Steal Equity in No-limit Hold'em
The following is from my current poker log. All comments welcome.
****************
I am math challenged. While at the cottage, to help in preparation of the “Manifesto†I asked a couple of simple question of the NOT so math challenged Brian Alspach. Armed with his “Idiot’s Guide to Probability†(my name, not Brian’s) I have been working through a series of calculations to help me get a better understanding of the efficacy of stealing in no-limit hold’em.
Basically, I have created three Calling Ranges – Tight (about 6% of hands) Average (about 13%) and Loose (about 50%). Then, I calculate the pot equity for every single hand against these ranges. Then, it becomes fairly simple to calculate what pot equity you need to have to make an all-in move of 10 x big-blind profitable. Once you know the pot equity required then you know the EXACT range of hands that will show a profit moving in. I am calculating this for one, two, three, and four opponents. So, I am looking at stealing from the small-blind, button, cutoff, and one before the cutoff.
I was concerned that one might have to differentiate against getting called by one, or more than one caller. But, I had an epiphany. Getting called is bad (with a hand like 9-7o for instance), but it’s worse to get called by only one player than two! For instance, against one caller from my tight range, 9-7o has 26.682% pot equity. Against two callers, both from the tight range, 9-7o has 18.739% pot equity.
What I am interested in calculating is the EV of every single hand assuming that you move all-in with 10 x BB. Remember, I am math challenged so I do this in an intuitive way, rather than a more clever way that math wonks might use.
Assume blinds of 1 and 2. I am the small blind. I have a stack of 20 after posting. I move all in with 9-7o. I will be called by a tight range (6.2% of hands). This means:
(1) I will win the blinds 93.8% of the time.
(2) 6.2% of the time I will be called.
(3) Of those 6.2% of occasions, I will win 26.682% or 1.654% of total hands. The other 4.546% of hands I will lose 10.
So… in 100 trials.
93.8 * 3 = A total win of 281.4
4.456 * - 20 = -89.12
1.654 * 23 = 38.042
Total = 230.322
Divide by the 100 trials and you get a profit of 2.30322 per attempt.
This means that EVERY time you move in with your 9-7o you will make 2.3 chips – a profitable play, to be sure.
Now, what happens if you make this move on the button? Now there are two potential callers.
The combinations are as follows:
Both fold = 87.98%
SB calls and BB folds = 5.8% of these I will win 26.682%
SB folds and BB calls = 5.8% of these I will win 26.682%
Both call = 0.42% of these I will win 18.739%
87.98 * 3 = 263.95
3.095 (one caller, I win) * 23 = 71.185
8.505 (one caller, I lose) * -20 = -170.100
0.079 (two callers I win) * 43 = 3.397
0.341 (two callers, I lose) * -20 = -6.82
Total = 161.812
Divide by 100 trials and you get a profit of 1.61812 per attempt. Not as good as being from the big blind, but still profitable.
What is, to simplify, we simply do % chance of getting called, at all, and the pretend it’s only one caller.
87.98 * 3 = 263.95
Called 12.027 of the time, against one caller I will win 26.682. So…
3.209 * 23 (I win) = 73.808
8.818 * -20 (I lose) = -176.36
Total = 161.398
Divide by 100 trials and you get a profit of 1.61398 per attempt. This is LESS than factoring in two callers. And, this makes life easy since we can do it this way and save a couple of steps. Again, the reason for this is that getting called by BOTH players is not as big a disaster as getting called by only of them. Your odds of winning go down slightly, but the amount you win goes up.
Phew.
So… calculate what percentage pot equity is required to make the all-in steal profitable and you will know EXACTLY which hands to move in with (provided, of course, that your estimated calling range is correct).
So, this is what I am working my way through.
Every single starting hand (all 169) v. three sample ranges.
And, although this is being done as “all-in†move, if you believe that you can outplay opponents post-flop it will also give a HUGE clue as to the range of hands that one should be stealing with. My guess is that even though I tend to steal a lot, it’s not enough. And, my guess is that I do not defend nearly enough.
Time will tell…
****************
I am math challenged. While at the cottage, to help in preparation of the “Manifesto†I asked a couple of simple question of the NOT so math challenged Brian Alspach. Armed with his “Idiot’s Guide to Probability†(my name, not Brian’s) I have been working through a series of calculations to help me get a better understanding of the efficacy of stealing in no-limit hold’em.
Basically, I have created three Calling Ranges – Tight (about 6% of hands) Average (about 13%) and Loose (about 50%). Then, I calculate the pot equity for every single hand against these ranges. Then, it becomes fairly simple to calculate what pot equity you need to have to make an all-in move of 10 x big-blind profitable. Once you know the pot equity required then you know the EXACT range of hands that will show a profit moving in. I am calculating this for one, two, three, and four opponents. So, I am looking at stealing from the small-blind, button, cutoff, and one before the cutoff.
I was concerned that one might have to differentiate against getting called by one, or more than one caller. But, I had an epiphany. Getting called is bad (with a hand like 9-7o for instance), but it’s worse to get called by only one player than two! For instance, against one caller from my tight range, 9-7o has 26.682% pot equity. Against two callers, both from the tight range, 9-7o has 18.739% pot equity.
What I am interested in calculating is the EV of every single hand assuming that you move all-in with 10 x BB. Remember, I am math challenged so I do this in an intuitive way, rather than a more clever way that math wonks might use.
Assume blinds of 1 and 2. I am the small blind. I have a stack of 20 after posting. I move all in with 9-7o. I will be called by a tight range (6.2% of hands). This means:
(1) I will win the blinds 93.8% of the time.
(2) 6.2% of the time I will be called.
