Some Triple Draw thoughts
Aug. 3rd, 2005 02:48 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
peterbirksAn interesting problem cropped up on a blog the other day, relating to Triple-Draw.
A number of assumptions are made here to simplify the mathematics (which remain complicated enough). They do not invalidate the conclusion (such as there is one), although one might argue about specific numbers.
This is the scenario. You pick up KQ732 on the button in a 5/10 game. UTG limps, two folds, you limp. SB folds and BB checks. $17 in pot.
Round 1: UTG draws 2, you draw 2, BB draws three. You pick up K7432. UTG checks. You bet. BB folds, UTG calls. $26 in pot.
Round 2: UTG draws 2, you draw 1. You get J7432. UTG checks, you bet, he calls. $46 in pot.
Round 3: UTG draws 1. What do you do?
Here’s where some assumptions come in. Given that UTG has checked and called, we’ll assume he is drawing from a smooth nine down to a smooth eight. That gives him 12 outs from 40 unknown cards. Given the lack of raising, we can make his draw slightly better than that. Say, 12 outs from 36 unknown cards.
You also have 12 outs from 36 unknown cards. So, should you draw?
The “tree” here is (if you play your hands as they merit)
You draw, you hit, he checks, you bet, he calls and loses: +55: 12%. = +6.60
You draw, you hit, he bets, you raise, he calls and loses: +65: 1%: =+0.65
You draw, you hit, he checks, you bet, he folds. +46: 20%: = +9.20
You draw, you miss, he checks, you bet, he calls and wins -10: 0%: = 0
You draw, you miss, he bets, you fold. 0: 21% = 0
You draw, you miss, he bets, you raise, he calls and wins -20: 0% = 0
You draw, you miss, he bets, you raise, he folds +55: 0% =0
You draw, you miss, he checks, you check, you lose: 0. 23%, = 0
You draw, you miss, he checks, you check, you win: +46: 23%, = +11.0
Total EV for drawing: +$27.45
You stand pat, he bets, you call and lose: -10 0%
You stand pat, he bets, you call and win: +55 0%
You stand pat, he bets, you fold: 0 33% =0
You stand pat, he bets, you raise, he calls and wins: -20 0%
You stand pat, he bets, you raise, he calls and loses: +65 0%
You stand pat, he checks, you check and win: +46 60% =27.6
You stand pat, he checks, you check, he wins: 0 7% 0
You stand pat, he checks, you bet, he folds: +46 0%
You stand pat, he checks, you bet, he calls and wins: -10 0%
You stand pat, he checks, you bet, he raises, you call and lose: -20 0% = 0
You stand pat, he checks, you bet, he raises, you call and win: +65 0% = 0
You stand pat, he checks, you bet, he raises, you fold: -10 0% =0
Total EV for standing pat: +$27.6
This is the “simple scenario”, where you both play according to the strength of your hand. Except, of course, people don’t (which is just one of the reasons for position being so important in TD). This most applies in scenario 2, where you have a distinct advantage over your opponent.
I posited in the relevant thread that there was an argument for ALWAYS standing pat, and ALWAYS betting if opponent checks.
Matt Matros pointed out that you only need to stand pat sometimes, and only need to bet sometimes, rather than do this always. I accepted this point, but it set me wondering. How often should you stand pat, and how often should you bet when you stand pat and your opponent checks?
What we have here is a misleading scenario, in that when the above sequences of betting take place on the turn, you will not always have Jack-to-a-perfect when you are willing to play. In fact, you will normally not have such a hand. Sometimes you will have a 10, sometimes a nine, sometimes an eight and sometimes a seven.
TD is unlike ordinary draw in that there isn’t much “either you hit it or not” (as when you draw to a straight or a flush in ordinary draw). There are margins. So you have to assign a likelihood that your opponent will call to a range of hands.
The easiest answer here is that what you do “depends on your opponent” and, indeed, there is a hell of a lot of this in TD. Similarly, what he does will depend on his analysis of you. But what DEFAULT percentages do you apply?
Well, if we take it that the stand pat move has a better “surface” return”, one could argue that this is the best default tactic. Ahh, but wait! Since opponent will miss 67% of the time, you can improve your percentage by standing pat and always betting, because opponent will pass a 10 high and perhaps a bad nine. This is the “he checks, you check and lose” scenario. This boosts the EV of standing pat to $37.8.
The problem here is that your opponent may shift towards calling with those tens and nines. Then your EV drops dramatically, not just back to $27.6, but further, down to +$20. All of a sudden the play of drawing looks better, as does the play of checking behind your opponent!
