I recently explained to a friend that in certain spots, for instance CO (Hero) RFI vs BB, according to solvers it's a reasonable strategy to use a very small cbet sizing on boards where Villain's range is composed of many snap-folds. My reasoning was that Villain can't just fold all his weak hands against such a small bet and therefore is forced to continue OOP with at least some of the weak holdings. I then realized that I am actually just regurgitating what I've read. It does indeed sound logically. However, I couldn't have proved it if my friend had asked me to. So my question would be, is there some sort of calculation to show it mathematically?

Another example would be that you can't just bet only your good hands OTF as the pre-flop aggressor but instead have to strengthen your checking range by moving some of your value into it. Again, can I calculate this to prove it? Cause, if someone would tell me that his RFI range is AA, KK, AK and were to ask me to prove to him that his strategy of betting only AA and KK whilst checking all his unimproved AK is inferior to a certain mix between betting and checking with value/air, I could'nt do it.

You can solve it with a solver and that would indicate whether or not you are correct.

Whenever you come up with a strategy, think about what's the worst thing that could happen if your opponent knows what it is

Interesting, so you guys couldn't prove it without a solver either? What about the second paragraph?

It depends on the constraints. You can do multistreet toy games for a perfectly polarized situation and it will need giveups/checks to maximize ev/achieve equilibrium or it will be overbluffing. You can just think about Anero's question/statement also. Equilibrium is achieved when neither party is incentivized to deviate from their strategy, so if you can think of a way to exploit the current iteration of a strategy, then it likely is not at an equilibrium.

You can look into mathematics of poker. I think they have a multistreet toy game example if I remember correctly.

you can't calculate this mathematically (there is math behind it, it's just too complex for us to write down) so much so that before solvers no one was betting small on flop.

the bet AA KK strategy is self evidently wrong if you follow the logic in my previous post
- If your betting range is AA KK your opponent will fold most hands and continue almost exclusively when you're beat
- If your checking range is AK your opponent will valuebet for multiple strets until ending up all in with anything that beats AK and bluff a proportional amount of combos
- If your opponent has this counter-strategy, betting flop with AK and checking AA KK would be the best counter-counter-strategy, therefor what you proposed can't be equilibrium

In mathematics of poker they speculated that small cbet strategy might be optimal in NLH and that is book from 2006.

You can't prove it without solving the spot.