I'm glad this is being discussed, as I also thought mixed frequencies were roughly the same EV.

But then I noticed this while reviewing a hand.

This isn't the same as what we are talking about though. You have 1 value hand and 3 bluffs. The value hand never goes small.

Those EV's aren't the same because the flush is 100% frequency in the bet big sizing and the bluffs are mixed between big/small.

Value EV is going to be bigger than bluff EV so it makes sense they aren't the same. And when you look at the aggregate EV, the bet small sizing EV will be lower because it never has value.

You have to look at every hand individually, you can't group them together and take the average EV. That will always skew the results when you have value + bluff hands together.

Okay I kinda figured that might have something to do with it. Thanks

Multi-sizing will always be closer to true equilibrium, though, as long as you get sufficient accuracy and include the sizes you're comparing in single size sims. I figured the discussion was more to figure out whether this is really a thing at equilibrium. At least I'm more interested in that first before finding out whether I should implement these sizes into my own strategy.

I would speculate anyway that the highest frequency size in a multi-sizing sim would yield the highest EV for a single size sim but not sure about that.

The only reason this happens is not enough accuracy for convergence. At equilibrium a lower EV action is NEVER chosen for any particular combo. Every combo in the range takes the action that maximizes EV 100% of the time. Mixing only occurs between options that have exactly equal EV.

If you let the solver think long enough it will eventually completely stop taking lower EV actions.

Individual combos should have exactly the same EV when mixed. Different actions as a whole will have different EVs because different ranges want take those actions. Generally bigger sizes will have higher EV but not always. For example sometimes medium strength vulnerable hands want to jam earlier streets for equity denial, while nutted hands would lose value taking that line.

I don't think this is true. Larger bet sizings generate more EV but have a higher risk-reward-ratio and hence get used less frequently.

This would only be the case in perfect GTO, but the solvers can't solve to such accuracy, so they simlify in the way I described.

I wasn't talking about the perfect equilibrium there, just about how the solvers work. If solver picks the "A" 3bet size 95% of the time and the "B" size 5% of the time, it's pretty safe to say that A is the more profitable size (assuming we should only have one 3bet size in the spot), as the longer that the solver keeps running, the larger the gap between the frequencies will get. At some point the solver will use the size A 99.5% and B 0.5%, and so on. We can assume that the perfect equilibrium of this spot would use the size A 100% of the time and B 0%.

I agree. It would be really weird if that wasn't the case in a certain spot.

I don't think that solvers really care about the risk-reward-ratio. Just the EV.

+1 to this.

I've noticed this as well, try running OB spots for flop cbet sizing's to test this theory.

How we can we know that?

I think assuming a frequency that is 90%/10% or 80%/20%. Will be 100%/0% in true GTO, is too big of an assumption.

Solvers create an interesting dynamic by giving us outputs without giving us reasons, then we try to fill in the gaps and guess what the solver is doing ad hoc.

It becomes circular logic because the conclusion of our outputs is true (we have the solve), but the premise needs as much evidence as the conclusion.

The premise being whatever reason we attach to the outputs.

Here is a simple toy model which shows why large sizings generate more EV despite being used less frequently.

Board: 22233
Pot: 10
Stack: 100

OOP Range: KK (50% freq = 3 combos); 66 (100% freq = 6 combos).
IP Range: AA (25% freq = 1.5 combos); QQ (100% freq = 6 combos); 55 (100% freq = 6 combos).

OOP - pacified.
IP bet sizings: 25% pot and 1000% pot.

-

If you solve this, you will find that the smaller sizing is used more often but the larger sizing generates more EV.

DooDoo asked me to post so just a couple things
- a solver always maximises the EV of any hand. Given the parameters you give it it will calculate an optimal strategy. That‘s all it does. It doesn‘t “simplify”.
-now if you give the solver various sizings options it will calculate a strategy based on those parameters. Good. You now have the solver for strategy for multiple preflop sizings. This however doesnÂ’t tell you which single size would yield the highest EV. In order to get that youÂ’d need to solve size after size.
A rather silly amount of effort once you realise that a wide range of possible 3 bet sizings yield basically the same EV. Which itÂ’s why you should just pick one and learn how to play it.