So, what percentage of times should you bet the Jack to make it irrelevant whether or not your opponent folds, and to keep your EV above $27.6?
And what percentage of times do you draw to the jack, rather than stand pat?
Damned if I know. But I’ll tell you what I like:
70% pat
30% draw
If pat, bet to a check 40%, check to a check 60%
If pat call a bet 70%, fold to a bet 30%
If draw and miss, bet to a check with Jack or better (40%?)
If draw and miss, call a bet with Queen or better. (55%?)
That gives a total probability tree (assuming opponent bets half the time you stand pat):
pat/bet to check: 14%
pat/check to check: 21%
pat/call the bet: 24%
pat/fold to bet: 9%
draw/bet to check: 6%
draw/check to check: 10%
draw/call the bet: 9%
draw/ fold to bet 6%
So, in answer to the question I set myself, I see my total likelihood of standing pat and betting to a check as 28% of scenarios where my opponent dutifully checks, and 14% of all scenarios where I have the J-7-4-3-2
Now I ought to go through all these percentages, work out the EV, and see how it pans out. Then I ought to see how it varies with different percentages of my opponent betting over checking. Alternatively, I could go and play some cards instead.
A number of assumptions are made here to simplify the mathematics (which remain complicated enough). They do not invalidate the conclusion (such as there is one), although one might argue about specific numbers.
This is the scenario. You pick up KQ732 on the button in a 5/10 game. UTG limps, two folds, you limp. SB folds and BB checks. $17 in pot.
Round 1: UTG draws 2, you draw 2, BB draws three. You pick up K7432. UTG checks. You bet. BB folds, UTG calls. $26 in pot.
Round 2: UTG draws 2, you draw 1. You get J7432. UTG checks, you bet, he calls. $46 in pot.
Round 3: UTG draws 1. What do you do?
Here’s where some assumptions come in. Given that UTG has checked and called, we’ll assume he is drawing from a smooth nine down to a smooth eight. That gives him 12 outs from 40 unknown cards. Given the lack of raising, we can make his draw slightly better than that. Say, 12 outs from 36 unknown cards.
You also have 12 outs from 36 unknown cards. So, should you draw?
The “tree” here is (if you play your hands as they merit)
You draw, you hit, he checks, you bet, he calls and loses: +55: 12%. = +6.60
You draw, you hit, he bets, you raise, he calls and loses: +65: 1%: =+0.65
You draw, you hit, he checks, you bet, he folds. +46: 20%: = +9.20
You draw, you miss, he checks, you bet, he calls and wins -10: 0%: = 0
You draw, you miss, he bets, you fold. 0: 21% = 0
You draw, you miss, he bets, you raise, he calls and wins -20: 0% = 0
You draw, you miss, he bets, you raise, he folds +55: 0% =0
You draw, you miss, he checks, you check, you lose: 0. 23%, = 0
You draw, you miss, he checks, you check, you win: +46: 23%, = +11.0
Total EV for drawing: +$27.45
You stand pat, he bets, you call and lose: -10 0%
You stand pat, he bets, you call and win: +55 0%
You stand pat, he bets, you fold: 0 33% =0
You stand pat, he bets, you raise, he calls and wins: -20 0%
You stand pat, he bets, you raise, he calls and loses: +65 0%
You stand pat, he checks, you check and win: +46 60% =27.6
You stand pat, he checks, you check, he wins: 0 7% 0
You stand pat, he checks, you bet, he folds: +46 0%
You stand pat, he checks, you bet, he calls and wins: -10 0%
You stand pat, he checks, you bet, he raises, you call and lose: -20 0% = 0
You stand pat, he checks, you bet, he raises, you call and win: +65 0% = 0
You stand pat, he checks, you bet, he raises, you fold: -10 0% =0
Total EV for standing pat: +$27.6
This is the “simple scenario”, where you both play according to the strength of your hand. Except, of course, people don’t (which is just one of the reasons for position being so important in TD). This most applies in scenario 2, where you have a distinct advantage over your opponent.
I posited in the relevant thread that there was an argument for ALWAYS standing pat, and ALWAYS betting if opponent checks.
Matt Matros pointed out that you only need to stand pat sometimes, and only need to bet sometimes, rather than do this always. I accepted this point, but it set me wondering. How often should you stand pat, and how often should you bet when you stand pat and your opponent checks?
What we have here is a misleading scenario, in that when the above sequences of betting take place on the turn, you will not always have Jack-to-a-perfect when you are willing to play. In fact, you will normally not have such a hand. Sometimes you will have a 10, sometimes a nine, sometimes an eight and sometimes a seven.
