I'm trying to figure out theoretically, what determines the decisions made by PokerSnowie to be GTO or close to GTO.

At any situation, given a list of possible hands with their frequencies I can either bet/raise or check/call or fold. I can have multiple bet/raise sizes for my bet/raise range.

Question 1
In GTO, when I'm determining what should be my bet/raise, check/call and fold ranges, what do I care about? Is it

1. maximising my combined EV of all my range such that EV of bet/raise range + EV of check/call range + EV of fold range = maximum?
2. balancing my EV such that maximum EV of bet/raise range = EV of check/call range?

I think the correct answer is 1.

Question 2
If I only care about maximising my combined EV of all my range, and there happens to have multiple bet sizes that will accomplish the same maximum EV, do I prefer the bet size that would balance my EV such that EV of bet/raise range = EV of check/call range?

Question 3
If the answer to Question 1 and 2 are both yes, then does it mean that when we try to balance our ranges, we are theoretically trying to maximise our combined EV of my ranges?

Some of this is explained on the Snowie website. e.g. https://www.pokersnowie.com/about/te...-training.html

It used an artificial neural network and basically "brute-forced" its results by simulating billions of hands in order to create a database of EV values for each combo on each board texture against the "random" agents it played against. When it learns that an action loses money (i.e. is -EV), it removes that action from its decision tree (i.e. it folds hands that it thinks are unprofitable as bets/calls/raises).

True GTO appears to utilize multiple bet-sizes with different parts of a range. Snowie, however, uses one bet-size for each situation, and it's the size that is best for the range as a whole, rather than for a particular combo. e.g. The nuts on the river might maximise EV by shoving 3x pot, but if Snowie thinks the entire range gets a better result overall on average by betting small, then Snowie will pick a small size for every hand it bets.

I think a GTO solution is the one that maximises the EV of your range, or your entire strategy. Solvers will do this by "range-splitting" (having multiple bet-sizes, with each sub-range being balanced to some extent). Snowie simplifies by picking one size. This presumably makes it slightly exploitable/sub-optimal, since in the vacuum of one hand it could be "leaving money on the table" by betting small with a hand that could win more by sometimes betting bigger.

The short answer to question 3 is YES. When you open AA and JTs for the same size UTG, it's because you want to get max value with aces (and you'll get action, because villains know you might have JTs and other weaker hands) but you also have a slightly positive EV with JTs, because villains know you might also have aces, so they have to give your open some respect (e.g. you might steal the blinds UTG with jack high). In effect, both hands are +EV, because your range is balanced. If you only played aces, or used different sizes for AA and JTs, there would be no balance and you wouldn't get action.

So, is there any resources on the Internet about how to decide what kind of bet size is the best for what kind of situation? And how to balance my range optimally? Any recommendations?

The problem with PokerSnowie is that there's no intuitive way to understand what it's doing to mimic it.

I think Matthew Janda wrote at least one (free) article for Upswing about bet-sizing, but I don't have time to google it. His second book goes into a lot more detail.
Other people have learned how to split their ranges by putting in a ton of hours with solver software like Pio or GTO+.

I'm trying Snowie for the second time. Having trouble warming up to it again.

My original issue was that every villain seemed to play like Snowie suggests. This is nothing close to reality at smaller stakes.

I'm giving it another try after a number of months. Now I've noticed that it doesn't seem to consider stack sizes. It's often suggested I bet 1/4 pot on flops or turns which would often leave me with less than a 50% pot sized bet behind by/on the river. I don't currently view that as good poker.

Thoughts?

Thoughts? You're probably wrong.

Snowie's style probably isn't ideal for the (soft) games I presume you're playing, since it has trained itself to play against "strong" opponents.

It is training by playing itself over and over and over...

Whatever it counts (nothing), IMO, it was hand coded to GTO similar play. I also saw hand fed ranges. If it learned by playing against itself also, I guess it did that also but I can't prove it.

All good chess engines (but the Google's perhaps) is a lot hand fed also, and it does its search in a coded manner. And I think Snowie is way closer to these chess engines than the Google's one. There are many Google's type of poker and chess engines and they are all just so wild (bad), counting out limit holdem heads up engines in cases where they learned it by trial and error.