Shoving mathematics

[Barry Greenstein]

The way it works in poker, once we believe we are making the correct play, it is hard to convince ourselves that anything other than bad luck is the explanation for the times it doesn't work. And when it does work, we have the positive reinforcement to keep playing incorrectly.

I was listening to Jimmy Fricke mention that Dan Kelly figured out that it is +EV to shove with 22 big blinds with K8 offsuit, when in the small blind and everyone has folded. I have played with both of these gentlemen and they are good players, but I take exception to their application of mathematics here.

As is often the case when I read mathematics applied to poker, the problem is often incorrectly solved and it’s the wrong problem anyway. I will post my top ten reasons that I have objections in this case. Hopefully Dan and Jimmy will have some comments and counterpoints in this thread.

Top Ten Problems with these kinds of shoving calculations (some are related, so I actually have 13 items)

1. I haven’t seen Dan Kelly’s calculations, but I assume he used something like pokerstove, and gave the big blind random hands after taking K8 out of the deck. In a nine-handed game, the big blind is stronger, on average, than it would be in a head up situation, because hands actually get stronger as people fold, since hold’em is a game where high cards are correlated to better hand strength.

2. So these calculations are right for when we are playing head up and we have the button, right? Wrong. There is less in the pot in the way of antes if you are only playing heads up. Players with these incorrect shoving and calling ranges often forget that we don't have eight or nine antes in the pot when we play head up.

3. These calculations and plays are dependent on the ante structure. If someone wanted to run a simulation they should use 1/5 ratio for ante to small blind for online tournaments (that's what PokerStars uses for later stages) or 1/4 ratio for live tournaments (typical for big live tournaments).

4. So except for ante considerations, the calculation is right, isn't it? No. As stated in item 1, our opponent's hands will be stronger than random. As I showed with a simulation, you will get pocket Aces around 1 out of 134 times instead of 1 out of 221 times, if everyone folds to you in the blind in a nine-handed game.
http://barrygreenstein.com/aces.txt
http://barrygreenstein.com/out-aces.htm

This is approximately the number you would get dealing with 4 aces in a deck of between 40 or 41 cards without an Ace instead of the normal 50, which people are using for their pokerstove calculations. So we can use this for a pretty reasonable approximation of the situation and figure out that our opponent will have an Ace in the big blind 4/40.5 + (36/40.5)(3/39.5) instead of 4/50 +( 46/50)(3/49) which is approximately .166 versus .136. So your opponent will wind up with at least one Ace in his hand around more than 20% more often.

5. You may want to avoid pushing all your small edges if there are bigger ones available because your opponents are weaker players than you are. Don’t bet your whole tournament on this small edge.

6. These are all chip equity considerations which are fine for side games, but in a tournament, chips are worth more per unit when you are the shorter stack. They are two very different problems when we are the one with 20 BB versus when our opponent is and we have him well covered, not just mathematically but also psychologically. You can shove any two cards against many opponents when they are the shorter stack and be +EV when you have them well covered because people are averse to going busted.

7. So how should you play this hand? Against normal passive opponents, ones that you may be able to shove any two cards profitably with under 20BB and who won't shove against you with unexpected hands if you make your normal raise, I think it is more +EV to make your normal raise and fold to a shove.

8. However, right before the tournament bubble or a large pay jump, you may want to shove as the bigger stack but if you are the shorter stack it may be right to fold.

9. Another benefit to folding is that you are about to get six or seven hands where you don't have to put blinds in and the first few, you will have good position.

10. There are some opponents who will almost always raise if I limp there as the shorter stack. I will make more off them by limping and shoving after they raise.

11. The times you shove when effectively playing for 20BB, you should do it for poker reasons. If you are against a tough aggressive opponent who is on the shorter stack, and one who may shove on us with Ax with a small kicker if we make a normal raise, but will fold this hand out of fear of domination or facing a middle pair, then we can profitably shove because of the hands we will bluff out.

12. Similarly, another poker situation where shoving may be correct: If your opponent isn't aware that you are shoving this light he will play incorrectly against you. But you probably shouldn't shove against players who are aware of your tactics and won’t throw away a weak Ace.

