Not sure if this topic has been discussed ad nauseam already, so bear with me, but how can we be balanced and unexpoitable with multiple bet sizes?

If we have different ranges for betting 0.25 post sized bet, 0.5 PSB, 1PSB and overbets, our smaller bet sizes will (generally speaking, not all the time) have weaker value. Isn't this exploitable by opponents?

For example, this is from one of Janda's books but he says that Pokersnowie will tell us to bet 0.5 PSB with AK, 1 PSB with sets and overbet with our straights on the turn of a KJ49 (flush draw) board. How do we reconcile this?

Different hands make it into the different bet size categories based on which bet size gives the hand the most EV in relation to the overall strategy. For example, if you 3bet TT from the sb vs btn call and flop is 952r, TT benefits more from a larger size moreso than AA would. TT likes folding out QJ+ whereas AA doesn't mind a call. So TT might be bet for the larger size more often than AA.

In a true GTO solution there could be an infinite amount of different bet sizes (or at least down to the smallest betting increments available).

In my experience, it's better for humans to focus on 1 or 2 bet sizes per decision point based on which size provides the most EV based on board texture; things start to get really wacky you start thinking about putting which hands in which bet sizes at what frequency. Especially when it becomes things like 20% of the 77 combos, 14% of the QJ combos, etc; it's not very easy for humans to use multiple (more than 2) bet sizes.

Got it. So what you're saying is that a solver/GTO strategy that has various bet sizing strategies IS actually balanced at each bet size, but they way it is balanced is really complicated wrt frequencies and not practical to use.

So to go back to the example from Janda's book with pokersnowie. He says pokersnowie recommends on a certain board: 1/2 PSB for our top pair hands, 1 PSB for sets, and overbet for our straights. What you're saying is that a lot of complicated detail that he is being left out. So for our 1/2 PSB there are a lot of other hands mixed in with our top pair hands to make it "balance", and that include mixed strategies such as 20% of our sets? E.g. Janda is likely simplifying it when he says snowie recommends betting PSB with sets.

Snowie is not a GTO solver. From my understanding it is not even designed to play GTO, and in some spots can actually be quite unbalanced. Also, I didn't think Snowie was capable of suggesting different bet sizings with different hands. It picks a single sizing for the entire betting range as a simplification. Someone with more knowledge of the software correct me if I'm wrong.

A true GTO NL strategy will have multiple bet sizes in many spots. I do not think that there is any spot where there would be an infinite number of bet sizes because ranges are finite, but I could be wrong. It is different in toy games. In the single street [0, 1] game with infinite stacks where the first player can bet and the second player can only call or fold, I believe the solution involves betting a different size for each "hand" in the first player's range, resulting in an infinite number of bet sizes being utilized, each sizing balanced with bluffs at the appropriate frequency.

There are an infinite number of possible GTO solutions, and any one of them could utilize the maximum number of bet sizes (limited by the smallest denomination chip) at low mixed frequencies

You are correct. One of my few quibbles with Janda's second book was his mischaracterization of the way Snowie 'plays'. To put it bluntly, Snowie does NOT bet different sizes with different hands. It might suggest that AK has a higher EV if if bets smaller (whilst sets maximize EV with a larger bet), but Snowie actually uses one size for its entire range. If it says AK has an EV of 3bb when betting small, I think that's only the case if sets (and bluffs) are also betting small. In effect, what it "loses" in theoretical EV with some combos, it gains with others. Indeed, it would be exploitable if it bet big with the nuts, and small with one pair, so it uses the same size for both types of hand.
This strategy leads to an approximated attempt at GTO play (with fewer branches on the decision tree), but we've learned from solvers that "range-splitting" (using multiple sizes, and any number of balanced ranges) can squeeze out even more EV. (IIRC, Libratus used up to 10 different "random" sizings in a given spot).

A couple quick questions then.

1) Are you implying that GTO is not the highest EV strategy? That makes sense to me, as GTO is more based on being unexploitable than maximizing our EV in a readless vacuum. To a certain extent, they are "balance" and "EV" not mutually inclusive?

2) A more philosophical question. Obviously GTO is not solved, even with one bet sized evidently. Is it possible to have a GTO strategy with multiple bet sizes then? How would that hypothetically work?

1. I'm not implying that, no. At equilibrium, GTO is the highest EV strategy, against an opponent that is also trying to maximize his own EV. It's just that apps like Snowie, or even solvers like GTO+, can't get to the ultimate GTO solution for deep-stacked poker in a timely manner. We have to cut off branches of the decision tree to simplify the game (e.g. limiting the number of bet-sizes), such that the "solution" can be approximated via artificial neural nets using reinforcement learning (e.g. Snowie), or computed on a laptop (e.g. Pio, GTO+).

2. In a sense, there is a solution (note: a solution, not the solution) for a version of the game that is restricted to a single bet-size. GTO+/Pio can fairly quickly compute a solution for a game if you only give the software the option of "pot" or "check", for example. It's just that in real world poker, you can bet any size up to the size of your stack. If you have enough CPU cycles (and some spare time), I presume that GTO+ could solve a river spot where it has one hundred different bet-sizes. Most of these sizings would have (near) identical EV, which is why we can narrow to the choices down to a few bet-sizes without sacrificing significant EV or becoming exploitable.

Just because different strength hands bet different sizings doesn't mean smaller bets with weaker value hands are exploitable.

Appropriate number of bluffs are dispersed in all bet sizes.

That's what makes each individual bet size "unexploitable".

If our opponent attacks our smaller bet sizes thinking the highest value hands in that bet size are capped, he would not show profit at equilibrium.

The bot would defend its small value bet by calling down at the right frequencies against a raise with the right combinations of hands. FWIW, some nutted hands will still be part of the small value bet size.

You see linus pull this move a lot. Bet small on the river with a medium strength hand and call down a river raise. "Inducing" in poker terms.