Suppose you're playing heads up NL against a 200BB perfect GTO player/bot. It's playing an equilibrium strategy so will never try to exploit your weaknesses. How would you adjust to lose the least? Can you exploit the fact that it's not exploiting you? What I mean is are there places where you can break even by playing very exploitably and just always folding, always calling, or always bet/raise/shoving because it should be playing a strategy that makes you indifferent to your actions (at least some of the time)?

Don't play it

Yes. The perfect counter-strategy to a GTO strategy isn't necessarily a GTO strategy itself. However, you wouldn't be gaining anything over using a GTO strategy yourself, the EVs would be the same in both circumstances (zero).

Assume you're getting paid to play it... Maybe 50BB/100?

If you played the same strategy vs. it it would net breakeven right,

Well you can play different strategies which likely still result in breakeven (or close to it). So basically just trying to play as balanced as possible would be best.

You should make this quote into a T-shirt.

If your goal is simply to minimize your losses against a GTO bot then this is not good advice imo, there is no advantage to being balanced in this scenario.

You would want to design a strategy that gives up minimal EV to GTO, while being more feasible for a human to play.

You could reduce the complexity by always picking the most straightforward line whenever a hand is originally played mixed in GTO. e.g. simply fold all your mixed-strategy bluff catchers, always bloat up the pot with your nut hands. As long as the line you picked is played some non-zero percentage in the GTO solution then your change will be EV neutral against GTO. The resulting strategy may be terribly unbalanced, but much easier to play.

Yes I agree and you are correct

Once upon a time I went deep down a similar looking rabbit hole. I found no fruit there.

http://forumserver.twoplustwo.com/15...91/?highlight=

Can we shove AA - or any hand for that matter - preflop from button at 200BBs? I'm guessing making raises that are 100 pots big is never a part of a GTO mixed strategy, but maybe I'm wrong?... But we do know that a GTO bot would have to call more often than with just AA (or we could profitably shove ATC).

I would bet 1bb when i'm bluffing & pot when i'm value betting & expect the A.I to never be able to adjust.

GTO doesn't need to adjust.

You do realize that, while the GTO strategy is static, it has different calling ranges against different bet sizings?

I haven't done the math, but I think I've deduced the answer as follows (and if my reasoning is flawed, I'd like someone to say so). Your bluffs will have 0 ev against the bot, because it will call at the optimal frequency. When you do have something like AA, you won't profit as much as the bot when the bot has AA. The result is that you'll lose money. So no, shoving every button will not neutralize the bot.

I didn't mean "can we shove any two cards?"... Obviously I know that 23o is going to lose money and we can't magically turn water into wine...

My thinking was basically that maybe AA maximizes its GTO EV by playing postflop, and we lose a lot by shoving pre... but maybe AK or KK or even worse hands like JJ and AJ, because they have lower EVs, can become mixed strategy hands, and shoving them is close to GTO against a GTO bot... My intuition, though, is that if there is a GTO shoving range, it must include the top of our range - I.e. Aces. But that doesn't necessarily mean we can't have hands like AJ that play close to GTO EV (even if they're not quite there) by shoving into a GTO bot, right?

Basically we know that a GTO bot in BB facing a button shove is going to have some fairly tight calling range such that we can't have any holdings that surpass their "true" GTO EV by shoving.... It's gotta be tight enough so it doesn't lose a ton to aces (if it called all the time, then aces would have an enormous shove EV), but it's gotta be loose enough so that we can't profitably shove any two cards... My thinking is that maybe a hand like AJs, because it's relatively strong and blocks the top of the GTO range, can become a profitable shove, and because it's not *as* profitable (it's "true" GTO EV) as AA or AK, that maybe it gets close to its GTO EV by always shoving.

You can still lose a lot of EV by deviating from GTO. I think the 200BB push/fold from the BU example will lose quite a lot, for example. It is tough to find an equilibrium strategy by hand, but I have tried.

Make the assumption that both blinds are .75BB, if one blind calls the other never will, and both blinds use the same range. If BU shoves something like {KK+, AK, A5s-A4s, A3s(25%)}, The blinds will call with something like {KK+, AKs(17%)}, and BU is not doing very well.

EVs of Each hand in BUs range follow, given the above blind defense range:

AA: 2.676
KK: .411
AKs: .436
AKo: .215
A5s: .047
A4s: .024
A3s: 0
EV of our entire range: .017 BB/hand

This is pretty dismal considering our fair share of the pot is .5BB, and we have the advantages of position and no forced blinds to post. The result for AA seems not too bad, but KK/AKs are simply blind steals, which is pretty sad.