13. Many of us move all in with less than 10BB, except when we have a big pair. Against strong opponents who are familiar with our game we keep them off balance by betting different hands similarly. Therefore we look for ways to balance these strong hands where we don’t shove with weaker ones. It will often look stronger in the flow that has been established if you make a normal raise with K8 instead of shoving.


Summary: In many instances, people are fooling themselves thinking these calculations are important. The ones who do well are good players who make good judgments and have good betsizing. The EV they generate from figuring out these ranges is small and not necessarily positive. The poker considerations usually override whatever they may come up with.


Barry

[Dan Kelly]

Hey Barry,

I'll do my best to respond to all of your points.

1/4. I've always wondered how the fact that everyone has folded affects the strength of the BB's unknown 2 cards, but unlike what some people think I'm not actually good enough at math/programming to be able to figure it out. I've always assumed that the BB's hand will be a bit stronger on average but not by that much since Ace-rag hands are almost always folded in early position and most off suit Ace-rag hands will be folded in middle position. Some people will open Ax suited in middle position, but there are still a lot of people who will fold it.

That point brings me to your program, which I think has at least one flawed assumption. If I'm reading it correctly, you are assuming that A2s through A9s will be played from 1st and 2nd positions? I Just can't see the majority of players opening those hands from UTG or UTG+1, especially the lower half of that range. I'm not sure how much making those adjustments would affect the final result, but I definitely think it makes the chance of the BB waking up with Ax smaller. Please correct me if I'm misunderstanding the program.

Another thing that I'd like to add is that even if your numer is correct, it doesn't change the fact that 99% of tournament players will not adjust their calling range adequately. They'll be folding many of they're Ace-rags anyways, so it's not a big deal. If the play is +EV with your cards face up, imagine how +EV it is when you're cards are unknown... The "everyone folding" effect might reduce the ev of shoving by a little bit but nowhere near enough to render the play useless.

2. Yes, the number that Jimmy was referring to is for a 9 handed game. Obviously less antes means you can't shove as wide, but you can still shove pretty wide HU, especially if you're opponent is not good at calling ranges.

3. Online antes have gotten much bigger over the years. Full Tilt has the biggest antes I think, but Stars are also decently sized. Rarely is the ante smaller than 10% of the big blind. And of course live antes are pretty much always big.

5. I think this line of thinking is a slippery slope. In my mind tournaments are about finding edges and pushing them over and and over again. Even the small ones add up. When people start overvaluing their tournament life, they start playing scared, which is just about the worst thing you can do. Players who are playing scared will just get eaten up in a tough field and are basically drawing dead to win. Plus, why pass up one edge in favor of another when you can just take the immediate edge AND the future ones? I think people vastly overestimate how often you will bust when shoving 20 BBs when you are usually getting called between 10% and 20% (sometimes less) of the time (not to mention you're pot equity is usally at least 30%).

6. The cool thing about tournaments is that the ICM effect most significantly impacts the caller not the shover.

7. Yes of course. I never said I would ALWAYS shove 20 BBs in these types of spots. If the big blind is super tight and is not going to reshove me, then I will just make a standard raise because that is clearly the superior play. But against tougher (this could mean either better players or simply someone who hates to fold) blinds where it is not clear whether a standard raise is better than a shove, I usually prefer to take the easy chips by shoving rather than playing out of position.

8. Yep.

9. Okay, but why not take the immediate edge and also take the future edges? Tournament poker is about accumulation after all...

10. In my experience these types of players are actually kind of rare in tournaments. Most tournament players like to just see the free flop. But I'm not against going for the limp reraise against the right player.

11/12. Yes, the best part about shoving 20 BBs is folding out good hands that would have either shoved or reraised you if you had just open raised. Those hands that you are folding out SHOULD be snap calling against you're actual range, but instead they are making a pretty signifcant mistake by folding. Isn't that a classic example of example of Sklansky's Fundamental Theorem of Poker?

13. For the most part in tournaments, where your oppositon is often complete unknowns and generally weaker, balancing is completely unnecessary and flat out suboptimal. But yes against good players who know me well, I have to mix it up.

