This example doesn't work since your 4 bet range would always be AA I imagine a range PPs and suited connectors could be found for implied odds when starting with 7:1 pot odds while knowing the GTO players hand.

Even just folding very hand except AA to a 4 bet would make 4 betting AA less optimal than calling and just changing the percentage of the AA hands doesn't change that fact.

It doesn't use itself as a best response. Itself is NOT a best-repsonse until the GTO is found. It finds an actual best response and the finding of that best-response is trivial compared to finding the GTO.

And where did you get the idea that any fixed strategy breaks even against a GTO? It doesn't.

CFR doesn't get stuck in a local maximum. It finds the actual maximum and it knows when it found it by checking with the best-response.

You don't need to "train" anything to find the "extreme exploitative strategy". You simply calculate it.

I would have expected a 4-bet range of AA plus many other hands. A GTO solution requires you to call a 4-bet with many more hands than just AA otherwise ton's of hands would 4-bet bluff.

+1

In theory, it seems to me that any non-adaptive strategy would eventually completely reveal itself after observing enough hands. Therefore it should be fair to optimize any exploitative strategy vs a known GTO strategy when proving GTO is our goal.

In effect the GTO player will broadcast all ranges on all streets and all post flop action for the entire post flop decision tree since all of this information would be available if the complete algorithm were known.

Given this information, It seems to me that the best approach would go something like this:

With perfect range information we could run equity calculations for the entire post flop decision tree without any loss of certainty.

Another words we could optimize exploitative GTO post-flop play. We only need to run an equity equation for 47 turns and 46 rivers and allowing for maybe 6 action combos per street and using 4 starting ranges it requires about a million equations per hand that should provide optimal exploitative post flop results for the given pre-flop ranges. So it seems well worth the trouble

47x46x6x6x6x4~1 or 2 Million equations to solve per hand.

fold equity can be calculated for precisely when folding ranges are known for all aggression on all streets.

Exact implied odds can be calculated.

the thinnest possible value bets can be calculated for known calling ranges.

Even calculate for possible floating situations given known ranges on turn/river runout combos.

Include folding and then choose the best EV.

For Pre-flop action:
I would start with 4 initial conditions for defining 4 pre-flop strategies.

Range A would be to passively call ATC for the cheapest possible flop.

The idea would Be to see just how far the perfect post flop play vs defined GTO can take us.

Range B would be opening with a loose range while capping aggression at 2 bets and making wide 2-3 bet calls with uncapped ranges.

Range C would cp aggression at 3bets and uncapped calls for 3-4 bets.

Range D would be a tight range capping at 4 bets and uncapped 4-5 bet calls.

Once we deal the flop for the predefined pre-flop action we can Run the million equations to get the optimal results for all post flop action.

Then we could then repeat this for a decent sample of flops. In order to get the accurate win-rates for the four ranges.

The total number of flop scenarios when including both hands would be:
C(52,2) * C(50,2) * C(48,3)
1326 * 1250 * 17296 ~ 28 Billion

Note that this is much smaller than all possible hands to the river so I would assume fewer hands are required for the same level of confidence but I never did bother to learn the precise way to calculate for this as I was never an online cash game player. maybe 30 minutes per iteration. Would give us results for a couple thousand hands. We only need to be accurate enough to establish the direction which tweaking the pre-flop conditions take us.

Next we tweak the four pre-flop ranges to get the next set of win rate estimations and repeat as needed.

We will only be changing starting ranges so the impact and trends should be clear.

We can increase the sample size around the optimums to get better win-rate estimates.

To run all 28 billion flops if we assume it takes a second to run the million or so equity calculations it would take a PC about a thousand years to finish, or take the NSA less than a second.

However I'm sure running around 10 K hands should be a decent sample size which others can hopefully verify. Remember it only involves variance to the flop, not the river.

Once any strategy is found to be long term winning, we would know it's not GTO and we could identify all strategies that may cross this line as well as by how much fairly accurately. While spending only a fraction of the CPU time for training the GTO strategy in the first place.

It's the advantage that those huge equity calculations provide us.

Yup. It's easy to compute a best-response to any strategy. That's how the algorithm used to create cepheus knows when it's Found a close approximation to a Nash Equilibrium (A, B).

When
- Best-Response against B does almost no better than A does vs B
and
- Best-Response against A does almost no better than B does vs A.

Calculating best response is simpler than that. You don't need to deal with ranges. You can calculate the best action one hand at a time. You don't need to worry about being balanced because you are dealing with a non-adaptive opponent. Your range doesn't matter. Only your hand, board, betting, and stacks matter. And you can do better than perfect post-flop play, you can also play perfectly pre-flop against it.

Right, but regret minimization algorithms find a strategy against which no strategy exists which is winning more than some tiny amount and it verifies that by checking the best-response.

i just beat it twice and ive never played limit before

Cliffs:
Cepheus played trillions of hands in order to support the thesis: "FLH is essentially weakly solved".
OP played 200 hands and thinks this is enough to disprove the thesis.
#lolsamplesize #variance

But you forgot #dontarguewithagenius

but ..what about an infinite number OPs ?

Not only it's beatable but it's very easily beatable by anyone who is willing to put an effort. Here is the recipe:

-harvest the strategy from Cepheus' site
-calculate max exploit vs it (this is fast and can be done on the fly for every flop)
-...
-profit 1mb/hand

The problem is that strategy is probably exploitable for way more than 1mb/hand... maybe it's 100mb/hand. Then Cepheus only needs to play max exploit vs max exploit vs it randomy in 1% of hands. Then you can adjust... and it looks like we are back to playing poker here.

my favourite post of '15

FYI by genius i meant more like idiot-savant. And by idiot-savant i mean more like just plain idiot. I've decided that I have better things to do with my time and I concede that Cepheus is probably not beatable for more than 1bb/100, though I'm not certain, the point is I will be spending my time playing for real money instead.

I think 4betting in Limit makes no sense because there can never be any fe. As such we can only 4bet with a value range, which then becomes exploitable so it must be better to include those in the calling 3bet range.

this however is a very interesting and valid question.

But i think there are many things that we consider standard that are simply wrong.

Yeah I mean basically you are better off just accepting what they say as true and trying to beat real money ganmes. Limit Holdem not really the best place to try these days.

This post did not age well