Shoving mathematics

[Eric Lynch]

Barry - you and I talked some about this at the Bellagio $1500 event in December. I agreed (and still do) that there are some gross misapplications of mathematics in poker, but wanted to comment some on your specific points.

On points 1-4 - I really disagree with the hand selection criteria you are using, I find it VERY rare to play at tables where everyone is playing Axs UTG and NOT playing 87s or 66. Your hand selection criteria is really heavily weighted towards aces a lot more than I think a typical table is and is therefore amplifying the so called 'bunching' effect. I'm on my laptop at the Borgata so I can't change up the hand selection criteria at all and see how it changes things, but I'm not sure it's all that important honestly. As in most situations where you find yourself with ~20BBs the rest of the table has very dynamic stack sizes in the 10-30 BB ranges that are significantly impacting their hand ranges as well (I am assuming I'm not so out of touch with today's game that we're not opening A2s with 15 BB but might with 30? ditto for 22?) that would make any assumptions beyond saying the big blind's hand may be a little bit stronger than average in these situations a bit reaching.

Really in both our pokerstove simulations (because it completely disregards the other 7 players discards) and in your simulation (because it assumes everyone is opening a similar range weighted towards aces and because it does not take into account how players will significant adjust their ranges because of stack size) there are imperfections.

I think studying this sort of pokerstove calculation has its usefulness in the way Justin already stated:

Here's a common example on the usefulness. Let's say player X is open folding K8o in the SB with 14 BB's in a short-handed cash game. Now DJK shows you can shove it profitably FACE UP with Y stack (where Y > 14). This proves that folding is a mistake. It doesn't show that shoving is always optimal, but it's a start in figuring out what the best play actually is, and corrects a very fundamental mistake that many players make (folding K8 in this spot).

I do agree with Barry that this often becomes very misapplied though. I mean I can prove in pokerstove that shoving AA UTG with 100 BBs is +EV but that doesn't mean I want to actually do it at the table. Ryan and Barry have already discussed in this thread that it's all about finding the MOST +EV play to make at the table that's important, and one factor I think that's very important here is the concept of edge.

Barry's background (I believe) is mainly live tournaments where your edge at the table is often far greater than it is in something like the $100 rebuy or $1k freeze out tournaments online. The bigger your edge is at a given table, the more the concept of passing up thin edges for future edges when your risk/reward is fairly skewed (as it is in this situation where we are probably risking 22 BBs for what probably amounts to a 1-2 BB edge, I haven't seen the calculations and don't have pokerstove on this computer) becomes important.

Dan's background comes more in the tougher $100 rebuy and such online games, where your overall edge over the table is much lower than it is in your typical live event (due to the limited information and typically poorer structures in the online games as well as fields that typically play more 'fundamentally' sound). The lower your overall edge is the more you have to push these thin edges to contribute to your bottom line.

The other usefulness in studying some of these calculations IMHO is there are times for whatever reason (although admittedly for the posters here much less frequently than most players) where we find ourselves with a 15-25 BB stack at a table full of extremely tough players who we may have little to no edge against, or because of our table position we are unable to exploit edges at the table that we would be much more easily able to exploit if we were to shift chips at the table from directly to the left into our stack. In these situations it can be helpful to know how to implement this sort of strategy.


A rough hypothetical example. I'm playing a tournament and have 22 BBs in the middle to late stages of the tournament (let's say 1000 players, 100 paid spots, ~200 left). I'm in the 1 seat and there are some extremely weak/tight players in the 3 and 4 seats. In the 2 seat however, is Sorel Mizzi (using him as an example, there are many players that would work) with a 30-ish BB stack. Sorel knows my game well enough to be using a pretty optimal strategy versus me when I have ~20 BB here, and him sitting between me and the weak players is preventing me from gaining the edge I would easily have if he were not sitting there or significantly shorter than he is right now.

Now, if it is folded to me in the SB and I have K8o and ~20 BBs I'm much more likely to open shove because I am not only exploiting a thin edge, but I am creating a situation that will, when I do get called and get lucky, create new +EV situations that wouldn't otherwise exist (the ability to now exploit the weak players without Sorel's interference). I wrote about this once on 2p2 I think, but it creates a 'compounding' effect when the table situation is right. Basically it's sort of the inverse of passing up a thin edge for future greater edges, in this case it's taking the thin edge because it creates future edges.

This concept doesn't come up all that frequently and I think many players misapply it, but I think for true students of the game knowing some of these situations, even if the actual need to apply it comes up very rarely, is something that is worth studying.

