frosty

Against hands we're ahead of we should be indifferent to raising or calling. For it to be worse for us to call than to raise against these hands we would have to be giving villain a profitable bluff 3-bet spot, which is something we can prevent from happening by folding a GTO portion of our range to a shove. Therefore I think it's appropriate to only consider villain's value hands that call or raise when conducting this analysis (and balancing value hands with bluffs as appropriate).

That said, I did oversimplify things, as we lose even more to the hands that beat us than the additional amount we put in the pot when we raise instead of call. Assuming villain shoves all that beats us (44 and 77 seem a little thin for value, but let's keep them in the shoving range for now), we can calculate our overall EV as follows:

Let's say we raise $30, leaving $42.50 behind (villain has x total combos):

-EV of raising vs. villain's b/c range instead of calling: $30*34.5/x=1035/x

-EV or raising vs. villain's b/r range instead of calling and then calling a shove (assuming villain is balanced and has 42.5/(31.5+22.5*2+30*2+42.5*2)=19.2% bluffs=3.8 combos, which would make EV folding=EV calling): -$72.5*16/x+72.5*3.8/x=-884.5/x

Total EV of raising=(1035-884.5)/x=150.5/x>0, so I still think it's a pretty clear raise. For this not to be a raise I think villain would have to be massively outplaying us when he shoves, e.g. have approx. <10% bluffs if we call his shove. If villain calls 44 or 77 this becomes even more of a raise.

I think it makes sense to think about this spot like this, but if you still disagree, NoHubris, I'd be interested to hear why.

Due to the fact that there are more hands that he folds (no point in reraising) or 3bets (hands that beat you) than hands that call and you beat, I'd suggest you just call.

I don't think this is true. Let's break down villain's value range into what bet/calls and bet/raises given the assumptions in your post about what he does with his range:

b/r: QQ, TT (sometimes checks flop?), QT, 77, 44 for a total of 1+3+6+3+3=16 combos

b/c: AA(50%), KK(50%), AQ, KQ, QJ, Q9s, KdJd, KdTd, JdTd, Jd8d, 8d6d, 6d5d for a total of .5(3)+.5(6)+6+8+8+2+1+1+1+1+1+1=34.5 combos

Given this analysis, it looks like a pretty clear raise to me. 

Should have refreshed before posting my comment below, could have saved myself some typing... I'm happy we came to more or less the same conclusion though.

Would you still be calling a CRAI on rivers that complete more draws, e.g. 3s rather than 3h? While zach shouldn't have many straights/flushes on any river, some rivers complete some of our semi-bluffs, which makes bluff CRAIs with Tx less attractive.

Nice video Fedor. I like your approach to the game and I'd be interested to see analysis of what you believe to be more commonly occurring "interesting spots".

As for the last hand...

GTO/Balance-wise I think we need to call here.

Let's assume Gambler's value range is {KK (1 combo), TT (3), 44(3), KTs(2), AK(8), KQ(8), KJ(8)} for a total of 33 value combos. If he is balanced in this spot, giving 3.15:1 to Zach means he should have 33/3.15=10.47 bluff combos assuming Zach's range is {bluff catchers}. Now, Zach's range is not {bluff catchers} here, and in this instance I would not expect Zach to get to the river like this with hands which don't beat our bluffs almost ever, while he will often play his nutted hands this way. My understanding is that we should be bluffing somewhat less against such a range, but I'm not an expert on the topic. Let's set bluff combos at 10.

If AK is the best hand we fold, we fold all our bluffs, AK, KQ, and KJ (total 34 combos) and call KK, KT, TT, and 44 (9 combos, with a T blocker this is reduced to 6). Zach risks 12282 to win 7565 so he needs us to fold 61.9% of the time for a shove to be profitable as a bluff. We fold 34/40=85% of the time here if we only call with better than AK, so a bluff shove with a T is ridiculously profitable for Zach. We would need to call an additional 12 combos to make a bluff with Tx break-even. Just in case Zach is doing something weird with AK or worse for value I would call AK, call some KQ fold some KQ, and fold KJ if we're trying to add 12 combos to call. If we are value betting wider than my initial range (or we check back KK OTF) then we would need to call even more combos.

Exploitatively, I also prefer a call.