I hope some of that made sense!

[Justin Bonomo]

Barry, I'm not sure who your post is aimed towards. You are referencing a hypothetical situation without actually giving us the situation or a link to the conversation.

Regardless, knowing how to calculate and apply these things is super helpful.

Personally, I'm in the school of thought that poker is in a primitive state, and even some of the most respected players in the game don't understand some VERY fundamental concepts. As a professional, I take it upon myself to learn the game to the highest extent I possibly can.

The way I do this is to analyze every situation I possibly can. I replay hands in my head, discuss hands with other pros, and do math equations to every situation I can possibly think of applying them.

Summary: In many instances, people are fooling themselves thinking these calculations are important. The ones who do well are good players who make good judgments and have good betsizing. The EV they generate from figuring out these ranges is small and not necessarily positive. The poker considerations usually override whatever they may come up with.

Basically, I can't believe you would try to deter a professional player from doing anything to study the game further. Even if this hypothetical calculation will never come up in a real life scenario, it still helps players gain a feel for their equity in a wide variety of situations.


Edit: 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). Discouraging players from figuring this stuff out is ludicrous.

[Todd Terry]

Yikes, Barry, I haven't read any C code in 15 years.

A few thoughts:

1. I think the nature of the fast-paced structures in online tournaments (plus the fact that a greater % of the field is playing fairly "optimal" short stack poker with stacks between 15 and 20 BBs) makes it more correct to push the marginal edges djk is talking about than it would be in a live $10K event.

2. It is the nature of a Nash equilibrium that you will end up making marginally +EV shoves (assuming that true EV, as opposed to chip EV, can be measured). If both you and your opponent demand a certain premium above break even to make a play (here you are shoving and your opponent is calling), the following cycle is going to occur. You will shove tighter than equilibrium against your opponent's given calling range. Your opponent will then adjust by calling even tighter. Which means that you can now obtain the desired premium by shoving wider, which causes your opponent to call wider ... The only "stable" situation will be a Nash equilibrium (by definition, where each opponent's strategy is the best response to his opponent's strategy). And in a zero-sum game, this means that you'll be shoving all hands that are even marginally profitable (since to pass up on an edge will give it to your opponent). Of course, one point is when chip EV != $EV (which is definitely the case late in a tournament, whether it is the case throughout the tournament and to what extent is an oft-debated point), you're not playing a zero sum game with your opponent. And this may also be the case with other weaker players at the table, if two strong opponents go all-in against each other, the rest of the table probably gains. But as djk indicated with his reference to ICM, when the prize pool affects the $EV-cEV balance, the shover has the advantage and will be able to shove even wider than cEV-equilibrium (assuming his opponent plays $EV-optimal strategy, if he doesn't by calling too loosely, it will force the shover to tighten his range).

3. As for the "clumping" effect of the players in front of you having folded, in the book Killer Poker By the Numbers, Tony Guerrera did some simulations with much less precise ranges than the one Barry used (although he used a range of ranges to get some perspective on the spread of possible effects) and concluded it does have some impact, without drawing any precise conclusions about how much.

[Justin Bonomo]

Good post Todd.

3. As for the "clumping" effect of the players in front of you having folded, in the book Killer Poker By the Numbers, Tony Guerrera did some simulations with much less precise ranges than the one Barry used (although he used a range of ranges to get some perspective on the spread of possible effects) and concluded it does have some impact, without drawing any precise conclusions about how much.

After looking at Barry's algorithm, it's obvious many of the assumptions of hands played are unrealistic. A long time ago N82 (Nat Arem) and I were talking about this subject. He actually analyzed hundreds of thousands of real flops at 10/20 NL to calculate the frequency of each rank card coming on the flop (obviously deuces are significantly more likely to be flopped than aces). I forget exactly how he determined how likely aces were in the BB when folded to the SB or BB in a 10/20 cash game, but I do remember the results being very close to the 1/134 number Barry cites.

[Barry Greenstein]

Dan,

I expect people play Ax suited in early positions when the stacks are deep. That comes from my cash game roots. I know that a lot of guys playing tournaments online have had to play more of a short stack strategy. But yes, you are reading my chart correctly. It is basically my pot entering strategy for limit hold'em.

