I have been playing around with solvers and have this questions:

1) The ev they show for every hand for one player is asuming they are playing against other player playing the same GTO strategy?

2) Some hands have a mixed strategy with same or diferent ev values. This means we should follow the frequencies to play the most GTO strategy?

3) What would happen if we forget about the frequency and just stick to the most profitable option for each hand?

4) So for example if player A always select the most profitable option with each hand, and player B always play the recomended frequencies, would player B do better against a GTO strategy than player A would be? Or inverse? Or how is the thing?

5) Same with the hands with same ev value for two options but one recomended 100%, why this happen why should we pick the recomended one instead of the other if both options have the same ev value?

THanks in advanced!

I'm not an expert but I'm interested in Game Theory ,someone correct me if I am wrong:
1).Against an optimal player(Snowie for example is playing vs itself)
2)Yes if u don't wanna be exploited
3)That is what u want,most profitable way is optimal way to the point of Nash Equilibrium when either player can't do better(in same spot someone will be winning ex. 5 bb the other will be losing 5 bb and neither can't do better than that as long as they play optimally)
4)I guess I answered that up there
5)If solver recommends 100% that means that is the optimal strategy(let's say Aces preflop are always betting because that's the optimal way and there is no reason for mix strategy for that holding)

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1) As a default, yes, assuming the inputted ranges are correct

2) not running it to low enough exploitability

3-4 apply to (2)

5) the differering frequencies is due to balancing reasons.

Let's see if I am clear:

1)
Player A will chose always the most ev+ line.

Player B will play always with the recomended frequencies.

GTO bot plays the recomended frequencies too.

Player A will do better against the balanced GTO bot strategy than player B.

But player A will be explotaible and player B will not be. Then if the GTO adapt to exploit the leaks, he will be possible to exploit leaks from player A, but not to exploit player B since player B is balanced.

Is true this statement?

2) I have another question and is about sizes. Let's say you are in certain spot postflop with certain range facing a GTO Bot with certain range.
Will there be an bet size or multiples bet szies that is/are better for all your range in terms of ev than others?

3) Can this best bet size strategy if it do exist be mixed in sizes between hands and streets?

against gto the max ev strat is to also play GTO.

when both players are playing at the equilibrium any deviation away will lead to a decrease in EV.

lets say A and B are playing GTO and A decides to move away from GTO

+ A will decrease his EV, player B's EV will increase

+ B's EV has gone up, but GTO is no longer max EV. he can move away from his GTO play to exploit A and further increase his EV.


GTO isnt all about balance. it seeks to maximally exploit the stategy that maximally exploits its own strategy

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at the equilibrium if a hand is played with a mixed strategy that means both lines have the same EV. if one had a higher EV then GTO would take it - there are no loss leaders in GTO.

if you are playing a GTO bot and you have a hand that calls 50% and raises 50% you can raise or call 100% of the time and it wouldnt affect your EV. both lines are indifferent, and although choosing one line 100% of the time might cause board coverage issues on later streets, the bot isnt going to adjust its strategy on those boards in order to exploit you.

I have been playing around with some solvers and clearly not always a mixed strategy means same ev for each options, they differ a bit often

SOLVE THE HAND MORE COMPLETELY --- LESS EXPLOITABILITY %

Solver wouldn't use a mixed strategy, if one option had higher EV than the other.

I think OP is failing to understand this point.

The "solutions" he's looking at aren't perfect, perhaps because they weren't computed for long enough, or he only used a small subset of flops/turns/rivers, or bet-sizes, rather than all possible outcomes. If the solver doesn't have access to all the variables, its EV results will only be approximations, and haven't converged on accurate numbers*, a bit like when you run an Equilab calc for 1 second instead of 2 minutes.
To emphasize what others have said, if you could compute a full solution, (with a "Nash distance" of zero), hands that utilize mixed strategies will have the same EV for all those strategies.

* This is somewhat akin to someone saying "Aces have 80% equity vs kings", when the percentage is actually a couple of points higher. That extra 1.95% of equity can make a small but significant difference in EV calculations.

It is stranger for me that in a strategy where you are trying to play optimal, you don't have little ev sacrifies with some hands to make all your entire range more profitable and balanced, and that two options have exact the same ev in so many hands.

But I Got it, so all mixed strategies have in reallity the same ev and differences are from Nash Dist. I am using simple omaha and it says a Nash Dist of 0.05 on what I am looking

Thanks!

This makes more sense now.

I think I get it. If the GTO bot doesn't adapt doesn't matter our choice in mixed strategies, if he does it matters because he will be able to exploit us if we don't respect the frequencies

THanks !

What about bet sizes? Can different bet sizes increase the ev of some hands in our range, making some betsizes better than others for our entire range and strategy?

Can we mix them between hands streets spots? Are they necesary and part of the GTO solutions?

For example:
This shows a mixed strategy for folding calling and raising with differents EVs. Should I just assume it is all wrong and the EV of all the three options is 0?

It simply means the solutions aren't fully converged.

Solvers use iterative processes and try to improve the strategies gradually

If you had a process that tried to max exploit on each iteration the strategies would fluctuate wildly. You can see this process in a program like icmizer. Try setting a tight call open shove range and calculate the max ev shove range. It will be wide. If you then try to max exploit the wide shove range it will lead to a loose call range, leading to a tight shoving range, causing our original tight call range to reappear etc etc. The strategies will bounce around with little convergence to an equilibrium.

But if you adjust gradually they will settle to The equilibrium solution.

This is why solvers won't immediately jump to taking an action 100% of the time whenever there is a tiny ev difference.

Yes it is likely these actions are indifferent.
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Wrt bet sizjng having more bet size options will increase the ev for a player, but it makes the solutions more complex both to solve and implement.

this isnt true btw, all gto does is it guarantees that player A will never increase his ev, however not all deviations will decrease his ev. We dont know how many of those non-ev-decreasing deviations are there, or even if there are any, but they are theoretically possible.

to use the simpliest example of rock-paper-scissors, all deviations from gto have the same ev as gto.

(obv all of the above assumes that player B plays gto)