Let's assume Zach's value range is {KT (3), TT (3), 44 (3), K4s (1), K3s (1)} for a total of 11 combos. We're getting 2.24:1, so Zach needs to be bluffing with >4.9 combos(=11/2.24) for this to be a call. To be generous to the folding argument, let's assume Zach's only potential bluffs are Tx. If we give him AT, QT, JT, T9s, T8s that is 9+12+12+3+3=39 combos. Therefore, to have 4.9 bluff combos he must bluff Tx 4.9/39=12.56% of the time. I would guess that Zach is bluffing Tx at least this much, and given that he sometimes raises value hands before the river and he may have other potential bluffs I prefer a call exploitatively (and by this same analysis with KQ/KJ as well)

To do a quick classification to make this analysis more useful in-game, this just seems like one of those spots where villain has a high ratio of potential bluffs:value hands, and it would appear to a thinking player that his line looks strong enough to get some very strong hands to fold (e.g. we're considering folding AK) plus he has good blockers with his potential bluffs, therefore leading to villain taking a reasonable % of those potential bluffs and deciding to bluff. This leads to his overall bluff % being somewhat high and means we should be inclined to call.

I'm more on board with leading flops like this where villain is likely to check back than most flops. I guess if you think he's bad you're thinking he'll likely cbet a lot anyway, in which case I don't think you'd want a leading range as you'll have a ton of hands to x/r him with here. As for what would go into a leading range... FDs, pair+SD make sense, and you'd want some strong value hands as well. I think I would prefer straights as a lead and sets normally as a x/r b/c a good reg's cbet range should be pretty FD-heavy and sets are doing better against FDs than straights, but that's a pretty minor point. Also, we can't be leading every FD here or we'll be unbalanced.

As for the rest of the hand, I could maybe find a fold OTR since this would be a weird line for villain to take as a bluff, but we are probably pretty high up in our own range here.

This got long but I think these are kinda cool spots:

OTT I generally prefer a raise. Ignoring villain's 14% call BB v BTN for the moment, we're usually looking at a range with a lot of 8x, some 97s, and stabs with two overs no draw, all of which will fold to a raise. Admittedly, a number of these hands are drawing to only 4 (two overs) or 6 (two overs plus gutter w/ 7c blocked) outs, but assuming villain's going to give up enough OTR with air on a brick to avoid giving us a profitable bluff catch given how many busted draws there are and how rarely we have air here, then we'd prefer to raise for protection. Against FDs, we'd rather be raising as well, this time for value/protection. Given how squarely the 5 hits us (55 and 65 are definitely in our range) as well as the fact that we have fewer draws in this spot due to our flop check (we're likely firing with most of our backdoor draws), I don't think we're facing a 3-bet semi-bluff all that often unless we're up against a combo draw, many of which we block with the 7c, so when we get 3-bet villain has a lot of sets and straights, meaning I think we can call, fold non-club rivers and get value on club rivers (only the Tc pairs the board, so we don't have to worry too much about dirty outs). Against Tx a raise likely buys us at least a free showdown OTR, and depending on whether villain thinks us capable of a protection raise we often have a profitable bluff spot on most rivers (maybe not 9, J, Q where our opponent rivers two pair more often). 

That said, I don't think a call is bad, and I'd be more inclined to do it against this villain than in general since his 14% call range BB v BTN seems to contain more Tx as compared to air in this spot and a WTSD of 39% suggests he's less aggressive so he's likely not pulling multi-street bluffs a lot and we can exploitively call turn/fold most rivers here.

OTR when we raise we hope that 1) villain is value-betting Tx (assuming all overpairs 3bet pre) and 2) will turn Tx into a bluff-catcher and look us up. Villain has a lot of potential bluffs here when the board bricks off so he could definitely bet some stronger Tx for value and hope to get called by maybe 4x, 77-99 (which sometimes bet flop), 33, and A5 (which could also bet flop), as well as potentially some weaker Tx. However, 6x makes up a decent portion of our range, so he risks value-cutting himself. On the other hand, the small bet sizing does seem to bring more Tx into his range. As far as bluffs he could put us on, when we check back flop and call turn we often have some showdown value. For us to be bluffing we'd primarily need to turn 4x, 33 into a bluff. I'd be bluff-raising 4x and 33 here a decent amount, so I'd be going for a thin value raise sometimes with 6x and I'd prefer to do it with the 7 blocker than 96s e.g. At these stakes I'm a little torn. On the one hand, I doubt villain's running into opponents doing this too much, but on the other hand players are a little stationy sometimes at these stakes. I'd go with: higher stakes reg: raise, these stakes reg: call, these stakes fish: raise.