What is the "nut pre-flop range" in ultra-deep stacked NLHE? (Or: what's better than 100% AA?)

(It’s possible this is all answerable by a solver, in which case, please share your results 😀).

One of my favourite theoretical poker questions is “If one player is always dealt AA and the other any two cards, who is the favourite?” The “obvious” answer is that AA must be the favourite as it’s a heavy favourite over any other two cards. But as discussed in this old thread, it’s not at all clear - it’s possible that (at sufficiently deep stakes, and where AA cannot move all-in preflop), AA’s transparency is such a drawback that it becomes an underdog.

I’m proposing a slightly different question, though. Suppose you’re playing HUNLHE for extraordinarily deep stacks (e.g. 1 million big blinds deep). You (and your opponent) can select to take any range at all to the flop (that is, you could nominate e.g. 100% of the combos of AA and 10% of the combos of 76s, then you’d be randomly dealt cards according to those ratios, then go the flop and continue as normal). What should your range consist of?

(In case this question seems too distant from actual poker, it's supposed to approximately answer the question "will continuous reraises always converge to an AA-only range preflop?")

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It’s quite clear it’s *not* going to be optimal to have a range that is 100% AA. A simple proof: suppose that it were the Nash equilibrium to have a 100% AA range. Then if we take that as given, the optimal strategy for each player would be to move all-in on any flop that contained two or more cards that shared a suit with one of their Aces (as their opponent would necessarily have the other two Aces and be freerolled). But in that case, either player could unilaterally improve by adding some other cards (e.g. a variety of pocket pairs) to their range, which could extremely profitably set-mine to the massive overbet. Therefore, 100% AA can’t be a Nash equilibrium.

Even if AA stops moving all-in on the flop, it’s still very vulnerable to facing huge triple barrel bets against a range that has hands that beat it, some bluffs, and many “chop bluffs” with the other combination of AA.

So both players need to include non-AA combos. But what would they look like?

AA would still surely be predominant. But how so? More like 99% of total combos in your range? 80%? 50%? Less?

As for the other combos, what do they look like?

Given you can have anything you want, I think other pocket pairs (for set-mining) are more promising than suited connectors. They have a considerably better chance of outflopping AA and are generally more disguised, so have potential as three streets of large value bets.

Suited connectors are probably still worth running, but their value is less obvious than pocket pairs, as it’s hard for them to outflop AA, and when they do it’s often a scary board for AA.

I’m not sure if it’s worth running any off-suit cards at all? You’re happy to play them in regular poker because you have to take what you’re dealt, but is there any reason to play AKo if you could instead swap it for more combos of AKs? The only reason I can think of is maybe the ability to threaten certain hands on certain boards, like AKs can’t threaten top pair and the nut flush draw on a Kc7c2c board that AcKx can, but I don’t think this makes up for making fewer flushes in total (and plus that hand is still behind AA anyway).

Would the optimal ranges for the in-position and out-of-position player vary much, and if so, how?

I think it’s necessarily the case that (in an equilibrium) all of your hands would have the same EV (0EV, or whatever the positional advantage/disadvantage is), otherwise, you’d keep playing more of the +EV hands. I think knowing this helps us give a rough idea of what proportion of your range should be non-AA hands. It can’t be a trivially small amount, otherwise e.g. all your pocket pairs become +EV on an overbet triple barrel strategy against a mostly AA-range. Your ranges have to have enough non-AA hands that there’s a real risk that your overbet strategy just might have some of its value hands get coolered by even better non-AA hands, but your opponent still needs enough AA in their range that the overbet strategy is worth doing. What that eventually boils down to, I’m not sure, but I’m guessing it means non-AA hands make up at least 10% (but probably not more than 50%) of the optimal range, and bet sizes are usually quite large but are never massive all-in overbets.

Finally, it's clear that with small enough stacks, non-AA hands are not able to take advantage of the transparency of AA sufficiently, and AA will be 100% of each range. What would be the minimum stack size where AA is no longer 100% of both players' ranges?