Shoving mathematics

[Barry Greenstein]

Eric,

I concede that your ranges more accurately depict what will happen with the cards concealed. Either of the options isn't too far apart which means we have to consider the other factors in my opening post. And, as you mentioned, one consideration we haven't factored in mathematically is how we will fare with the betting lead, but out of position, if we only get called. I'm +EV in those situations (it's not a zero sum situation from the point of calculation), but I don't know if it's more +EV than if my opponent would have just folded initially.

Barry

[Barry Greenstein]

When I set up the hand ranges on PokerStove that I compared to K8 offsuit, I had to choose between using the Monte Carlo method (random boards generated from a random number generator) to evaluate, or I could “Enumerate All” of the possible hands and boards. Since there are more than 2 million five card boards for each comparison and there are a few thousand comparisons to do (each K8 offsuit vs the hundreds of hands in the range) that gives us billions of combinations. Evaluating each hold’em hand with each board and making comparisons will probably take more than a hundred instructions at the assembly language level. This means that it will take on the order of a trillion instructions to do a complete enumeration. (Actually, the program probably uses one of the K8 offsuits and then multiplies the results by twelve.)

When I was a programmer, I wouldn’t even think of attempting to run a program that had to execute a trillion instructions, because it would never finish. But apparently with today’s processing power, calculations like these can be done in real time and when I tried it, it seemed to take less than a second. There must be a bug in PokerStove because the complete enumeration yields a very different result than the Monte Carlo method.

Here are the results I get when I choose to enumerate all of the approximately 6 billion comparisons:

------- equity --- win ---- tie ---- pots won ---- pots tied
Hand 0: 34.195% 32.89% 01.31% 2067854244 82156890.00 { K8o }
Hand 1: 65.805% 64.50% 01.31% 4055412264 82156890.00 { 22+, A2s+, K8s+, A2o+, K9o+ }

Assuming the bug is in the Monte Carlo part of the program (since Justin Bonomo told me that he verified the Enumerate All on another program he uses), this changes our calculation for EV(shove) to:

EV(shove) = (3/4) 4600 + ¼( (.342)(42,600) – (.658)(39,000))
= 3450 +( ¼) (14569.2 – 25662)
= 3450 – ( ¼ )(11092.8)
= 3450 – 2773.2
= 676.8

This is not significant as far as the discussions go, but I didn’t want to have the wrong calculation.

The moral of this story is: A trillion just isn’t what it used to be.

Barry

[Todd Terry]

I was aghast at the possibility of there being a bug in PokerStove, so I ran the Monte Carlo with the given ranges quite a few times and always got a result within .01% of 34.19%. Maybe Barry just experienced one of those 50 standard deviations from the mean events, similar to those that occurred about 70 times last year in the financial markets.

[Barry Greenstein]

Originally Posted by Todd Terry

I was aghast at the possibility of there being a bug in PokerStove, so I ran the Monte Carlo with the given ranges quite a few times and always got a result within .01% of 34.19%. Maybe Barry just experienced one of those 50 standard deviations from the mean events, similar to those that occurred about 70 times last year in the financial markets.

I must have a buggy version. My version is 1.21. It never gives a result near the apparently correct one, and it always has a different number of ties for the two hands.

Barry

[Isaac Haxton]

Originally Posted by 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

You really did all this work to prove that if our opponent will either jam over our open or call our jam with exactly every hand that beats us and no others and never flat calls its better to make a small raise and then fold rather than shove our whole stack in? Your assumptions seem picked especially to make shoving as bad as possible and not very realistic.

[Barry Greenstein]

Originally Posted by Isaac Haxton

You really did all this work to prove that if our opponent will either jam over our open or call our jam with exactly every hand that beats us and no others and never flat calls its better to make a small raise and then fold rather than shove our whole stack in? Your assumptions seem picked especially to make shoving as bad as possible and not very realistic.

