Thanks, I'm not sure how to word the question in a better way -- you're essentially saying the implication "in the GTO calling range ==> they are higher EV than folding pre-flop" does not hold in the other direction, if I read that right.

That's why I put this in there, but it might not be clear what I meant:

"Now, want to find a game-theoretic calling range from the Big Blind. (I use the term "a" calling range and not "the" calling range because I want to keep things simple for myself at the moment, though I understand that our calling range will technically be unique.)"

Are you saying that they aren't necessarily in the Nash calling range because of the way that we've selected the hands? (I.e. we've taken a subset of the complement at random without considering the entire thing, and the EV of our calling range depends on all of the hands in there, so it may well be the case that we have left some out?)

I guess you could think about the question as, "how do we find a calling range that's for sure not going backwards in our quest for more EV?"

While the Nash calling range will be unique, the ranges in the line above should not be.

What I'm saying is that when you take a random collection of hands as your calling range, even if all of them turn out to be (relative) +EV calls, some of those hands may not be in the gto calling range.

Suppose, for instance, we selected the 10 best combos not in our 3bet range and then 7c2h as the only two hands in our calling range. I suspect that 72 is still going to be higher EV than folding as it's going to get to make up a bunch of equity by bluffing lots since your calling range is now fairly strong.

Yeah, that's the type of perversion that I was looking to avoid with the assumption "2" above, but as you aptly point out, I need to do more than that:

"2) I have assumed that we know which hands to 3-bet so that we avoid some possible perversions in the solution (i.e. calling a hand like Aces distorts the EV of the range in such a way that with how I have worded the questions, we may "calculate" that we can end up calling very weak hands.)"

So, constructing the calling range (assuming we know the 3-bet range) is basically a case of marginal range building as Alex Sutherland calls it?​

If you're looking for a practical approach to the problem there are some tools you can use to guide you.

I see you've mentioned Alex Sutherland. He's the guy behind Gtorb, right? I'm not too familiar with their software. I have played around with PioSolver a bit though.

With Pio there are subsets of flops that it can appropriately weight to "model" the full set of flops. You can run simulations on all those flops and then get an aggregate report and a preflop ordering from such. With this all preflop hands (in the possible range) will be ordered by their weighted-averaged ev across all the flops simulated (and if they've done their modeling correctly this should be close enough). You can then use their ordering for adding more or fewer hands to your ranges. (though iirc they may already include the weighted-averaged ev in the report so you can just pick all the +ev hands as your calling range.)

I could be misremembering something about the way this works as it's been a long time since I worked with it. But it works in some way along these lines.

For practical purposes, I'd say it's more important to play your range well (and have an easily implementable strategy) than it is to get the perfect range. Further, given that you're a smart player any range you're coming up with and solver is telling all hands in are +ev is probably quite good.

Thanks for the detailed response -- I'm basically just getting back in to poker to play for a little bit. I like the part about playing your range better being more important than playing the perfect range, which is why I'm looking for something that's obviously not going backwards.

I'll check out the PIO thing you mentioned -- I guess it basically lets you get some ranges for when it folds around to you in the Big Blind.

Alex Sutherland is the GTORB guy -- he uses a different program as an adjunct to GTORB.