- if you do wanna find the optimal single betsize on the flop the process is far more reasonable. My approach to that is to start with a complex solve to figure out the EVÂ’s for each player using multiple spread out sizings. Then i reran the sim with a bunch of different single sizings to figure out which single size yields the highest EV (and if itÂ’s close enough to the original sim ofc. If not you should consider adding an additional size)

Betting frequency is not an exact indicator of EV: For example a strategy betting 25% only could and likely will involve more betting than betting 100% of the pot only. Does this tell us anything about the EV of both solutions?

I solved it. The large sizing generates more EV for AA and is therefore used 100% of the time. For QQ 25% if more profitable and therefore used 100%. For 55 everything would be the same EV in equilibrium.



I've ran many sims and this is roughly what the frequencies always look like for 3bet sizings over time:

"Start of the sim"

A: 33% B: 33% C: 33%
A: 25% B: 50% C: 25%
A: 15% B: 70% C: 15%
A: 5% B: 90% C: 5%
A: 1% B: 98% C: 1%

"End of the sim"

It's very obvious to me that if we kept running the sim, the solver would at some point be using the option B at 100% frequency.

That's just not true. Here are some examples:

1. This is from the earlier sim:


Here, the highest EV acton for 55 is to check, and yet PIO doesn't maximise the EV of the hand by always checking, it actually bets the hand at some frequency, which contradicts your claim.

2. This is from a Simple Preflop sim:


Here, the most profitable action for ATs is to 3bet, and yet, the solver calls with it at some frequency too.

If the solver picks option A 99% of the time and the option B 1% of the time, it's pretty safe to say that the option A yields the highest EV, especially if we assume that solver always maximises the EV with its frequencies. Using the size A with our whole 3bet range should be more profitable than using the size B.

Which is why it seemed pretty clear to me that in the below sim, the 22bb 3bet size was more profitable than the 10bb one:



If I kept running the sim longer, the solver would just keep increasing the 22bb frequency until at some point the 10bb size would be used at 0.00% frequency.

Yeah but I don't think those are the real EV's. They are the same EV, we just didn't calculate it too enough accuracy to show that they are 0 EV instead of .06 or whatever.

Once you start getting into 1/100 or 1/1000 of a decimal, the EV's won't be the same because of computer limitations.

What is more likely:

1) You stumbled upon some holy grail revelation that mixed strategies aren't the same EV even though every poker book/high stakes pro says they are the same EV

2) You aren't correct

1) wrong you just didn't let it run long enough. therefore your sim is worthless.
2)again either these EV numbers are off or you didn't let it run long enough.
3a) a solver doesn't "maximise EV with it's frequencies" it maximises the EV of every hand. It will always play every hand to it's maximum potential. There's no "loss leaders" in GTO.

3b) This is not at all what this says. It says "given the parameters given to me this is the optimal strategy". That optimal strategy now includes all of your preflop 3bet sizings. It does NOT say one size is better than the other.

It says "this is my strategy split into X 3bet sizings" and "this size is most commonly used within that sim" and the reason for that is very likely your parameters. We already established that EV's will be very close no matter what preflop size we pick.


Honestly I'd recommend you stop obsessing over "perfect preflop sizes" and focus on what really matters.


edit: the only accurate way to establish if one size is really better than the other is to run single size sims. then compare the EV of both.
you can also verify this by using the example Lethi gave you.
Remove bet small: how much ev do you lose?
remove bet big: how much ev do you lose?


it's really not as straightforward as "this gets used more often, therefore it's the better size"

Those are the real EVs as far as the solver knows.

I never said that the mixed strategies aren't the same EV in GTO. I just said that the solvers aren't able to calculate to such accuracy, which is why they are forced to simplify and mix strategies even when they don't have the exact same EV.

That's the whole point of what I'm saying. The EVs are different and yet the solver is still mixing - this means it's simplifying.

