the likelihood of filling them. There are 4 aces +2 jokers, and 5 aces pays extremely well compared to a flush, even though it's harder to fill. So the strategy is, in the long run, you will make more money holding for the 5 aces (and lesser hands that can result from holding that way, because you're drawing 3, then 1 if you need it), than holding for a flush, (assuming you don't have a straight flush draw, where you're drawing 1, then 1 if you need it), because even if you make the flush more often, it's the only hand you're drawing to, and it's one of the lower-paying ones.
EDIT: answered the same time as JB, but letting it stand anyway. Thanks, JB!
Anyways, here are some hand ranks for everyone 2ND DRAW...
Values are based on a 4 unit bet
8.1) WW+A2s
7.1) WW+8Ts
6.0) FLUSH
5.6) W+678s
5.2) WW+KTs
5.0) STRAIGHT
3.7) WW+59s
3.6) WW+55
3.3) W+AKQs
-.32) W+A56s
-.88) W+567
-1.6) W+679
-1.7) W+479s
-2.8) PAIR
-2.9) STR4i
Quote: JBThe strategy is somewhat inaccurate because it is based solely on the average return of each play in hands where that play was the best play; it doesn't take into account hands where it exists but was not the best play, because that would have been an enormous undertaking.
So for example, in all of the hands where Joker+Ace was the best play, the EV is 2.01265; in all of the hands where 4 to a Flush including a Joker+Ace was the best play, the EV is 1.98682 or lower. Since I sorted the strategy list by EV, this puts Joker+Ace higher than 4 to a Flush w/Joker+Ace, but in reality, Joker+Ace is the better play only if you don't also have 4 to a Flush.
Darn it, JB, now you've got me confused. If Joker+Ace has an EV of 2.01265, and the best 4 to a flush including those cards is 1.98682, what is the element that counteracts this and makes 4TAF the better play?
Quote: beachbumbabsDarn it, JB, now you've got me confused. If Joker+Ace has an EV of 2.01265, and the best 4 to a flush including those cards is 1.98682, what is the element that counteracts this and makes 4TAF the better play?
I will try to explain it better. When my analyzer outputted the strategy results, it listed the best play and its EV for each possible starting hand. From that list, I averaged the EV's of each type of "best play" made.
The end result is that the average EV of Joker+Ace is 2.01265, because it only considers hands where Joker+Ace was the best play. In other words, the output did not list Joker+Ace as the best play in hands where 4 to a Flush was the best play, therefore the value of Joker+Ace in 4-to-a-Flush hands are excluded from the Joker+Ace average.
In generic terms, the EV for "Play A" does not take into consideration the value of "Play A" in hands where "Play B" is the better play. Ideally it should, in terms of developing the strategy, but for Double Draw Poker this would have been a huge undertaking. So the strategy just shows the plays made sorted by their EV. The result is that there are some anomalies such as the Joker+Ace vs. 4-to-a-Flush scenario.
The same situation occurs when developing video poker strategies, but I spent a great deal of time making the VP strategy generator on WizardOfOdds.com take this into consideration, so that the resulting strategies are as accurate as possible.
Quote: tringlomane...
And the 2nd draw strategy is pretty much...do you have a chance to win? Call.
It seems to me that this isn't quite right. For example, one can show that you should fold an inside straight draw if you've already discarded one of your "outs" from the deck. Here's a way this could happen in the game: You are dealt 4c, 5h, 6h, 7h, 9d. On the first draw, optimal play is to keep the 5,6,7 suited and toss the 4 and 9. Then you draw 2d, 3d. For the second draw, the only possible play is to hold either 2,3, 5, 6 or 3, 5, 6, 7 and try for the inside straight. But since you've tossed one of your fours, then you have 5 ways of completing the straight (three fours + two joker), with 47 cards remaining in the deck. The EV of the second draw is thus
5 * (5/47) + (-4) * (42/47) = -3.04. But the EV of folding is -3, and so this hand should be folded.
The situation above is probably rare, but there might be others where folding is right. Does anyone know the percentage of 2nd-draw situations in which there's a positive probability of winning that instead should be folded, for a single player at the table playing all decisions optimally?
If it helps, the Bally rep at the trade show I first played this said a little more than 97% of hands should stay in for the first draw, and a little more than 92% should stay in for the 2nd.
For each of the 58 hands, the error if you didn't fold was less than 0.04 units and was typically around 0.02 units. Since these are so rare, a strategy of always playing a hand in which you have a chance of winning will be very nearly optimal for the 2nd draw.
