Pretty sure this "debate"--if there is even a debate to be had here--has been settled already (see: https://forumserver.twoplustwo.com/1...hlight=removal)

But I was talking to a buddy of mine whom I run PIO sims with and this idea of card removal (especially in BvB scenarios) came up.

Essentially, if we assume that A, K, Q are overrepresented in earlier player's opening ranges, then by the time get seven folds (in a full-ring game) the starting hand ranges of SB and BB are skewed from a full range to one where A, K, Q are now overrepresented in the blinds' ranges due to assumed underrepresentation in the prior folders' folding ranges.

There's really no mathematical validity to this, is there? And any possible effects would be trivially small or nonexistent.

In other words, if we wanna model a BvB spot, we're perfectly fine and mathematically OK to just assume SB and BB have all possible combos in their ranges and go from there. That's correct?

My argument was it would be futile to even try to model this, go ahead and use full starting ranges. And if we did want to prove it one way or the other we could in theory look at a database of BvB spots, but it would be impossible to get a statistically significant sample.

To clarify--my question specifically is in reference to computing Nash equilibriums.

I can accept that we can compute some conditional probabilities of, say, BB getting a walk given we know BB's hand, which some very well-done results in the tread I linked to seem to imply.

I could also accept that, conditional on it being folded to the SB, the composition of SB and BBs ranges without them acting is slightly different from ATC--perhaps weighted towards high cards that otherwise would tend to find their way into an opening range.

But I'm having a hard time, admittedly without proof one way or another, believing that these considerations would make any difference at all when modeling a BvB Nash equilibrium versus just using ATC.

Does anyone disagree with the following statement: It's perfectly "safe" to model a BvB full ring spot with both players starting ranges being ATC.

I think this is more of an "empirical" question, not a "theoretical" one. By that I mean it seems like the answer can be found by playing around with PIO.

Remove low cards from SB & BB ranges (say, remove all 2's, 3's, and 4's) and see if that changes PIO equilibrium strategies.

My guess is that it would have an effect.

Thanks, WN.

I'm starting to think so as well.

Using your sim results for # of cards of each rank folded I'm gonna come up with some weights and rerun the BvB trees I've currently got.

I agree this is prob more empirical than theoretical. I was hoping to spark about a discussion about whether, in theory, this is something we should worry about. But I guess that's just empirical too

BTW thanks for your contributions--awesome stuff! And thanks for not excoriating me for misspelling "Removal"

I've tried a simple push fold game using CardRunnersEv to show some effect as crev does handle card bunching/removal effects. This is only a simple 10bb game but it may throw just a little light on how much an effect there may be.

I used HRC to give approximate NASH ranges for 9 handed all 10010 chip stacks, bb=100 ante=10 game.

I plugged the 9 handed open push ranges given by HRC exactly as is into CREV (Note, the HRC ranges are not perfect nash as this tool like most others doesn't take card removal into effect.)

In CREV if an early player pushes eg, utg pushes 14.7%, then in the CREV tree I just fold all the remaining players to keep the tree small(ish). It is only the open push SB vs BB part of the tree that is needed for this. (This is actually an 11.1% event, ie, 88.9% the earlier players wake up with something using these ranges.)

Here is an image of the result:

The left part is the SB result from HRC chip ev (all 9 players 1010 chips, bb 100, ante 10).
The middle and right are SB and BB respectively if I put all the utg to BB ranges into crev and do 30,000,000 sims.

Clearly if card removal is used but all ranges stay exactly the same it would be better for SB to tighten up and not push the bottom T6s to T2s etc that HRC recommends and the BB should not call a couple of the weakest hands like T7s,97s.

Although it seems that SB has to tighten up a little more than the BB does I think to get to equilibrium the SB probably has to tighten up approx half of the -ve hands shown and then BB would also have to tighten a few extra that may show up as just positive here. If the SB push tightens then the BB has to call tighter, but a tighter BB call allows a wider SB shove. I don't really know the true Nash freqs

Even adjusted as mentioned above it is still not NASH as the utg to Btn hands have been left to the opening frequencies calculated by HRC.

If the above is correct it seems like both SB and BB have to tighten in this approximation. Also you can see that the high hands do earn more chips.
eg, for SB, AKo is 327 chips in HRC image (note this is given in bb) whereas when adjusted for utg to Btn opening frequencies it is worth 357 chips the SB shown is the middle part of the image)

Nice work Base.

Also, are there any "short deck" GTO apps out there? Of course, short deck is 6+ so it essentially (kind of) already set up the deck for the full-ring BvB scenario. Hand rankings are different in short deck, but possibly some insights can be gleaned.

I don't know of any.

I think with deeper stacks and wider opening ranges using 2x to 4x size, the effect on the small blind will be negligible, until he looks at his cards. If he has Qx+, then his opponent is somewhat more likely to have a bad hand.

Ps I would limp the T7s in the small blind there. T8s is close and depends on reads.

i would limp k7o in the sb here. AA is close but it depends on reads.

Yes, this is a thing and BenCB789 from RYE talks about it in some of his Youtube videos.

Nash calculators do not account for reverse removal and it is more likely the BB is dealt a high card. He suggests adding a small form of 'tax' to the nash EV. I don't know how much, probably not a lot.

Essentially, it should be quite an easy math problem to solve. You could assign weights to all of the cards remaining in the deck after each action (UTG folds etc) and use these to calculate the actual effect after 7 ranges had folded.

For example (and this is a complete guess) there might only be 3.92 deuces in the deck, where as there might be 3.98 Aces

And that's an average. Of course you couldnt actually have 3.92 Deuces in the deck, but it would account for the times when there are only, two, three, one or even NO deuces live!

The rank frequencies due to card removal have been investigated. See the link in Post #1 above.

But it is not clear that any GTO solvers handle (or can handle) such frequencies. That is a much harder problem.

The latest HRC beta has full support for card bunching / passive card removal when using the new Monte Carlo mode. EVs and ranges in late positions change quite significantly compared to the regular mode.

Excellent work, cheers.

Sounds like we're all agreed that there is an effect here and I wanted to put some numbers to that. I have a custom code base that works well for these sorts of questions and used that to model the following results.

For my model, I used the Sklansky-Chubukov ranking method and assumed that UTG opens the top 10% of those hands and that the button opened 35%; for positions in between I scaled the opening percentage up proportionately. So opening frequencies looked like this:

Then I took 100,000 random decks and dealt a 9-handed game out of each. I found that 14.3% of the time it folded to the SB, meaning none of the positions had a hand that fell within the opening threshold shown above.

In those 14.3% of cases, I found that the SB and BB each had a 23% chance of having a top 20% hand and an 11.5% chance of having a top 10% hand (again using Sklansky-Chubukov).

So empirically, yes: bunching is real. First strategic implication that comes to mind here is that if you're in the SB and it folds to you, the BB is a titch more likely to have a decent hand and presumably more likely to defend against a steal. How that affects equilibrium, no idea and I'll leave that for any subsequent discussion.