Even AAs EV will approach 1.5 (slightly less, actually) as stacks get deepen, because the opponents' defending range approaches AA only as stacks deepen.

It's playing a static strategy having bluffs/value in every spot & expecting me to have bluffs with almost every sizing & therefore will over-call vs my pot size bets even though i'm devoid of having bluffs vs it. So yea, being really unbalanced & bluffing with the minimum bb's possible n always betting the maximum with value i think is the way to combat a A.I bot imo.

But it's going to call a wider range when you bet small. You can't really gain anything by always value-betting big and bluffing small.

IMO, the idea should be to somehow limit the number of decisions we're making per hand (as each decision gives us the opportunity to compound our EV mistakes). We could do this by folding 100% of the time when the bot looks polarized and we have a bluff catcher. Or bluff shoving any time that seems reasonable. And also probably want to bloat pots if we have strong hands (as noted earlier itt, we cant just start shoving pre, but we can set ourselves up to shove on flops or pot flop, shove turn).

I wonder at what rate Libratus loses to true gto.

As usual I am confused and uninformed about GTO - I used to think that the only sensible strategy vs GTO is not playing or also playing GTO. (I still do but I think I need to have the GTO only play a GTO or it isn't really a GTO solution.)

I think my old mental image of the strategy was confused though... I pictured the strategy like a massive/infinit checkerboard pattern where each square could represent a state the game could get into, on each square was written the appropriate next response including what square you get to next. The instructions on the square may contain some randomization (eg, 100% bet AA to 2.2xpot,.... 0.3% bet 32o to 5xpot).
So, in this model, there is no need to adjust to the villains moves just get to the appropriate state square and follow the instructions on it. This approach never counts or adjusts to the frequency of the opponents play, no adaption.

I now think (I am likely wrong.) this checkerboard state solution only exists for opponents also playing perfectly GTO. If the opponent is not playing GTO your own GTO is not actually valid.

If a GTO does not respond to an opponent (this seems like my fixed checkerboard state model) I now think it can be beaten, so clearly not GTO.

I haven't checked or thought this through but as a thought experiment lets say, always 200bb deep at start of each hand, Heads-up play standard 1.0, 0.5 BB/SB.

If the opponent always pushed allin or folded as the first possible response move, knowing the GTO response call frequencies, can get the frequencies for this by playing millions of hands, or get it from solving GTO yourself [more than one solution but can eventually find which one is being used and this never adjusts].

The opponent could work out the 'perfect' shove frequency response against the 'GTO' fixed call range using all hands exceptions being JJ+, and AQ+, and when dealt these exception hands play the best multi-street poker the opponent can muster (note, this GTO bot is never adjusting so it's states are always expecting the type of hands another GTO would throw at it not these top 3% hands in multi-street play.)

I think the pushed/folded hands could get quite close to break even vs the GTO and playing good poker with these other top JJ+ AK/AQ hands would possibly be enough to overcome any small losses the approx 200bb push fold lines cause.

A human would very quickly spot that the only times an opponent played multi street they held a great starting hand and adjust but this gto bot won't.

I am now not so sure that a fixed 'GTO' could not be beaten, as mentioned I know little about this subject though and I suspect that the definition of GTO requires a perfect opponent for it to be a GTO solution. (I am confused???)

You would get wrecked in all in pre situations by using that strategy.
Not having premiums when you shove will mean you get massacred every time you get called.

Yes, you're confused. The definition of a nash equilibrium is a strategy pair where neither opponent can improve their outcome by deviating from the strategy. So if your GTO opponent is playing his side of the NE strategy, your best outcome comes from playing the other side. If you deviate from that, you can not profit.

There may be deviations where you come out with the same EV, but you can't beat him.

Your first checkerboard formulation is correct. The GTO bot will have a fixed (yet possibly mixed/randomized) strategy for every position it gets itself into. However, note that when we play against the bot, we can't know exactly what checkerboard it is in, because it partially depends on the bot's hole cards. We can, however, calculate probabilities for each of the hole card combos based on the bot's strategy.

Anyway... Let's say I have a (non-GTO) checkerboard strategy like this. I write it all down, and give it to you. Now let's say you create a strategy to maximize your EV against me, creating a different checkerboard strategy. You write it down, and give it to me. Then I go ahead and maximize my strategy against yours, and you again against mine, etc. Eventually we'll reach a point where neither of us can get any further improvements against the other. That's a Nash Equilibrium, or GTO strategy, and by definition is unbeatable.

GTO is actually easily exploitable