-Rizen

[Barry Greenstein]

Originally Posted by Eric Lynch

On points 1-4 - I really disagree with the hand selection criteria you are using, I find it VERY rare to play at tables where everyone is playing Axs UTG and NOT playing 87s or 66. Your hand selection criteria is really heavily weighted towards aces a lot more than I think a typical table is and is therefore amplifying the so called 'bunching' effect.

Eric,

This program was written for my book in the days when limit hold'em was more popular than no-limit, not only out in the live world, but in the early days on the Internet. I didn't write it to skew the numbers. In limit hold'em, players often throw away their small pairs in the first two spots, but limp with their Ax suiteds.

It would be easy for someone good a programming to adjust my table. (I would do it, but it always takes me too long to set up my environment for programming.) I'm pretty sure it would change to number to between 140 and 150, but the exact number is not important. What is important is that the effect of people folding is significant and these calculations, that people are using like they are exact, are most likely off by a couple of whole numbers.

By the way, I see a lot of my opponents coming out of the hole with Ax unsuited in earlier positions than I play this hand, so the number isn't as off as people are saying anyway.

Barry's background (I believe) is mainly live tournaments where your edge at the table is often far greater than it is in something like the $100 rebuy or $1k freeze out tournaments online. The bigger your edge is at a given table, the more the concept of passing up thin edges for future edges when your risk/reward is fairly skewed (as it is in this situation where we are probably risking 22 BBs for what probably amounts to a 1-2 BB edge, I haven't seen the calculations and don't have pokerstove on this computer) becomes important.

My background is in deeper stacked cash games, and I play my Ax suiteds and all my small pairs. I would never think of making any of these shoves in that environment except in Bobby's room where we usually played NLH with 1/2 ante to small blind ratio.

The other usefulness in studying some of these calculations IMHO is there are times for whatever reason (although admittedly for the posters here much less frequently than most players) where we find ourselves with a 15-25 BB stack at a table full of extremely tough players who we may have little to no edge against, or because of our table position we are unable to exploit edges at the table that we would be much more easily able to exploit if we were to shift chips at the table from directly to the left into our stack. In these situations it can be helpful to know how to implement this sort of strategy.

There is definitely usefulness in studying these things but, with a lot of this generation of players, math miscalculations have become the new religion or superstition. Someone says a number is true and he gets a lot of followers. But Todd's number is different than Dan's and I have heard from people who have used pokerstove or other tools but claim other numbers are correct. And none of them have actually even done the right calculation, because as I said, the bunching effect is real and tournament considerations affect the decisions.

Almost all experienced tournament players have a basic strategy for entering the pot and for shoving with different hands a certain number of blinds deep. But none of us really knows how accurate our system is. Over time, with good empirical results, we assume that our strategy is reasonable. I think mine is, but I certainly see a lot of good players with basic strategies that are clearly different than mine, and I notice the ones who are successful are simply the ones who are the best players in poker judgment situations.

A rough hypothetical example. I'm playing a tournament and have 22 BBs in the middle to late stages of the tournament (let's say 1000 players, 100 paid spots, ~200 left). I'm in the 1 seat and there are some extremely weak/tight players in the 3 and 4 seats. In the 2 seat however, is Sorel Mizzi (using him as an example, there are many players that would work) with a 30-ish BB stack. Sorel knows my game well enough to be using a pretty optimal strategy versus me when I have ~20 BB here, and him sitting between me and the weak players is preventing me from gaining the edge I would easily have if he were not sitting there or significantly shorter than he is right now.

As I said, I shove deeper against good opponents, but more often against good opponents who don't seem to call with higher frequencies.

This concept doesn't come up all that frequently and I think many players misapply it, but I think for true students of the game knowing some of these situations, even if the actual need to apply it comes up very rarely, is something that is worth studying.

It's certainly worth studying, but don't be so confident that your numbers are right. I still think my points are much more important than the number and these numbers for each hand are approximations that people are worshiping as exact.

At this point I have gotten ranges from people claiming to use different programs that show it is marginally +EV for shoving K8 offsuit with anywhere from 12 to 32 BB. (I usually don't completely trust programs I didn't write.) But as I said, I haven't seem anyone who has actually done the important EV calculation for playing against typical passive opponents where I would make a normal open and fold to a shove.


Barry

[Eric Lynch]

Originally Posted by Barry Greenstein

Eric,

This program was written for my book in the days when limit hold'em was more popular than no-limit, not only out in the live world, but in the early days on the Internet. I didn't write it to skew the numbers. In limit hold'em, players often throw away their small pairs in the first two spots, but limp with their Ax suiteds.

