I've made reference to the Fundamental Theorem of Chasing in this thread a few times, but it neatly covers this question:

"Given sufficient action yet to come, optimal strategy must threaten the nuts in any sequence."

So if there is sufficient money behind, then you're right that optimal strategy would never pare down to just AA, for example, because of exploitive destruction by the other guy. The key is "sufficient." What's sufficient? It might be pretty big.

Nevertheless, I would expect that at every deep stack size, there are probably value raises and bluffs (which are almost certainly blockers), as long as the money remaining makes it unprofitable to narrow one's distribution too much because of exploitation by the opponent. I don't think that the "seven-bet means aces" which has some kind of consensus is optimal, it's just that we all suck at balancing in practice.

This kind of problem isn't trivial to solve, even if you have programming skills. The idea isn't hard to understand, but I'm not going to flesh it out entirely.

Say it is a HUSNG where the other guy jams every hand. You don't know the equities for various stacks, though. So maybe the stacks start 50-50, but you don't know how much your edge is. You do know what your equity is if one guy has all the chips. So assume the following: that if you have X% of the chips, you have X% of the equity.

Now solve the game exploitively for the shove every hand strategy, assigning yourself 100% if you win, 0 if you lose, and 49% or whatever if you fold. Find your implied equity starting from this situation. Now that's your 50-50 equity estimate. Do this for several stack sizes (90-10, 80-20, 30-70 etc). OK, now you have rough estimates for your equity starting from various stack sizes. Now iterate that process, but instead of using the chip equities for future stacks, use the estimates instead. If you get to stacks not in your table, interpolate or something. Now iterating this process a few times should converge on something. That something is your implied equity given your edge from a particular stack, and conveniently, the strategy that realizes it.

I assume you know about the "Jesus Problem", where villain exposes AA after raising to $5 in a 1-2 game with $50k effective stacks and the hero can call with 100% of his range. I've always assumed that 50k was a random number.

Do you know what stacks need to be for villain to call with 100%?

What is hero's calling range when effective stacks are 5k? 1k? $500?

Tom Weideman proved some kind of bound on the stack size at which calling 100% was right back when this problem was posed on RGP. I've never really been that interested in this problem, though, so I don't really know any of the answers.

A reasonably complex model suggests a Nash strategy has characteristic X.

A) There is a weak consensus amongst top HUNL players that a robust strategy does not have characteristic X.

B) There is a strong consensus amongst top HUNL players that a robust strategy does not have characteristic X.

Under scenario A and B would you implement a strategy which had characteristic X. (Obviously implementing and seeing what happens is costly)

Can you skype me cgalley77 is name would like to see if you could help me a bit thanks

Well, if there is a such a consensus, that means they think that the characteristic a) doesn't add value to the strategy b) is exploitable. So if I think that it does add value to the strategy, then I would figure out how they plan to exploit it.

If they can't, and it does add value, then I would implement it, until such time as they exploit it.

That's the interesting thing about the current crop of HU-LHE bots. Like you I came into the game with a strong prior that things like never limping and never capping were just easy ways for us humans to keep our ranges wide and probably not truly optimal.

And the first bunch of bots did in fact limp and cap, but as the bots have gotten much much better they've stopped doing these things entirely. (The 2012 winner, Slumbot, doesn't abstract on any street prior to the river.). At this moment I'd probably bet at reasonable odds that when the true equilibrium is found that it doesn't include these strategies preflop from the SB.

I find the tentative evidence, that perhaps we humans can use strategies on some streets that are both simple and very close to being optimal, really exciting.

Some simplistic models I came up with suggests you can do ok by never raising in in position before the river in many situations, for example I believe (but I am in no way convinced) that you can only call 3bets in position in NL if the stacks are sufficiently shallow.

Thanks for MOP.

Do you think (part of) the poker playing ecology will or is by the process of evolution converging to GTO strategy?

i.e there is convergence or it is 'cyclical' changes

By 'cyclical' I mean e.g in exploitative {R,P,S} where an initial proportion of popn. play only R and participants play exploitable so play P and we end up with ecology changes where dominant exploitative strategy out there cycles though R->P->S->R.

Edit: * as in we're 'going round in circles' and getting no closer to GTO by just playing vs each other and changing our strategy to just not be exploitable by the current flavour of the month.

punter

or how about when stacks get shallower? say against 40bb stacks surely you need the option to 4 bet all in because calling a 3 bet in position leaves such a low SPR that position isn't worth as much as with 100bb stacks?

what do you think is the ideal SPR on the flop for your never raise in position before the river model?

What makes you think it's obvious ?

This is a pretty good question; of course the answer is complicated.

I think that play is moving toward GTO, but pretty slowly. Compare modern HUNL play to that from a few years ago, and it's kind of obvious in that sense. I don't think, however, that play will converge to optimal or even good unexploitable strategies without computer involvement. And the presence of very good exploitive players probably makes this process slower, because they may not only exploit your strategy but also other stuff like timing tells or physical tells (live) that game theory won't help you with. I also think it's possible that the process just stalls out because there are players playing strategies which are hard for humans to exploit because beating them just involves putting yourself in all kinds of uncomfortable situations and playing accurately in them, and basically humans aren't really up to that over the longer term.

If we call a 3 bet in position against a 40bb stack as opposed to a 100bb stack then there is obviously less room to use position to outplay them. Even floating the flop can be dangerous because it's easy for them to bet flop and turn all in with only 2 streets of postflop betting.

Are you suggesting that never 4 betting in position v midstacks may be ok?

Can it really be profitable to call with say, A9o sharing a suit? My understanding was that even with clairvoyance and very large stack sizes, the best you can do per street is to add bluffs (almost) equal to the fraction of winners you have. For instance, if you have a bluffcatcher and I have a draw to 90% dead hands and 10% nuts, my optimal strategy is to bet all-in with my nut hands and enough of my dead hands such that you are indifferent to calling. At very large sizes, my bluff fraction approaches 50%. Since you are indifferent to calling these bets, we can calculate my equity by assuming you fold all the time, meaning I effectively win the pot just under 20% of the time. With another street, I can count those 20% hands as "winners" and bluff accordingly another 20% of dead hands.

But with A9o sharing a suit, you only have 5.9% equity. Even if all five community cards were dealt before the three streets of betting, you can only "convert" that to 47% winners (and I don't think you can leverage tied pots, for which there are many in this case, in the same way). As they're not, this is most likely an overestimate of A9o's equity, and so it might not be getting the pot odds to call depending on AA's raise size.

One possibility is that there are a significant number of situations where the non-AA player has "too few" bluffs in his distribution, such that the marginal value of extra dead hands is positive. I'm not sure of a quick heuristic for quantifying this so I'm not sure how important this is.

But in any case, this highlights my concerns about the "Fundamental theorem of chasing". Large stacks have sharply diminishing returns, so even a trillion big blinds isn't going to be enough to induce very weak nut-probable hands to call.