I hate you guys for making me think this hard, but here's a disproof by counterexample of the idea that mixed strategies can always be varied without penalty.

Imagine a game between players A and B, where A has either been dealt a 1 or a 0, with 50% probability each.

In round 1 of the game, if A has a 1, he may choose to turn it face up or keep it hidden. If he chooses to turn it face up, B has to pay him 1 coin.

In round 2 of the game, B may wager that A has a 0. If he is correct, A has to pay him 10 coins. If he is incorrect, he has to pay A 20 coins.

It's clear that A cannot turn up his 1 100% of the time. If he does, B can always wager in round 2 whenever A does not turn his card up, always winning 10 coins off A. What A has to do is ensure that B cannot exploit him in round 2. To break even in that round, he needs B to be wrong once for every two times he is correct. Therefore, he needs to have 1 in 3 odds of having a 1. He can achieve this by mixing his strategy such that he only turns over his 1 50% of the time. This means that heading into round 2, he will have a 0 50% of the time, a hidden 1 25% of the time and an exposed 1 25% of the time. This prevents B from having a profitable wager in round 2. A's expectation with this strategy is 0.25 coins per iteration, gaining 1 coin on the 25% of occasions he turns his 1 over.

So here we have a game where A is playing a mixed strategy. Yet it is not the case that it doesn't matter what his frequencies are in round 1. If he unilaterally changes his strategy such that he never turns over his 1, his expectation in the game is now zero.

I've x-posted that in the Poker Theory thread, maybe move discussion there as I feel like I've ebolad up LLSNL enough

If the drunk player is a loose passive station I don't mind a lead here at all with v1's strong value. This board misses most of hero's iso range and hero isn't going to Cbet a lot 3 ways vs a station if he misses.

Can people also explain to me why they don't like this turn bet with villain's hand? Hero is going to have a ton of overpairs to the flop, draws, and pair+ draw type hands that he should be calling down with since the ace doesn't really change anything. I would have to imagine a pair of aces is way ahead of hero's flop and then calling range. If hero is aggressive, then betting a marginal made hand here is difficult, but if hero isn't playing back at villain lightly, then it's a perfectly fine bet. I guess it does bloat the pot a tad and I'm not sure what his plan was for brick rivers.

This is incorrect. If we have a balanced bluff and value bet ratio and they don't call as much as a proper minimum defense frequency would dictate, all else being equal, a GTO strategy would profit off of villain's mistakes.

There's also a lot of value in understanding GTO strategies because it helps us identify what things our opponents do that are exploitable.

EDIT: I noticed after I posted this that plexiq gave the same solution in the Poker Theory forum thread.


I think you made the assumption that B will always guess 0 on the second round. This can't be right, though, because then A would exploit this by hiding 1s more often. OTOH, if B never guesses 0 on the second round, then A would show 1s more often. So B will mix between sometimes guessing 0 and sometimes not.

Assuming B faces no penalty for not guessing, B will guess 0 on the second round exactly 5% of the time. If A has a 1, then A's EV will be 1 for showing in the first round, and 1 for hiding it (5% * 20 coins = 1). A is now indifferent to showing or hiding when he has a 1. And equilibrium is reached. (A made B indifferent to guessing 0, and B made A indifferent to showing.)

Now fix B's strategy. B guesses a 0 exactly 5% of the time on the second round... If B does this, then A will have the following EVs
EV(show 1) = 1
EV(hide 1) = 1
EV(hide 0) = -0.5 [5% * -10 coins]

This yields an overall EV of (0.5 + -0.25)=0.25, like you said.

However, since B's strategy is now fixed, A can choose to show or hide the 1 however often he likes. A does not need to stick to the strategy of hiding 50% of the time when B is not trying to exploit A.

On the flip side, if we fix A's strategy so that A hides 50% of the time, then B may choose to guess 0 at any frequency B wants.. Because EV(guess 0) = EV(no guess) = 0.

BUT... If A and B continue trying to maximally exploit each-other, they would remain in equilibrium only by mixing their strategies at the optimal frequencies.

Now that this is resolved...

I think your intuition was based on the conflicting ideas that (1) GTO is the strongest type of strategy, and (2) Being allowed to choose frequencies as we wish against a GTO opponent (and not lose anything by doing so) would seem to make that GTO opponent relatively weak.

GTO will still win quite a lot against any human regardless. The fact that we can choose frequencies as we wish doesn't actually make our life all that much easier. We have to know exactly what hands are always a fold or always at least a call or always a raise. And it's not like every situation will actually allow a mixed strategy. Perhaps most hands/situations don't have a mixed GTO strategy (as most frequency balancing would happen across the bot's range of hands, not within a single hand like JJ).... So knowing the correct or "permissible" actions would be quite difficult. This is especially true in NLHE, where it might be correct to bet or raise only a very precise amount of money.

Wasn't really intuition, more incomplete reasoning I guess. I think I now understand logically, but it still doesn't make a lot of intuitive sense. I think you might be right that mixed-strategy hands are rarer than you might expect at NL.