Yes, I could keep running the sim for one more year to get the "perfect GTO" results (the 10bb size would be used at 0.00% frequency), or we can just use common sense to predict what the results will be after a year. Since the frequency of the 10bb size kept decreasing for the whole length of the sim, it's fair to say that it's not all of a sudden gonna start increasing for no reason. It will keep decreasing until it reaches 0%.

If it uses one size 95% of the time, it is pretty clearly saying that size is better than others. In every preflop sim there is only one optimal 3b size and to me it's very clear which one it's going to be in that sim.

We never established that. I find the proper sizing of the preflop 3bets much more impactful than the exact range we do it with.

Why would I want to reduce the EV of my strategy by simifying?

OP your feelings don't matter. If you plug any reasonable 3b size into a solver it will give you a solution for that. Whether that's 12 or 20bb. You will get a strategy for that. The resulting EVs will be very close for both strategies.
Therefore it's a fact that either preflop size is playable.

As for your last point. It's an example to show that the EV will drop if you remove the big option even though it's not the most commonly used size.

Big sizes add value to your strategy even they're only used infrequently.

Apologies for the double post but OP has posted a bunch of nonsense
A solver doesn’t simplify. It calculates an optimal strategy.

A solver can’t really distinguish whether b21 or bb22 is better. It can calculate the optimal strategy for b21 only b22 only OR for B21 and B22. All of these will give you more or less the same result. Unless you messed up the sizings.

There is no such thing as an “optimal preflop 3b size” (and his process of finding it is flawed anyway)
Even he was somehow right since the EV of various preflop 3b sizes is nearly identical the only logical conclusion is that what matters isn’t the size of the preflop 3bet but rather how well you execute the strategy.

OP where are you getting your solver knowledge from? Did you actually talk to the creators of simple pre/atleast an experienced high stakes player or did you listen to a YouTube video?

Seems to me like everyone agrees on the following points.

• At equilibrium, the strategy will only include the highest EV actions for each hand.
• On the way to equilibrium, the solver will increase the frequency with which it takes higher EV options.
• On the way to equilibrium, the solver will still be taking lower EV actions at non-zero frequency.
• Solvers are rarely used on models which are simple enough to be solved to complete equilibrium.
• Removing a sizing option from the solver will either reduce or have no effect on the EV of the strategy.

The toy model I suggested was exclusively addressing the question of how removing a sizing option from the solver will affect the total EV of the strategy. In particular, I was referring to the aggregate frequency of a sizing within the whole range and the aggregate EV of the entire strategy. I actually think I agree with all of the opinions in this discussion and it was just this one partial comment which I thought was a little inaccurate: "highest frequency size in a multi-sizing sim would yield the highest EV for a single size sim".

Really nice thread ZKesic and huge respect for your approach to poker and also to your engagement in this discussion. In particular, thanks for sharing so many results from your sims and the following conclusions.

The next quote is a gem which shouldn't be lost in the flood!

The problem is that it isn’t inaccurate. It’s flat out wrong. A complete misapplication of what the solver outputs are saying to him.
The quote is brutal as well. There is again no such thing as an optimal sizing with a hand.

What a 6 sizing sim is saying to you is “use all these sizings at X frequency” not “this size is used more often therefore it’s better”. It’s the same fallacy you easily disproved with your simple example. If one size was really better than the other it wouldn’t be mixed. It would be pure.

Tbh if we disagree on something as simple as that there’s really not much point in continuing this debate. OP wants “the most common size” yet doesn’t understand that this is simply not the same as “the highest ev size”.

Edit: the best part is that this doesn’t even matter. You can totally get away with playing a strategy of 20bb 3bets from the BB at 200bb. It honestly won’t perform terrible in theory and likely will do alright in practice as well if you’ve studied it.
Does that mean it’s the best 3bet size in theory? No.

The answer to this comes from experience with hw solvers. This is because of position. ALWAYS. Btn is last to act. its why I personally adjust my SB strategy, hasn't failed me yet.

Dude, you are out of line stating sim is worthless. unless you developed GTO (and not a solver using GTO), who are you to say that a sim is worthless.

I have run single size sims and multiple size sims, and the ev is obvious. However, multiple sizings are more accurate. Period, if you want to chose a sizing.