It would be easy for someone good a programming to adjust my table. (I would do it, but it always takes me too long to set up my environment for programming.) I'm pretty sure it would change to number to between 140 and 150, but the exact number is not important. What is important is that the effect of people folding is significant and these calculations, that people are using like they are exact, are most likely off by a couple of whole numbers.

By the way, I see a lot of my opponents coming out of the hole with Ax unsuited in earlier positions than I play this hand, so the number isn't as off as people are saying anyway.

I wasn't at all trying to imply that you purposely skewed the numbers or anything, just that I think there are some fairly significant flaws in the method you are using to calculate the so called 'bunching' effect. To be fair, I also agreed with you that there are flaws in only using pokerstove for this calculation as well.

I would agree a lot with your ideas a lot more if we were ~22BB deep early in a tournament when everyone else was a lot deeper, but in the middle to later stages when everyone is 15-40 BBs deep so much of a person's opening range depends on their stack size that the impact it will have on the potential holdings of others is nearly impossible to figure.

Really the only point I was trying to make is that I believe the method you were using as a counterpoint is flawed as well. Of course this is easy for me to do on the internet especially as I'm not really offering a better solution. The only real way to solve these sorts of problems is to have truly accurate ranges for all of the players at the table, which is pretty much impossible to do and even more impossible to do at the table. We can make reasonable assumptions, which I think is what you were attempting to do, but my personal opinion from the experience I have (which is admittedly far less than yours) is that the ranges being used are very inaccurate for a 'typical' table and skew towards Aces far too heavily amplifying the 'bunching' effect.

My background is in deeper stacked cash games, and I play my Ax suiteds and all my small pairs. I would never think of making any of these shoves in that environment except in Bobby's room where we usually played NLH with 1/2 ante to small blind ratio.

My apologies. I never meant to assume what your background was, but I think the point (that you play primarily in a live setting where the information you have at your disposal is far greater) remains. The more information you have at your disposal the greater edge you can create, and the less need to exploit thinner edges when tournament considerations are on the line. In an online setting the information you have is much smaller (not to mention many players are playing multiple tables compounding the information loss) which should equate to a smaller overall edge at the table that would make taking advantage of thinner edges much more necessary in that setting.

There is definitely usefulness in studying these things but, with a lot of this generation of players, math miscalculations have become the new religion or superstition. Someone says a number is true and he gets a lot of followers. But Todd's number is different than Dan's and I have heard from people who have used pokerstove or other tools but claim other numbers are correct. And none of them have actually even done the right calculation, because as I said, the bunching effect is real and tournament considerations affect the decisions.

I absolutely agree 100% about math miscalculations. We talked about this some at the Bellagio. I think you summed it up very nicely below:

Almost all experienced tournament players have a basic strategy for entering the pot and for shoving with different hands a certain number of blinds deep. But none of us really knows how accurate our system is. Over time, with good empirical results, we assume that our strategy is reasonable. I think mine is, but I certainly see a lot of good players with basic strategies that are clearly different than mine, and I notice the ones who are successful are simply the ones who are the best players in poker judgment situations.

It's certainly worth studying, but don't be so confident that your numbers are right. I still think my points are much more important than the number and these numbers for each hand are approximations that people are worshiping as exact.

I would whole heartedly agree than anyone worshiping these numbers as exact is making a mistake, but I do think that running some of these numbers and using them as starting points for ideas and discussion are healthy for students of the game. Poker situations always trump pure numbers (something I have often stressed in my training videos), so we're in total agreement there, although I have some disagreements with you on how much of an impact 'bunching' has on decision making at the table, but that's probably a discussion for a completely separate thread.

At this point I have gotten ranges from people claiming to use different programs that show it is marginally +EV for shoving K8 offsuit with anywhere from 12 to 32 BB. (I usually don't completely trust programs I didn't write.) But as I said, I haven't seem anyone who has actually done the important EV calculation for playing against typical passive opponents where I would make a normal open and fold to a shove.

We're completely in agreement here, and I really think we're largely in agreement overall Barry. I mainly replied in this thread because as someone who learned the game online I used programs such as pokerstove a lot during my early learning period I have seen firsthand the usefulness of exercises such as the one Dan is doing. I do agree that there are many that worship at the alter of mathematics who are likely miscalculating or not even using the numbers to solve for the right problems, but programs such as pokerstove when used as a tool (along with good poker judgement) can be a healthy part of the development of one's game.

-Rizen

[Barry Greenstein]

Now that we’ve had time to digest the discussion, it might be helpful if we solve the related math problems. Let’s analyze the situation where we have K8 and we are playing for effectively 20 big blinds when everyone folds to us in the small blind. If we use blinds of 1000 and 2000 and the ante 200 in an eight-handed game, this means we have a starting stack of 40,200 in chips.