TD is unlike ordinary draw in that there isn’t much “either you hit it or not” (as when you draw to a straight or a flush in ordinary draw). There are margins. So you have to assign a likelihood that your opponent will call to a range of hands.
The easiest answer here is that what you do “depends on your opponent” and, indeed, there is a hell of a lot of this in TD. Similarly, what he does will depend on his analysis of you. But what DEFAULT percentages do you apply?
Well, if we take it that the stand pat move has a better “surface” return”, one could argue that this is the best default tactic. Ahh, but wait! Since opponent will miss 67% of the time, you can improve your percentage by standing pat and always betting, because opponent will pass a 10 high and perhaps a bad nine. This is the “he checks, you check and lose” scenario. This boosts the EV of standing pat to $37.8.
The problem here is that your opponent may shift towards calling with those tens and nines. Then your EV drops dramatically, not just back to $27.6, but further, down to +$20. All of a sudden the play of drawing looks better, as does the play of checking behind your opponent!
So, what percentage of times should you bet the Jack to make it irrelevant whether or not your opponent folds, and to keep your EV above $27.6?
And what percentage of times do you draw to the jack, rather than stand pat?
Damned if I know. But I’ll tell you what I like:
70% pat
30% draw
If pat, bet to a check 40%, check to a check 60%
If pat call a bet 70%, fold to a bet 30%
If draw and miss, bet to a check with Jack or better (40%?)
If draw and miss, call a bet with Queen or better. (55%?)
That gives a total probability tree (assuming opponent bets half the time you stand pat):
pat/bet to check: 14%
pat/check to check: 21%
pat/call the bet: 24%
pat/fold to bet: 9%
draw/bet to check: 6%
draw/check to check: 10%
draw/call the bet: 9%
draw/ fold to bet 6%
So, in answer to the question I set myself, I see my total likelihood of standing pat and betting to a check as 28% of scenarios where my opponent dutifully checks, and 14% of all scenarios where I have the J-7-4-3-2
Now I ought to go through all these percentages, work out the EV, and see how it pans out. Then I ought to see how it varies with different percentages of my opponent betting over checking. Alternatively, I could go and play some cards instead.
Triple Draw
Date: 2005-08-04 03:35 pm (UTC)Just so you know, I am the poster BluffTHIS! on 2+2 and also on Bid Dave's blog. Regarding this TD problem, I would definitely stand pat on a 9 here and consider doing so on a T, but I would not with a J even though you have position. If the opponent draws anything reasonable at all, you will only be able to beat a pair, or A/KQ. You also have a draw to the nuts, which if hit might collect extra bets from an opponent who overplays a #2 or #3 if made. Plus with position, if you get a bad but not hopeless card, you can often get a checked down last round. (If you get a pair or A and are checked to then bluff.)
Keep up the good blogging, you are never boring even when talking about business. And gl and good skill at the tables.
BluffTHIS!
P.S. On the Drudgereport's website today there is reproduced an op-ed by Joan Collins in the Daily Mail bemoaning the state of Britain today. I would be interested in your comments on same.
http://drudgereport.com/flash3.htm
Re: Triple Draw
Date: 2005-08-05 01:37 am (UTC)Obviously the Jack scenario is chosen because it is provocative. The nine is a no-brainer and I think that the 10 has something like a 90%-stand-pat line.
Your point about the potential extra bets to be made if you hit a perfect and your opponent hits a 2-3-4-6-7 or a 2-3-5-6-7 should not be underestimated, although such an event is a small percentage of the total set.
To say that "you will only be able to beat a pair, A, K or Q" (plus a worse Jack, obviously!) understates the fact that this is precisely what your opponent will have well over 50% of the time.
The key to the positive EV for standing pat (compared to the positive EV for drawing) is how often your opponent will fold a better hand to your bet when you stand pat, and how often he will call with a worse hand.
The mathematics of the situation remain -- a Jack-perfect is a better likely expected hand than you expect to get if you draw. So if somehow we had a VERY LARGE pot, and a similar headtohead situation at the end, standing pat on the Jack makes mathematical sense. The question is, how small does the pot have to get for the draw's implied return ( the chance for betting on the river knowing that you have won ) to become more important than the chance of drawing and missing?
Very little about this is clear cut because the game is still young and the database on how people play isn't there. I still think that standing pat on the Jack a majority of the time is good, although maybe I'd pull it down to 50:50. And I still think that putting in that bet on the river is good. I just like the metagame benefits as well!