GTO was solved quite a while ago. Tools incorporate GTO so you can make study and decisions. I bet $100 you were not part of the Nash equilibrium development for poker.

However, we are playing against humans, and some with solvers. Consequently, we simplify the ranges and bets. I have a lot of experience with this, and it actually changes by player pool tendencies (5nl to 200nl).

Sure, you could use the same strategy at all levels, but it doesn't work depending on player pool. You need to adapt that into your solution (hence the simplification). Do we want to win, or just stare at sims? Seriously.

If a properly run sim mixes between 2 options those two actions have the exact same EV. If the sim mixes even though the EV’s are different you didn’t let it run long enough. The gap between 2 frequencies is NOT an indicator of one option being better than the other.
It’s a really simple concept. Are you trying to argue against that?!


What does simplification that have to do with what i said? The solver doesn’t simplify for you.

JustLuck is correct that solver just is finding the nash equilibrium with the constraints. Input an a lot of constraints/sizings and then it doesn't necessarily matter which one is used more than the other if they are mixed they should have equal EV when played against that other strategy. You'd want to compare apples to apples if you only want to use one sizing. You would have to do a lot of solves with only one solve to see which is a better simplification to one sizing. For flops in particular I really don't think this matters much/at all at low stack depth (100bb or lower). The EV and strategy difference does increase quite a bit as we increase stack depth and as we move to turn and river. It would no longer make any sense to be using small sizings in vast majority of spots on river and in position. So, with that being said there can be/will be "better" sizes and as far as I'm concerned two different strategies can have equal EVs with different sizings in the same spot.... so I'd probably call it "optimal sizingS" rather than there being a singular optimal sizing... and again, this doesn't matter particularly much early in the game tree in my personal opinion, and this would include preflop. As long as the sizing is somewhere within the range of reasonable (which I guess would be a different topic) then the EV should be very close.

Sorry, but I just can't agree with this.

I literally proved that this is not true with examples from two different solvers. Solvers very often use lower EV actions at non-zero% frequency, which shows that they are in some way "simplifying". If solver uses the 3bet size "A" 99% of the time and the size "B" only 1% of the time, it's very obviously implying that the option A would have higher EV in equilibrium for our whole range. We just aren't able to solve until perfect GTO with our current technology, so often suboptimal actions won't be used at 0% frequency, but at 0.17% or something similar. Using the 0.17% frequency 3bet size with our whole range, however, will surely cost us some EV. I also think it's very subjective of you to say things like:
I showed in the original sim that the difference in EVs of those two 3bet sizings was 0.37bb/100, which seems quite significant to me, as BB facing a BTN open is a very common spot to occur in NLH.

I already spent years focusing on "what really matters". I understand the basics. This is the next step.

I agree with all of this.

This comment is completely correct for the preflop sims, as they always only prefer one opening or 3betting size. So, if you ran the preflop sim to a high enough accuracy (for like a month+), all the low% 3bet options would at some point reach 0% and only one size would remain used. However, instead of wasting one month on the sim, we can instead just predict what the result would be, as it's usually very obvious in which direction the sim is going after just few hours.

It's different for the postflop sims, as there multiple bet sizes are used in some spots.

Thanks.

About to talk out my ass for a second with regards to the 99% and some of the solvers I do believe use some type of function that basically if EV A > EV B then action of B keeps decreasing, but does not necessarily go to 0% unless the EV A >> EV B. So, I do believe something along the lines of what you are saying does occur. This no longer really happens with PIOsolver 2.0, so I'm not entirely sure what they changed with regarding to the algorithm for that.

Does simple preflop give you an exploitability % for the accuracy of your sim? If I run preflop sims in PIO 2.0, then even if I run a preflop sim like this for 24 hours the exploitability/accuracy is somewhere around the value you're stating (0.37bb/100) and that would be with a significantly smaller game tree. If it's mixing two strategies and the EV difference listed is 0.37bb/100 as you say, it's entirely likely that the value is just falling within the error rate and that is why it's mixed .... not necessarily due to whatever "simplifying" thing you think occurs (which I don't necessarily disagree with in instances where its like 99-1%).