We are deciding between option 1, shoving all-in, versus option 2, raising to 6000 and folding to a shove by the big blind. Let’s assume that the big blind will call our shove if he has a better hand than ours or if we choose option 2, he will shove on us with better hands if we make the three times the big blind raise.

Let’s forget about the bunching effect that was discussed because it complicates the calculations and we’re just trying to get an approximate comparison of the two plays.

First of all, what hands are better than ours? All pairs are better, all hands with an Ace in it are better, and all hands with a King and a higher kicker are better. We can throw in K8 suited also.

We need to figure out what percentage of hands our opponent will be playing. After we take our K8 out of the deck there are 50 cards left, which means there are (50 x 49)/2 = 1225 possible hands for the big blind. There are 6 ways to get each pair except for Kings or Eights where there are 3 ways. There are 16 ways to get each Ax, but there are only 12 ways if the kicker is a King or an Eight. There are 12 ways to get each King with kickers above an Eight, and there are 2 K8 suiteds. The total is 66 + 6 + 160 + 24 + 48 + 2 = 306 hands that are better than ours, and 916 hands that are the same or worse.

Option 1. We shove for a total of 40,000.

Our opponent folds 916/1225, which is just under 75% of the time, and when he calls the other 25% of the time we will have around 33.8% equity as we see below.

Text results appended to pokerstove.txt

101,734,626 games 54.525 secs 1,865,834 games/sec

Board:
Dead:

------- equity ----win ----tie --- pots won -- pots tied
Hand 0: 33.807% 32.30% 01.51% 32855665 1538301.50 { K8o }
Hand 1: 66.193% 64.68% 01.51% 65803189 1538305.00 { 22+, A2s+, K8s+, A2o+, K9o+ }


So 75% of the time we win the pot of 4600 and 25% of the time we either win 42,600 or lose 39,000 from the point of decision.

EV(shove) = (3/4) 4600 + (1/4)( (.338)(42,600) – (.662)(39,000))
= 3450 +(1/4)(14399.8 – 25818)
= 3450 – (1/4)(11418.2)
= 3450 – 2854.55
= 595.45

So shoving is better than folding and it gains us around 595.45/39000 or around 1.5% of our stack.

Option 2. Raise to 6000, but fold to a shove.

We still win 75% of the time, but the other 25% of the time we lose 5000.

EV(Raise 3x and fold to shove) = (3/4) 4600 + (1/4) (-5000)
= 3450 – 1250
= 2200

So in our comparison of the two options, raising and then folding to a shove is almost four times more profitable than straight shoving.


Conclusion: These calculations seem to support the arguments in my opening post, and that is why I rarely would shove with King Eight offsuit when I am 20 big blinds deep.

Barry

[Justin Bonomo]

So 75% of the time we win the pot of 4600

Wait what... When we 3x, the big blind folds 75% of the time? Does our opponent know how to breathe?

Oh, you raised my blind? I guess my QJs is no good. I fold.

[Barry Greenstein]

Originally Posted by Justin Bonomo

Wait what... When we 3x, the big blind folds 75% of the time? Does our opponent know how to breathe?

Oh, you raised my blind? I guess my QJs is no good. I fold.

No, of course not. That is why my opening post was explaining that poker considerations override a lot of these calculations. There will be some situations where I will have to play after the flop also.

I wanted to get a initial comparison of what might happen against a passive opponent. Then I will use my poker ability to make adjustments.

Barry

[Justin Bonomo]

I wanted to get a initial comparison of what might happen against a passive opponent.

I'm being a nit here, so no response necessary... but technically, this is the least passive opponent possible (he never calls, fold or shove ONRY).

[Ryan Daut]

yea barry the entire point of shoving in these spots is if we are in a spot that we get shoved on a lot and have a hand that is bad for calling OR if we have a hand that we want to shove to add strength to our jamming range.

i.e. if i have 13bb in the sb i shove queens because i am shoving such a wide range in this spot i dont want people to think my range is 22-88 A2-A9 K2-KJ etc


the reason a shove is close to or more profitable as raise folding is because he is shoving QJs, QTs, JTs, T9s, 98s etc etc and other hands we arent calling if we 3x but get to fold by shoving and is sometimes folding better hands to our shove like K9 K8s A2o A3o etc

[Barry Greenstein]

Originally Posted by Ryan Daut

the reason a shove is close to or more profitable as raise folding is because he is shoving QJs, QTs, JTs, T9s, 98s etc etc and other hands we arent calling if we 3x but get to fold by shoving and is sometimes folding better hands to our shove like K9 K8s A2o A3o etc

Ryan,

I certainly agree with you, but the first calculation we do with these kinds of problems is figure out the EV where the first hand acts and then the other hand has to either fold or shove with both hands face up.

