How are optimal bet sizing's determined? what is used to determine the sizes? i.e stack sizes, ranges, position etc. (question meant for cash games)

All of the above, plus (if you're talking about optimally exploitative betsizes), the tendencies of your opponent.

Optimal river IP betsize can be calculated with.

BETSIZE = 1/(SQRT(2*(1-EQ)))-1



Eariler streets its way more complicated and the question is a lot less black and white.

^ What assumptions did you use to get this? For a full game this looks very...wrong. You can't determine the optimal betsize just based on range EQ, not even on the river IP.

Its optimal betsizing for hand equity based on 1-A def, no raising allowed, but if you wanna take raising into account, just include the raising% as a frequency that beats ur hand, even though it has some bluffs.

I'm confused. Are you suggesting we should use a different betsize for each hand based on the hand's equity against the opponent's range? (And no, that would not be correct - even if we exclude the possibility of raises.)

Well perfect player would use multiple sizes and balance every one of them. In practice you can often get away with one size earlier streets and maybe even OTR, but generally just sticking to one size OTR will handicap your game a lot.

If we don't allow V to maximally exploit us, there's certain size every single hand wants to bet, and those differ based on equity and blockers. So when you know what type of size your hands want to bet, you then do some estimations etc that if we only use 1 or 2 sizes, what size/sizes loses the least amount of EV.

Like if most hands want to POT but like 2% of betting range wants to 3xPOT, might just stick with POT, as the EV lost is marginal for your range. But if 30% of valuehands want to 2xPOT, and 30% wants to go 1/2POT, just sticking to one size will lose a ton of EV for your range.

When you refer to a "certain size every single hand wants to bet", what exactly do you mean there? Is this based on a hypothetical scenario where your range consists only of this specific hand plus zero equity bluffs?

Well its OTR, so bluffs are 0 equity and then Villain defends 1-A vs all sizes.

Ahh, the important point is that you only meant apply the formula you posted to your value hands with >50% equity and then balance each betsize with an appropriate amount of 0% bluffs. You skipped that explanation

One issue with that is that you may have a lot of hands with >0% and <50% equity, they can't be bet for value but also can't be used as bluffs for balancing the other hands, at least not without adjusting the formula. (This is important if you run out of 0% bluffs.)

Also, as you mentioned, this definitely doesn't work when raises are allowed. (Card removal is also an issue, but not nearly as bad as raising.)

Raising not an issue, just treat the raise freq% as valuehands. Like if V has 5% of hands that beat ur hand and will always raise, and he can fit 2.5% bluffs there, your hand equity is 92.5% instead of 95%. Like, the valuehands that beat you, is alreadyu taken into account with equity, the bluffs aren't, but you can just assume them as valuehands also, because when you face riverraise, your EV = 0

Blocking effect is a bit trickier, but should also be able to taken into account with some math.




But just because it isn't 100% perfect, doesnt' mean it's useless, this stuff can easily be compared to solver results etc.

Models aren't supposed to ever be perfect for them to be very useful for **** sake.

I'll have to look into raising tbh, but I'd expect that additional actions left to play (raises, re-raises?) will increase the value of merging different types of hands into a small number of betsizes.

Fwiw I certainly think this model is useful. From your first post it was just very unclear what you meant.

how did you get this?

how would you include raising frequency into formula?

Is that equation a variation on the one Matt Janda provided (either in his book or in a video)?
My notes say:
0 = 1 – (2Y)(1+2X + X^2) where X is our bet size in PSBs and Y is how often we "effectively lose".

If I remember correctly, Janda said the quadratic equation first appeared in Chen & Ankerman's MOP.

Yes. Doc's equation is the quadratic solution for X, bet size, where Y = 1-EQ.