I agree with you that a bunch of small edges add up to a big one. It's a mathematical mistake people often make when they justify passive play where they pass up a small edge because it's only, let's say, 5%, but they don't realize it's all these 5% edges that add (or is it multiply) to a big edge. I was just pointing out that selectively it's often right not to bust yourself on a small edge.

Justin,

I'm totally in favor of doing the mathematics to get a ballpark idea. But that's what it is. We have people who think this is the answer to the question of how to play poker, and it's not. They actually hurt their play with their mathematical solutions, which are imprecise anyway, and miss some big gains. I played with a well-known player who justified a terrible shove by coming up with a pokerstove calculation showing it was +EV. He should have made a normal raise and played postflop against a weak player.

My algorithm is probably a closer approximation than the pokerstove assumptions, but my premise is that exactness is not attainable nor more important than the other considerations I brought up.

Todd,

This is why computers are a long way from beating weak players as effectively as good players can. Nash equilibrium will give you small edges against all players, where a good player can cater to specific inadequacies of each opponent.

Barry

[Ryan Daut]

just working the numbers, you can shove about 35bb effective profitably with K8o assuming no dead cards, the ante is 1/10 of the big blind, there are 9 players at a table and the big blinds calling range is pretty damn wide (33+, A2s+, A4o+, K9s+, KTo, QTs type range). as dan said there are other things to take into account and def more optimal ways of playing the hand at certain times, but in a lot of spots if you have 22bb and K8o and you would otherwise fold in the sb why not just jam?

(for the record, with no antes and hu you can shove 18bb profitably with K8o)

the fact that nobody had a big hand before you is an interesting point but i think the combinations of Axo folded by many players make up for this fact a lot of the time. ill post again once i read through your code and see the assumptions you made.

[Ryan Daut]

pretty cool program. brings me back to some good days of programming in college. i really loved comp sci until we got into assembly language and other nonsense then it lost its appeal, but up through algorithms it was a ton of fun, and this looks like a project from that class.

your program looks too loose for online tourney play (and online cash game play given how nitty full ring games are in terms of people just folding suited aces and even A8o in the hijack).

given the nature of the stacksizes around the table people are folding hands they play in cash games like the small suited aces with 18bb i cant imagine a lot of people are playing that hand. other situations like if someone has 25bb in the hijack but cutoff/button both have 17bb most people are making some marginal folds like A5o even though they win the pot a fair bit, they just get shoved on a lot and dont have a good hand to call with. regardless of how much tighter people are playing theres no doubt that your program has a lot of merit and the number is somewhere between 133 to 1 and 220 to 1, but a fair amount closer to your number, id guess in the 160 to 1 range.

anyway, i think its kind of irrelevant given you can shove >30bb with K8o in the sb assuming 9 handed with 10% antes (500/1000/100a), 22bb will be a shove almost regardless of what people are playing/folding in front of you.

[Barry Greenstein]

Ryan,

As I keep saying, the relevant question is, "Is shoving more +EV than a straightforward raise?" With that in mind, you need to calculate what the expectation is for raising to 3 times the big blind and folding to a shove against hands that have you beat and winning the pot right there against hands that you have beat. Of course, that's not what will happen, but this is an important comparison we need as a first step.

Barry

[Ryan Daut]

Originally Posted by Barry Greenstein

Ryan,

As I keep saying, the relevant question is, "Is shoving more +EV than a straightforward raise?" With that in mind, you need to calculate what the expectation is for raising to 3 times the big blind and folding to a shove against hands that have you beat and winning the pot right there against hands that you have beat. Of course, that's not what will happen, but this is an important comparison we need as a first step.

Barry

of course. and when i do shove in spots like this its either as a range booster with a strong hand like 77 to make my overall range seem stronger if i get called or if its cause i thought i might get shoved on, have a bad hand for calling resteals, and because the expectation of shoving is better than folding.

but finding the optimal play is clearly the best idea, and its usually not jamming. its sort of a "if i would otherwise fold cause im gonna get **** on, lets just take a +EV play" route