I wonder how much EV it would lose against a maximum exploitative gto strategy, and if it would be useful to improve the understanding of the favorite roles of each hand in any given spot.

Is it possible?

Has anyone tried this?

Thanks

It's possible in Piosolver menu -> Tree -> Round Strategy.

I ran a simple tree and then rounded the strategies of both players over flop+turn+river and then checked exploitability.

Rounding | Exploitability
No rounding | 0.2% (1bb/100)
10% rounding | 0.8% (4bb/100)
25% rounding | 2.7% (15bb/100)
50% rounding | 8% (45bb/100)
100% rounding (pure strategy) | 30% (165bb/100)

So versus the Nemesis (clairvoyant and maximally exploitative opponent), a pure strategy would be extremely exploitable.

I usually round my sims to 25% to simplify the solution and remove the worst solver noise.
No way I could remember and implement anything beyond 25% rounding anyway.

I don't think this is exactly what he means.
If you actually solved with a pure strategy in the first place and didn't have to round it after, I believe exploitability would be pretty low. Probably about 0 on flop and maybe a little bit as ranges get narrow, but it wouldn't be too big of a drag on the whole strategy imo.
And no I don't think solvers can do this

It's possible in theory;;


Imagine... Counting to a million. Now if a metronome could go on forever, it would technically count to infinity



But infinity is a concept that only exists in theory.


A human could strategically beat this number counter, or rather, find weaknesses, if the seconds in between the metronome deviated between 1.000000000... seconds.


Bugs exist in bots.



Now...;;;;


If you played a bot that played rock paper scissors 33.33 recurring but did a different timing for rock than paper or scissors,,,,,


Win

I think it would still be very very high. Like you can exclude so many cards as the opposing player (almost all actions of the pure player reduce his range significantly). So probably best for the pure player to only have one sizing most (all?) of the time. So if you solve with a lot of sizings and then round the exploitability will obviously be humongous (if multiple sizings are actually played). Anyways the exploitability is not realistic in any sense since nobody would know your exact strategy (the exploitability shown is realized by the perfect counter strategy).

Use the tendencies of your villain to determine your action at mixed strategy nodes.

Yes, it's possible.

The keyword you're looking for is called "Pure CFR". It has been done by academics, but never implemented into commercial solver projects as far as I know.

https://github.com/rggibson/open-pure-cfr

How they reach equilibrium when it's not guaranteed with pure strategy.

The question is not clear. One way to interpret the question would be given a game tree (involving 2 players) find a least exploitable pure strategy of say player 1 though this question would be of purely academic nature (since the question (interpreted in this way) is NOT relevant in any way to actual play).

I'm not 100% confident, but pretty sure the "pure" in Pure CFR refers only to the strategy profiles being pure (ie playing best response like in e.g. Fictitous Play). The resulting overall ranges would still be mixed, as these are the (weighted) averages of the played strategy profiles over all iterations.

I thought the same thing but didn't write anything since I'm only 99.99% sure and I'd need to invest 1 hour to be 100% sure which I didn't want to do.

I am not sure if I understood your question, but do you mean playing pure strategies with each combo, or with range?

With range, I believe you always lose something, varying from small and insignificant to a lot.

With each combo, the issue I have (and maybe this is due to a still limited theory understanding) is that in a world where you are not going pure with your range, wouldn't always taking pure actions with combos leave one of your ranges (let's say, checking or betting when they are the possible choices) unprotected? I mean, equilibrium has to be somewhat exact, frequency wise, and poker combos are discrete in nature, like, you don't have 4.954 combos of 99, you have 6, 3, 1 or 0 depending on blockers.

You're right, my mistake. I was under the impression that 'pure cfr' would produce pure strategies, but the averaged strategies, what we see at the end, is still mixed.

It should still be possible though. The solver would just have to use other combos to balance. Like if AQ and AJ are both bet 50% then the solver could bet AQ 100% and AJ 0%. There would likely be an EV loss, but I wouldn't expect it to be too large. Any EV loss would likely be less than a human gives up trying to implement mixed frequencies.

You can do this with PIO but I don't think other solvers have this option.

I experimented with this many years ago. From what I remember I had results all over the place. In some spots, the EV loss was quite large and in others, it was small.

Blockers sometimes had a bigger impact on the non-rounded strategy.

That's likely because the solver is solving for mixed strategies THEN rounding. Like if you have a group of numbers between 0 and 1 and you are trying to calculate the mean (average analogous to equilibrium strategy). If you had a group of numbers... .01, .54, .6, .51, etc and you rounded them all to 0 or 1 before you calculated the mean, the results could be highly inaccurate. Or they could be close depending on the distribution of the numbers.

To get a more accurate solution the solver would have to be forced to only use pure strategies for every combo THEN calculate an equilibrium strategy, given that constraint.

Interesting - are there solvers that do this?

There aren't any that I'm aware of, but I haven't worked with with every solver. The closest thing would be the implementable strategies developed by Jonathan Little. He basically looks at solver outputs, and tries to match the frequencies with pure strategies.

For example he might say, "OK, solver is betting 60% of flush draws here," Then he chooses roughly 60% of the combos to bet at a full frequency and 40% to check at a full frequency.

He's sort of mimicking the idea, but the human-developed strategies won't have the same level of precision as a solver would.

The solver could do things to protect your range on different run-outs. Like if it always bet top pair with a queen kicker, it could also always check middle pair with a queen kicker. That way you don't lack two pair combos in either node when a queen comes on the turn (not that this is really necessary if your human opponent doesn't know your strategy anyway).

Even a solver ran this way to solve for pure frequencies would likely lose some EV vs. a mixed strategy. My suspicion is that the EV loss would be minimal in most spots though. The exception might be nodes with very tight ranges, where the solver doesn't have enough pure combos to effectively balance it's strategy.

It would be nice if a solver had a switch you could toggle on and off so you could compare the strategies and EV of pure vs mixed strategies, without the inaccuracies that comes from rounding every combo after the solve the way PIO does.