no limit holdem 4 handed deep stacks for all players.

cutoff raises 2x, folds to big blind, big blind 3 bets to 7, cutoff calls.

flop 29T

big blind bets 2/3 pot, cutoff calls.

turn J

big blind bets 1/2 pot

can the cutoff determine that since the J was not in the big blinds flop betting range, the big blind is more likely to hold a hand that was a value hand on the flop and is thus less likely to have a big flush here? Is this variable change at work in poker?

I am not sure that you could say he is more likely to have a value hand. I think that would depend on opponent and how they play OOP as 3better on this board.

I do believe that you can use future board cards to narrow and eliminate parts of a villains flop range. For instance, on this board BB does not have AsJs, KsQs among other hands that he would have threebet range. Additionally JJ is now half as likely a holding since combos aren’t reduced from 6 to 3. It doesn’t seem like a lot but board cards will n conjunction with what you hold can help you significantly narrow opponents range as the board plays out. With 9,T,J aloud one suit on board that encompasses a ton of Suited broadways that BB would threebet, this making it significantly less likely he has a flush. If you’re on the BTN with AXo with As you could even put a lot of BB range in a tough spot raising Turn and over betting river. I’d be inclined to do the same with 44 or 55 with a spade and f opponent is the type to 3bet A5s and A4s from BB.

If BB 3-bets an (arbitrary) 8% range of TT+,ATs+,A8s-A4s,KQs,AJo+ then he has 102 combos on the flop, but only 9 of them flopped a flush draw.
Let's assume he has a straightforward strategy of c-betting all his FDs (9), all his sets (3), overpairs (24), TPs (3), gutshots (4), and all his BDFDs that aren't in the above categories (26). That would mean a c-bet frequency of 66.7%, of which FDs make up 9 out 68 combos (13%)
Once the Js comes on the turn, BB has 7 flushes and 1 straight flush in his range. i.e. Flushes/SFs make up 13% of his turn range. This "low" amount of flushes doesn't necessarily mean he was overly value-heavy on the flop. Loads of hands (like AKo, AQo, AJo with the aces of spades) now have NFDs, but they were drawing quite thin on the flop.
Meanwhile JJ is now top set, and many of the overpairs now also have a FD.

Exactly which hands he barrels with is impossible for me to say. It's quite common to check the turn when the flush gets there in 3-bet pots OOP, but some of the hands he barrels with are very strong (possibly even the mega-nuts), some still have high equity (sets and NFDs), and others (e.g. heart-heart combos) are basically airballs.
I think it would be truer to say that he had more "air" on the flop (because everyone does), and more "high equity hands" in his turn-betting range, since he's going to c-bet the flop at a reasonably high frequency, but then dump a load of junk into his "give up" bucket on many turns, particularly this one which likely benefits CO more than BB.

maybe it's a bad example of what I'm trying to get at, or the wrong turn card. the point is to compare the card removal effects retroactively for the flop on different turn cards.

If the 4 falls on the turn, it will create less removal effect than other cards that are more prevalent in the big blind's flop betting range.

if the A falls on the turn, it will create more removal effect.

the tighter the big blind's preflop 3 bet and flop bet range become, the more retroactive removal effect will be seen on turn cards that are prevalent in the big blind's flop betting range.

seems to me that the difference is not negligible, but perhaps a bit insignificant.

I bolded the "low" part of arty's post to point out that it's a measure of relative range composition that depends on having a reference point, which I did not provide in the original post. I think using the A and 4 makes the intended comparison more clear.

so if the A creates a situation where the big blind's flop bet is retroactively different than if the 4 had fallen, perhaps we can use this against the big blind.

I'm trying to imagine the spectrum of all turn cards and how the cards could be ordered from (least removal effect) to (most removal effect).

on some flops and turns, there will be little removal effect, on other flops and turns there will be heavy removal effect. let's focus on the flops with more removal effect for the purpose of this exercise.

going back to the flop:

on the flop, the cutoff has the task of ranging the big blind:

(what are his value hands?)
(what are his draws?)
(pure bluffs?)
(avg price to make it to showdown including future streets?)
(avg price to improve a draw before the river?)

and traditionally, all of these questions will help us to form the top and bottom of our calling range. seems fine.

Removal effect from turn cards that eliminate the highest number of combos from the big blinds flop betting range as shown with card matrix pictures could be illustrative of what I'm getting at. I think this concept could be used the following ways:

assume static solid preflop 3 bet range, these are the big blinds options on the flop:

a) miss value bets, miss semibluffing value, gain checking value

b) value bet too thin, semi bluff too much, miss checking value

c) miss value bets, semi bluff too much, gain checking value(except on draw completing turns).

d) value bet too thin, miss semiblufffing value, miss checking value(except on draw completing turns).

vs (a)'s flop bet, I see fewer turn cards having heavy removal effect because the betting range is narrow. however the A likely causes a major range composition shift here.

vs (b)'s flop bet, i see many turn cards having a medium removal effect because the betting range is wide. the cards that would typically cause heavy removal are no longer so influential as the blocked hands individually represent a smaller fraction of the betting range.

vs (c)'s flop bet, he's naturally losing betting ev here, which is transferred to my bluffcatchers and draws(as draws can win with a pair more often vs bluff heavy opponent), but right now I'm confused as to how removal affects this opponent's ranges.

vs (d)'s flop bet, the big flush cards will cause major range composition shift, and probably the offsuit straight cards will have a significant effect as well.

