Really nice work. I'm happily surpised to see that there's a tool capable of guaging this effect without the need for a supercompter.

I'm gonna play around with deeper stacked HRC examples to compare significance. I remember the book Modern Poker Theory had an example where the frequency of the opening ranges seemed to be less affected by bunching as it got deeper, but I don't really understand the underlying reason.

You can see a significant shift towards higher cards (and pocket pairs, for some reason). The effect is actually much stronger than I anticipated!

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The reason you don't see solvers calculate for this is because you'd need to input the strategy for every player at the table. That's fine if you're running multiway preflop solves, but everything else will ignore these effects.

The simplest implementation would involve including a default preflop library of some kind. I think we'll see more solvers including support for this stuff in the near future.

This really depends on the calculation method a solver uses, it isn't quite as straightforward.

If you do full chance sampling then the solver essentially just simulates hands one at a time, each sample assigning specific hole cards to each player and a single board run-out. Considering card bunching is really trivial if you sample like that: All the card removal effects are exactly the same as in the "real" game and you don't have to take any special steps to incorporate bunching here, you just need the opening ranges of all players. (This isn't what the main 2-player solvers use though, there are more efficient algorithms for 2 players.)

For more elaborate calculation methods it gets increasingly complex to correctly consider bunching, e.g. when you cover multiple boards and/or full player ranges in a single iteration. It's still possible to incorporate bunching effects by approximation here, I think that's what skario referred to earlier. In any case, it's no longer quite as simple for these methods.

Great thread! its funny I never thought about this but its true. TBH its good to hear, I've believed that PIO is "good enough" while not perfect but perhaps poker has a longer way to go than I thought

Here are some preflop solves for 9max/6max/3max BTN opens with the same parameters:





These are really simple 100bb preflop trees with a ton of abstraction, no rake. But even so, I think it demonstrates that this effect is not "insignificant".

If you copy preflop ranges into a postflop solver you will erase all of this information. And this is the first move of the game. By the time you get to the river this effect will compound. This is the futility of super high-precision postflop solves.

How precise would a solve need to be, in your estimation, to be rendered futile by way of failing to incorporate bunching effects?

This is a fantastic question that I don't have a real answer to. I would guess that the EV gained from adding more than 2 or 3 sizes/node will be outweighed by the inaccuracy of ignoring this card removal. The utility of the river probably depreciates much more drastically than the utility of the flop, as errors compound each action.

How off would your sim be if it miscalculated equity/runout frequencies/subsets by a half percent or so for each card?

If anyone has Jesolver and wants to compare bunching vs non-bunching sims, hit me up.

I know it has been mentioned, but I think an important aspect of bunching effects is the frequency of getting a given hand from a given position. In 6-max, you will get a random 2 cards from UTG, but based on UTG's (and subsequent players') actions, no other position will have an equiprobable distribution of cards, as some cards will now be less likely than others. This means that you can't (very accurately) use the number of combos in range to calculate the frequency that certain actions are taken, as certain combos will appear more frequently than others. For example, if you are on BTN and it is folded to you, and you open AKs and 65s 100% of the time, you will have a higher frequency of AKs than 65s, as the 3 previous players are more likely to fold a 5 or a 6 than an A or a K. Pretty sure I saw Zkesic talking about this in his PG&C thread, where he did provide data from his pretty large database on the frequency with which this occurs.

Yeah, I did some research on this a while back. The results were pretty interesting. Here are two graphs:

Wow, nice work ZKesic. That's a significant difference.

The frequency distribution within those ranges changes substantially. I think this comment here really drives the point home

A lot of people will look at my sims above and say "well the opening ranges don't change much from 6max to 3max", but this ignores the fact that the frequency distribution within those ranges is already quite skewed, not to mention the postflop equity/runouts/EV and everything else that gets skewed by the bunching effect.

Generally speaking are you adjusting your strategy based on this bunching research or is it strictly academic?

Strictly academic for now. There are almost no postflop solvers that can account for it.

I see. However if button is 20% more likely to have AKs than 32s and thus probably by a similar percentage hands like 65s, in the most marginal spots the adjustment is pretty clear? If you defend BB against BTN open, the flop has an ace or a king, folding all the hands that would be bare floats is a reasonable adjustment? What do you think?

I think the best adjustment (for the distribution of préflop cards) is to change the frequencies of the préflop hands. So, say you're aiming SB Vs BB, you first establish how the 4 previous players folding impacts the frequency of you getting each card. Then the frequency of getting each hand. Once you have that, you modify your open range frequency on that basis, then your calling frequencies.

So, let's say you find that AKo is your most frequent hand. Then you set that to be 100% frequency open from SB. Then, assume you open JTo 100% of the time when you get it. But, you get it 5% less often than AKo. Then, you set the frequency for JTo in your open range to 95%. And do the same for all other hands.

