Let's say we are IP on the river, with SPR=2. We have a value hand, OOP checked and he's not allowed to raise after we bet. Most people know that a 100% equity hand (nuts) want to bet as big as possible, 200% pot (all-in) in this case. We also know that hands with better than 50% equity when called are good enough to value bet.

Let's say ranges are wide enough and OOP has mostly bluffcatchers. He can't raise, therefore we can be even more flexible with our sizings. We've a good hand but not close to nutted, that after going all-in and get called has 52% equity, therefore betting is the correct play. However, this combo probably will have higher EV by sizing down and having higher equity when called. We can size down a lot, maybe to the point that we win 95% when called. Well, according to my experience with solvers, this won't be optimal either. Usually the optimal size will fall in the middle, something like 70% equity when called seems to maximize.

My question is, is there any way to take a look at this in a more analytical technique? I find it interesting to know around what equity maximize EV after getting called and be able to put some logic behind that number.

Cheers

So something that I'm asking myself for a longer while now is how to determine optimal value-to-bluff ratios on flops and turns. Across the forums, you often read some rules of thumb mentioned and they often loosely refer to Janda, but I haven't seen anything systematic yet.

So on the river, my understanding is that the value-to-bluff ratio is completely determined by our bet size. For instance, let's assume the nuts/air vs. bluffcatcher game. If we bet pot, this implies a value-to-bluff ratio of 2:1. The defender risks 1PSB to win 2PSB and hence needs to be good 1/3 of the time to break even. In order to stop him from just folding out or calling off his entire range, the bettor hence chooses an exact ratio of 2:1 value hands to bluffs such that our value hands can get their pay-off. The bigger we bet, the more we can bluff. If we bet into an empty pot, we would always have an optimal ratio of 1:1. The bigger we overbet (say 100x or 1000x pot), the more our value-to-bluff ratio approximates this 1:1 ratio of betting into an empty pot.

Okay, but what about flops and turns? There are flops where the agressor can get away with betting range, if the equity and nut advantage is so severe that the defender fails to make the bettor's bluffs indifferent between betting/checking (see Will Tipton's first volume, chapter 4.2). So here, I guess the value-to-bluff ratio goes through the roof with bluffs heavily dominating? What does turn texture due to our value-to-bluff ratio? Is this different for static or dynamic boards? Is there a simple formula based on the bet size just as for the river for all scenarios where we can't bet range? If not, which are heuristics that help us constructing correct value-to-bluff ratios on earlier streets?

Not really a "question" but something I would be interested in are some videos on the most popular toy games, how we can solve them and what we learn from them substantially for our game. Kind of what you did in your last The Mathematics of EV video with the nuts/air vs. bluffcatcher game AA/QQ vs. KK.

What about the AKQ game? Or the AKQJ version? What can we learn from the different versions of the [0,1] game? What is this clairvoyance term that keeps getting thrown around? Explanations for these games can be traced e.g. in the Mathematics of Poker book or other sources, but explaining these step-by-step in some video(s) would make for some great content IMO.

Is having a low frequency limp range from EP ( LJ & HJ ) in nitty 6 max pools full of rakeback grinders that 3bet at a very high % but overfold to 4 bets bad?

Small frequency limp range would include:

22-77
Almost all suited aces
AA, KK, AK.

Limp re-raising the suited aces and AA,KK & AK, limp calling the pairs.

From what I can tell from my DB and notes on players they 3 bet or will iso a limp at very high % but play very fit or fold post flop. Also, they seem to give way too much respect to a limp re-raise, and are only continuing with AK/ QQ+ in most cases. I have been waiting for them adjust but its like they are bots or something very rarely deviating from their strategy.