67s makes a withdrawal from ATM

I'll defer to submersible when it comes to what solvers would do as OP here, and the theoretical explanation.

In practice, playing low stakes, I think betting super small on the river to induce a raise is fun when we have the nuts and know V is prone to spazzing. It's also okay to do with bluffs or thin value when V is likely to over-fold and / or unlikely to raise.

I hesitate to do it with thin value in spots like this, i.e., where we don't block the nuts, and won't have many nut combos in our range when we just flat call the EP raise over a call, from the CO, with three players left to act, and then check-raise the flop, second-to-last-to-act on this super wet and dynamic board, and then check turn.

How often are we going to have AXdd if we take that line, and then bet 10% pot on the river? I'd be more inclined to believe T8dd, but only if we just check-called the flop.

Maybe we check-raise the flop with A2dd, bottom pair + NFD, or A8dd / A5dd / A4dd, one over, the NFD, and a backdoor straight draw. Maybe we have 98dd (impossible on the turn) or 54dd for a combo draw. Maybe 85dd if we're playing really loose pre.

The fact that hero x/r'd out of the CO, after the EP PFR checks, is already pretty sketchy, and is just going to look like 2P and sets for value. Hero's bluffs would likely be combo draws, but the 9d on the turn blocks the best combo of 98dd, which weights his range more towards 2P / sets, and less towards flushes, at least not strong flushes. Maybe hero has 54dd.

If we're going to bet thin for value, I prefer to de-polarize, and bet a normal size that isn't likely to get raised by worse. A $175 bet into $290, in a 1/3 game, is going to get looked up by non-believing over-pairs, and hands like 9x or 7x, but unlikely to get raised very often.

why cant you just call with this hand if u bet 10% and get raised?

What size are we getting raised? Are we calling if he jams?

We're starting out $900 deep, and V covers us with $2k. The pot is $290. We x/r'd to $120 on the flop, so betting $29 on the river is going to look weird. I'd guess it looks super weak or super nutted, so in a weird way, it's polarized.

Betting $29 on the river is so small that it could be disregarded as a check. There are spots where I think it's massively +EV to bet small to induce a raise, but here, I think it's worse than checking.

If we check, maybe V bets somewhere between 1/3 pot to 2x pot. If we bet 10% pot, and V decides to raise, odds are he's never betting less than 2/3 pot. So if we're beat, and call, we stand to lose more than we would have if we just checked.

We also stand to lose more than if we just bet out for 1/2 to 2/3 pot. And, if V does just call our 10% bet, could we have gotten more? I tend to think so.

If we would otherwise bet $175, then that's the break-even point. It's fine if V raises to $175, but V might have bet that much if we just checked. So why put ourselves in the position of having to guess if we're ahead when V raises, when we'd know for sure if we'd just bet out for a normal size?

So, I think the question comes down to the frequencies and V's raise sizing. How often does V call a 1/2 to 2/3 pot bet with better or worse? How often does he raise a 10% pot bet, and how often is he doing that with better or worse, and what size is he raising to, when he raises?

I don't understand why we'd want to increase the variance in a spot like this, where we can go thin for value with a normal size that isn't likely to get raised by worse. I just don't see this as a spot where we want to get cute by betting 10% pot. It's a 1/3 game facing a rec-fish who is reported as VPIP'ing less than usual, chasing all his draws, and knows hero can "make a move on him."

Why not just play it in a straightforward way, with a hand that doesn't fare so well if it gets raised? What's wrong with betting $175 and folding to a raise, rather than betting $29 and possibly calling off more than $175 because we induced a raise and feel obligated to call with our 2P?

I dunno. I think this is an example of greed theorem in action. We should be trying to figure out the largest size we can bet that will still get called by worse, in order to maximize the value of what we win, not trying to figure out how small we can bet to get called by more of V's range, to get called more often.

Preflop I think its a huge error not to 3bet this combo when our opponent is opening way way way too wide. Its just an auto print to squeeze anything that doesnt profit more from going multiway (small to medium PP and weaker suited Ax). These smaller SC will profit a lot more from getting it HU IP vs weak opponents.

As played on the flop I would go a lot more polar with my check raise size. On a very dynamic/wet board I would go somewhere around 200 as thats going to help us with SPR on future streets.

Turn as played seems fine probably checking range on this card.

Then on the river I agree with others that I would choose the largest size that I think I can get called by a hand like 98s T9s A7 stuff like that. Somewhere around 1/2p seems good but its obviously player dependent.