Lol I'm writing an answer to your comment for like the fifth time because each time I reread you I get a different idea of what you meant. So when you say:

do you mean changing our knowledge of what our preflop range actually is, or are you already arguing for adjusting the actual opening strategy (and thus opening certain hands a different frequency)?

Let's just look at it from the perspective of the player who raises first into the pot. Let's take the example of a 6max cash game, and it folds to the SB. Let's say your range from SB includes no limping, for simplification. So, you open all pp from the SB.

Let's say you have established that if it folds to SB, you get dealt AA as the most frequent pp (as A is the least likely card for the previous 4 players to fold) and 22 as the least frequent pair. And you have also established that you get dealt 22 approximately 10% less often than AA. What you would do to account for this is when you enter your preflop range into the solver, you put AA with 100% frequency, and 22 with 90% frequency, even though you are actually opening both every time they're dealt to you.

After this adjustment (for every hand in your range, and every hand in villain's range), you should have solves which account for the preflop bunching effect. However, it still won't account for the postflop bunching effect, as certain cards are more or less likely to appear on each street, again due to the cards that have been folded or called.

Ah! I see. Makes perfect sense

Very interesting read. Esp as it becomes more range defining going down a game tree where narrower ranges clash, just as blocker significance goes up...

Can we somewhat generalise & say that the bunching effect somewhat counteracts itself preflop to postflop whereby more 2x > Ax is folded preflop and more 2x > Ax appears postflop (in the community cards).
Could we calculate how the bunching effect vs no bunching effect would skew the probabilities of say flopping a pair, w/ the 2 seperate ranges and folded ranges prior (bunching vs no bunching)
I understand this is mostly Futile and we can just use jesolver to compare outputs & it I believe someone touched on the calculations required being too complicated

The effect does cancel itself out a bit. Like if UTG-CO fold then the deck becomes quite top-heavy, but then if BTN opens the deck evens itself out a bit. The exact extent of the effect could change significantly from one spot to another.

There have been a few users that have written scripts to calculate this. It's not too hard to do with monte-carlo calculation. But no shared code yet.

Just looking at the rank distribution in the remaining deck that's correct, yes. Once a pot has been opened and called, the distribution will be less skewed, or possibly even skewed in the opposite direction. (That was actually found in the flop rank analysis in the other thread, iirc? ie fewer high cards than expected from a random deal)

But in the context of solvers, the card removal effects by the active ranges will be fully taken into account. So this won't cancel out (or even reduce) the bunching effect in that case, the deck will still contain more high ranks than the solver expects. (That's assuming that the solver doesn't account for bunching obv.)

I'm looking forward to seeing HRC come out with a postflop solver, so we can truly measure this effect and see how the solutions would change compared to a non-bunching postflop solver.

I had a colleague tell me that they tried to observe the effect in Jesolver and that it wasn't so drastic. But they only looked at one flop, and of course, the deck probability had to be manually calculated. I suspect like most card removal it could be more drastic in some spots than others.

From what I understand[1], the bunching effect is always captured by theoretically perfect pre-flop solutions. A hypothetical full sim of all streets ends up capturing the induced frequencies of the deck. e.g. flatting low cards LP on a multi-way board can be found +EV for the exact removal effects described

That would mean any post-flop solution is strictly limited by the resolution of the pre-flop strategy and therefore the ranges fed to the post-flop solver[2]. So if you give a post-flop solver good ranges I think you're fine and no further work needs to be done to fix the post-flop solver.

[1] forgive me if this is way off base, I'm very new to solvers and just joined this forum
[2] I suppose this requires a more formal proof...

That part is not correct, no. Providing perfect ranges for the remaining players is not sufficient to model the state of the deck. Most Postflop solvers only work with 2 players and are oblivious to any folding ranges earlier in the tree. When it comes to card-removal, these solvers are not even aware that they are solving a 2-player subgame from a larger multiplayer game.

Afaik between the publicly available solvers, only Jesolver allows you to manually define the card distribution of the deck and all others just assume a random shuffle. (Going a bit off rails here, but supplying the relative frequency of cards within the deck is actually not quite sufficient to fully capture all bunching effects. In all likelihood it's "good enough" for practical matters though, and it's certainly a huge improvement over just assuming a random shuffle.)

Ah yea, you're totally right. I was generalizing a HU claim.

This is a pretty interesting point. I quite can't figure out why that's true yet.

Completely unrealistic example, just to demonstrate the point:

3 handed game, BU folds.

BU has a folding range of:
#1 {22, 33}
or
#2 {32o, 32s}

Both folding ranges result in the same card frequencies in the remaining deck on average, i.e. there's one deuce and one three missing. The probability of SB holding 22 when BU folds is different in these two scenarios though. On average there are 3.5 combos to deal 22 for #1 while it's only 3 combos for #2.