Normally, you start your analysis by using hands open, and a simplified version of the game: in this case, fold or shove are the only options for our opponent. You try to show that your plays are mathematically reasonable if your opponent plays perfectly against you. Eric Lynch gave more realistic ranges for actual play.

The point of all this is that the EV for shoving and making a normal raise are similar, so we should look at other factors which I claim are the real issues for choosing the correct line of play.

Barry

[Isaac Haxton]

Originally Posted by Barry Greenstein

Normally, you start your analysis by using hands open, and a simplified version of the game: in this case, fold or shove are the only options for our opponent. You try to show that your plays are mathematically reasonable if your opponent plays perfectly against you.

Unless I'm misunderstanding something, your analysis is set up such that raise/fold is always better than shove except in those cases where your hand and your stack size dictate that you are committed to call the jam, in which case shove and raise/call are identical. Its impossible for your analysis to conclude that shove is the better option. Furthermore, your analysis suggests that you should raise/fold KK for stacks of 7BB+ or so. I guess my point is that assuming open hands in a comparison of shove vs raise/fold is a bad assumption and will often lead to conclusions that are very far from any realistic scenario.

[Barry Greenstein]

Isaac,

Your points are certainly worth noting, but K8o is an interesting case where open hands analysis is very practical. Not only is it the highest non-Ace unsuited non-connector, but it may be the unique hand that people play the closest to perfectly against in these shove/raise-fold situations. Most people will required suitedness or connectedness with their Kx to call and the QJ and J10s will have enough equity in the pot to get back to even against K8o.

Barry

[Jerrod Ankenman]

I'm home from the WSOP now, so I figured I'd post a couple of thoughts about this thread.

1) Pokerstove is a valuable tool for calculating EVs for exploitive play, but when trying to do calculation of exploitation-proof strategies I think it is generally not very useful. If you want to do this kind of thing seriously you need tools that are more flexibly automated, which means some programming language. My recommendation is Python, which is fairly easy to use, and for which a fast suite of poker evaluation tools has already been ported.

2) I did some work on the effect of card removal (from folded hands) on jam-or-fold strategy between the blinds a while ago. I found one surprising thing: that the problem of finding the hand distributions of the blinds given that the first n players have folded a specified set of distributions is NP-hard, which means that simulations such as the ones Barry has done are as good a method as any. When I did this, I found that there was very little change to JoF strategies under 10 BB, which is what I was concerned with at the time. If someone really desires it, I can probably dig up the old work and have a look-see at larger stacks.

3) I agree with the following points made throughout the thread, or which I made up:

- If you are going to fold K8 here, you should jam instead.
- You probably should not open-jam many hands at this stack size, and K8 is probably not one of them.
- If you are going to raise a small amount, it may be cleaner to assume your opponent always shoves, because then you can use pokerstove to figure out your equity. However, in real life, your opponent should call a lot and play the pot with 19 BB stacks in position. This is harder to analyze without making up a bunch of assumptions; the clean certainty is removed because there are all these streets and flops and so on.

4) Re: Barry's summary:

Using math to analyze postflop play is hard and pokerstove doesn't help. So he is right that in these situations where there is a choice between a certain mathematical alternative (JAM, because we are 100% sure it's +EV), and a uncertain one (raise to 3x, because we think we are making money from the combination of preflop and postflop play), it is often correct that the uncertain one has higher EV, and so the first fact isn't that useful. So Barry is right in that sense.

But the kind of analysis presented here (you can jam for 22 BB with K8 because it's +EV) isn't exactly the state of the art, and as we keep playing poker, the set of situations which are mathematically "solved" is going to grow until a lot of what Barry calls "poker considerations" are mere inputs or slight adjustments to known strategies. So it's a dangerous idea to think that "judgment and poker considerations" are going to provide a durable, persistent edge for players playing in tough games, just as it was dangerous to think that "fit or fold" was a reasonable way to play shorthanded limit holdem back in the day - even though it probably worked. Since AFAIK all of the mathematical published NL analysis basically concerns preflop play, this state of affairs may last for a little while, but I wouldn't count on it remaining that way.