(3) Of those 6.2% of occasions, I will win 26.682% or 1.654% of total hands. The other 4.546% of hands I will lose 10.
So… in 100 trials.
93.8 * 3 = A total win of 281.4
4.456 * - 20 = -89.12
1.654 * 23 = 38.042
Total = 230.322
Divide by the 100 trials and you get a profit of 2.30322 per attempt.
This means that EVERY time you move in with your 9-7o you will make 2.3 chips – a profitable play, to be sure.
Now, what happens if you make this move on the button? Now there are two potential callers.
The combinations are as follows:
Both fold = 87.98%
SB calls and BB folds = 5.8% of these I will win 26.682%
SB folds and BB calls = 5.8% of these I will win 26.682%
Both call = 0.42% of these I will win 18.739%
87.98 * 3 = 263.95
3.095 (one caller, I win) * 23 = 71.185
8.505 (one caller, I lose) * -20 = -170.100
0.079 (two callers I win) * 43 = 3.397
0.341 (two callers, I lose) * -20 = -6.82
Total = 161.812
Divide by 100 trials and you get a profit of 1.61812 per attempt. Not as good as being from the big blind, but still profitable.
What is, to simplify, we simply do % chance of getting called, at all, and the pretend it’s only one caller.
87.98 * 3 = 263.95
Called 12.027 of the time, against one caller I will win 26.682. So…
3.209 * 23 (I win) = 73.808
8.818 * -20 (I lose) = -176.36
Total = 161.398
Divide by 100 trials and you get a profit of 1.61398 per attempt. This is LESS than factoring in two callers. And, this makes life easy since we can do it this way and save a couple of steps. Again, the reason for this is that getting called by BOTH players is not as big a disaster as getting called by only of them. Your odds of winning go down slightly, but the amount you win goes up.
Phew.
So… calculate what percentage pot equity is required to make the all-in steal profitable and you will know EXACTLY which hands to move in with (provided, of course, that your estimated calling range is correct).
So, this is what I am working my way through.
Every single starting hand (all 169) v. three sample ranges.
And, although this is being done as “all-in†move, if you believe that you can outplay opponents post-flop it will also give a HUGE clue as to the range of hands that one should be stealing with. My guess is that even though I tend to steal a lot, it’s not enough. And, my guess is that I do not defend nearly enough.
Time will tell…
Comments
There are a couple of factors that are pretty hard to account for.
* If you are called by someone with others still remaining to act, does their likelihood of also calling not drop?
* Once people realize you will steal with almost anything, won't this increase their likelihood of calling?
* Do you need to factor in the size of the stacks remaining? A huge stack who knows what you are doing will look you up. A small stack who can't afford to give up any more blinds may do the same.
I'd say put it all in a spreadsheet then you can easily modify your assumptions to see what happens.
I'm curious why you'd pick 10xBB? Is this the new black? I'd figure you've still got plenty of play left in your stack. Raising to 3xBB presents a higher risk of being called but doesn't give you an 'all or none' result for the hand. You definitely have to make a move before your stack loses its ability to threaten. I'm wondering if a slightly lower xBB would be as effective? Then there's a question of what to do if you are slightly over that level? Oy!
I guess at this level of play, you are moving away from Sklansky-style approach and more into Harrington. He has a very nice write-up on Structured Hand Analysis in his latest volume that discusses a lot of what you are investigating. It is a lot of work but I'm sure the knowledge gained is well worth it. This line of thinking has really opened my eyes. Personally, I'm hoping for the movie to come out soon...
I find the conundrum that getting called by more than 1 opponent increases your EV quite compelling. I guess the risk increases logarithmically while the reward increases linearly. Or something like that. Still not something I'd look forward to anyway.
I'd be very interested to review your work when it is ready to share.
(3) Of those 6.2% of occasions, I will win 26.682% or 1.654% of total hands. The other 4.546% of hands I will lose 10.
You will lose 20.
Yes, it does. But, this does not change the baseline "percentage that I will be called at all?" which is the only thing that really matters.
Yes you do need to factor in these things. And, in combat, what I do is start with my baseline "Under normal circumstances I thing this player is tight (6.2%). But, he has a BIG stack and I have been stealing a lot. So, I think he calling range is substantially increased (maye to 13%)." Therefore I can move my steal range up or down accordingly.
The basic relationship remains -- percentage chance of getting called (which is a factor of the cards that you will get called with) v. your specific hand.
Calculate this for a few different situations and you can interpolate fairly accurately no the fly.
I don't generally move in with 10 x BB even though most pundits reccomend it. But, is a typical number and you have to pick some number. Most importantly, perhaps, is that it is the number at which a calling range of only 6.2% of hands seems reasonable. Get down to 5 BB and you are going to get called A LOT.
Win one set of blinds and you extend your life by ten hands. That's a lot more in which to hope to fluke off a premium and double up.
This was an epiphany to me. You are HOPING that if the first guy calls, the second guy does too. Kewl.
I was wondering why the 9 7 off. Would this calculation be better with the average hand, J 7 off. This would be a better calc IMO as some times you will have a better hand and your opponents will put you on a steal, and you may have a real hand.
Rob.
which would include J7o, or whatever other hand you might be interested in.
Obviously stealing with J7o will be more profitable than 97o.1 Stealing with AA will be wildly more profitable than stealing with either of the other two hands.
I think the exercise is best looked at in terms of individual hands. That is, for each individual hand (against each individual player type), is stealing more profitable than folding; and if so, by how much?
ScottyZ
1Okay, this is not totally obvious. It depends on how much the 97o can make up for its lack of high card strength with its connectedness and lower probability of being dominated by a hand from the opponent's calling range. In fact, my guess is that the 97o actually does perform better than J7o against the Tight opponent.