A slight deviation from this policy: I didn't include K8 offsuit in the big blind even though it would call a shove, because when it shoves the small blind won't fold. Of course, there is no fold equity for K8 suited also. And to be more accurate, we have to check to see if there are hands that are underdogs against K8 offsuit, but if they put in the 38,000 against a shove they will get back more from their equity of the total pot of 81600. If the equity is x, we want to know when 81,600x = 38000. We get x = .4730... I think the only hands worse than K8 offsuit but with more than .473 equity are J10 suited in other suits than the K8, which has .4821 equity, and J10 suited in the same suit as the Eight, which has .4772 equity.

These cases where the big blind is an underdog but can profitably call a shove or will be called if it shoves against the three-times raise affect the actual numbers slightly, but not enough to change our conclusions. Each J10 suited in other suits gains about 1333 in chips by calling, but they are each one case in 1225 so that affect for each on the EV is about 1. The K8 offsuits give up 2800 by folding, so not folding each of them decreases the EV for either option by a little more than 2. The total affect on the EV from all these cases is less than 10 and it's all going in the same direction, i.e., decreasing the EV for both options.

Anyway, back to Ryan's complaint: Once we see that we have the two plays available, we factor in the points I initially mentioned and also things like taking the playing ability away from our tough opponents and seeing flops against our weak ones.

Barry

[Eric Lynch]

Just for an alternative calculation that I think represents the situation a little better than playing K8o face up. If we assign our opponent a calling range of:

55+,A2s+,KTs+,A7o+,KQo

to call our shove with, which I think is somewhat close to accurate for a player who has some kind of idea of how we're playing here. That is 16.3% of hands (using pokerstove here, not accounting for any 'bunching')

In this case your calculation would look more like this Barry:

EV(shove) = (.837) 4600 + (.163)( (.32)(42,600) – (.68)(39,000))
= 3850.20 +(.163)(13632 – 26520)
= 3850.20 – (.163)(12888)
= 3850.20 – 2100.75
= 1749.45

Now, let's say if we raise to 3x our opponent will shove over the top of us with the following hands:

22+,A2s+,K5s+,Q8s+,J8s+,T8s+,98s,A2o+,K9o+,QTo+,JT o

That is 32.1% of hands, so the second half of your calculation would look like this:

EV(Raise 3x and fold to shove) = (.679) 4600 + (.321) (-5000)
= 3123.4 – 1605
= 1518.40

Now shoving is more profitable than raise/folding.

This isn't to say that shoving is more profitable than raise/folding here always, I just want to show you how when people's calling ranges become narrower and raising ranges become wider it has a profound impact of the EV of the play.

There are a lot of poker factors here we aren't considering (at some point, our opponent will be shoving enough hands that it's more profitable to raise/call than shove, although with K8o that has to be VERY wide). There's also the consideration that by pushing we will sometimes be out of the tournament, which carries value not measured by the math. We also don't know the true impact (if any) of bunching on these particular ranges.

We also still don't have any clue how it will mess with our EV when the big blind calls.

This is not a post saying shoving is more profitable than calling, I actually think against 95% of opponents 3x raising here is far better than shoving, and even with the 5% shoving might show a higher chip expectancy but our $EV will probably be lower provided there are softer spots at the table to get our chips from plus the fact that the chips we lose here are more valuable than the chips we will gain when we win.

I mainly posted this to show that when dealing with opponents who have a large gap between what they will call a shove with vs what they will shove over a 3x raise with, it can create situations that while K8o may be borderline, shoving KTs against this specific opponent would definitely show a bigger profit than raise folding.

Of course with KTs against this opponent can show that raise/calling is more profitable than shoving:

EV(Raise 3x and call to shove with KTs) = (.679) 4600 + (.321) ((.478)(42,600- (.522)(39000))
= 3123.4 + (.321)(20362.8 - 20358)
= 3123.4 + 1.54
= 3124.94

I haven't seen it as much in the live tournaments I've played in (although it's happening with increasing frequency) but certainly in online tournaments you deal with these sorts of players enough that knowing what kinds of hands play against these types of players then having a rough idea if open shoving, 3x/folding or 3x/calling is best can be helpful, although I will definitely concede that using pokerstove here is just a rough approximation and can give some general ideas, not something that should be taken as the gospel as many factors other than pure chip EV (cEV) impact your bottom line in tournaments.

-Rizen