Instinctively I think aces on the turn are almost always going to create more of a removal effect, as the 3-bettor's pre-flop range will be quite ace-heavy.
I can't say I've given much thought to specific removal effects on a variety of boards though, or how that changes the proportion of flop value-bets. :/

No problem arty. I'm gonna keep going with the analysis of factors and implications.

this is all relative to the unknown values that equilibrium would provide for (value betting ev), (draw betting ev), and (checking ev). since nobody plays perfect poker, we all fall into one of these categories.

my speculation about equilibrium poker and the retroactive influence of card removal breaks down like this in my head:

flop XXX

i think a pattern would emerge if we ranked turn cards from (significant removal <-------> insignificant removal), to the effect that the cutoffs flop play should face this set of retroactive flop ranges that manifest when the turn falls:

A
K
Q
J

etc etc

5 55
444
333

every line above(those implied by etc etc as well) will give the big blind a slightly different flop range that was actually held at the time of the flop play. once the turn card falls we can then narrow the range we're facing by eliminating blocked hands.

so if we look back in time to the flop, instead of him holding (flop range that we must counter) he's really holding (a shitload of individual ranges that manifest when the turn card falls).

There are 47 different turn cards we might see given the information we have. I'm wondering if the turn cards that improve our actual hand(thus making it more likely for us to continue to call or even raise) are most likely to benefit from knowing the opponents sub flop ranges. After all, if we're folding on the 3 turncard, well then removal doesn't matter.

those important cards in the case of the original post:

A
K
Q
J
8
7
AAA
KKK
QQQ
JJJ
888

ordered by my best guess. so there are 11 important ranges that the opponent might hold on the flop.

looking back at our possible opponents:

(a) will have extremely strong turn bets when these cards fall(but i would expect a lot of checks).

(b) will have relatively weak turn bets when these cards fall(and I would expect a lot of bets).

(c) will have very strong turn bets when these cards fall(and I would expect lots of bets).

d) will have relatively weak turn bets when these cards fall(and I would expect a lot of checking).

the reversal of cause and effect is really mind ****ing me right now.

is this real life? lol.

I'm not sure how to form any conclusion from this can of worms. Perhaps the solution is to fill the rabbit hole with the can of worms, cover with dirt, and leave it alone.

but yall know I'll be back later to ramble more most likely. probably with analysis of opponent (b) as this opponent seems to offer the most exploitive profit. the others just make me want to fold the flop a lot(a,c) or simply call more bluffcatchers and draws on the flop (d).

I think the benefits/exploitation are more pronounced when the information is not shared between players (i.e. having the As in my hand is overall better for me than having it on the board and not just because of the increased equity it would provide).

In addition I would think given the preflop action the turn card improving more hands in my range is likely better overall than eliminating hands from my opponent's range as my range is likely to be more narrowed and capped anyway given the preflop action my opponent having a few of his strongest hands removed due to card removal is not as beneficial as shifting the strongest hands to my range.

i agree. i'm proposing we get to use removal even when we don't hold key cards.
i also agree that three to a flush on the turn benefits the flop bettor more(if i remember correctly, three to a flush on the turn provides an extra 5% raw equity for the flop bettor, which is a 10% equity shift effectively, when compared to the same rank card but different suit).

however this doesn't equate to anywhere near 100% pot share for the big blind. much of my range still has playable equity and I'm going to take down some big pots in this spot. if I can identify even slight weakness here, it could result in winning many big blinds that i otherwise would not have won. if i can save that 2/3 pot turn call because I identify strength just a few times more than I would have before considering this removal effect, it could save many big blinds in the long run exactly with the more questionable flop and turn decisions. its the marginal hands that are most likely to have a game time play change from standard ---> exploit.

note that the pot is already about 24 big blinds after the big blind bets the flop. that's before you make a decision. the number of big blinds in the pot, combined with the heads up nature of the action, should imply that both players have somewhat narrow ranges. narrow ranges are more vulnerable to both transparency and card removal, which implies to me that this is something worth working on.

Solvers already take care of this, making sure ranges are just as strong as they need to be on every turn and river. Using solvers you can also see how removal by community cards affects strategy. For example imagine a UTG vs BB situation and the flop is AsQsX then the turn brings in Ts. Well at this point BB will lead a lot because it has all the Ksx, Jsx and low card flushes while UTG has very few. It's just one example of specific community card removal affecting strategy.

Yes, this is what I was going to say as well. Certainly turn/river cards have card removal effects. That is unquestioned.

The question is is there "additional" information available to Hero from card removal over and above the information reflected in Villains action (bet/check/fold/raise, betting amount, etc.) on turn/river?

In general I think there is and Bob148 has raised a very good point. In GTO equilibrium, my guess is that there is no "additional" information from card removal effects as this influence is already fully captured.