I am starting to study this topic, and I have some doubts wich I would really appreciate someone give me a hand.

1) In a theoretical situation where we could play perfect GTO, we would be winning money against any villain who plays against us that is not playing GTO by himself right?

2) Then, the only way to solve this from villain perspective would be play GTO against us, giving him a EV = 0 against us? (and we too because the game is zero sum right?)

3) I already got the frecuency of bluffing to make villain indiferent to call or fold his bluff catchers, but I can't get the calling frecuency we need to call our bluff catchers to make him indiferent to bluff. Could you give me a hand with this on a math friendly way?


Thanks!

3) I will explain here how this confuse me.

River.
Villain has 50% value / 50% bluff
Villain bets 1Potzise bet.
Villain bets 100% of his value hands and 50% of his bluffs for a balanced bet frecuency.
Villain % total of bet on river is 75% (50% value hands, 25% bluffs)

Till here everything fine right?

Now hero:

Hero's range is 100% bluff catchers.
why the theory says we should call 50% of the times here? Of where comes that frecuency and why?

1) For heads up not including rake, yes, or you'll at least break even. You can get a seat vs non-GTO opponent and be -EV when playing multihanded.

2) Not sure what you mean here. Villain won't always be at 0 EV with every decision he makes, even if in the long run he won't win money against us. For example, if he's dealt aces, opening doesn't have an EV of 0 no matter how perfectly you play. Even when villain is "bluff catching" on the river certain removal effects will be good against a GTO opponent.

3) You call at a frequency to make your opponent indifferent to bluffing. If you call more than 50% of the time, he'll never bluff. If you call less than 50% of the time, he'll always bluff. It comes from the fact that if villain makes a $100 bluff to win $100 pot his bluff needs to work half the time to break even (he'll win $100 half the time and lose $100 half the time by bluffing and ultimately break-even).

Again this ignores removal effects, which can be a really big deal.

2) I mean, if we are playing HU and no rake, we playing GTO and villain playing non-GTO, then our overall strategy vs his overall strategy should be winning on the long run. The only thing he could do, is play GTO too, then our overalls strategies would be break even against each other (ev = 0 for both), Am I right?

3) This is very lighting. What if villain Bet pot size, but bluff 34% of the times? Should we call with 100% of our bluff catchers then playing "explotative"? Every calls should give us a small amount of EV+ profit.

And if we want to give that little margin of profit to keep balanced, just still calling 50% of the time? From calling 100% of the time to calling 50% just for 1% of difference in his bluff frecuencys is really crazy.

This is mind blowing for me right now lol. Really thanks for the reply!

I prefer to exploit on the margins. Calling 100% of your bluffcatchers may be the maximally exploitive line against a 34% bluffer, but that strategy is quite exploitable itself.

For #2 there are super nitty technicalities but I wouldn't really worry about them.

For #3 yes you would always call. If you win $200 34% of the time but lose $100 66% of the time you'll win money.

Thanks! Really grateful

Thinking about all this, I got another doubt.

I can see the utility on knowing that we should bet bigger if we want to have more bluffs on our range and still unexploitable.

Given the last asumptions I just gave, like our range is polarized to 50% value and 50% bluff, and villain only have bluff catcher, then if we would like to bet most of our bluff hands we should overbeat, and this would be good for us on this spot because we will be betting a larger % of times, something good because everytime we check our bluffs villain scoop the whole pot with his bluff catchers.

But what would be the benefit of balancing our sizes, if for example we have 75% value and 25 % bluff on our range?

We should bet then 1/2pot to make villain indiferent to call or fold, but don't we get the same EV result if we bet for example pot?

What if our range would be something like 90% value and 10% bluff, should we bet 12% size of the pot? That doesn't make much sense. What if our range would be 100% of value, we should not bet then? lol

Just a fish with a lot of doubts!

^Don't worry my head hurts too.

Its definitely really interesting stuff though. And once you hear about it, you can't stop seeing it everywhere.

Now some of my friends like to quote "I'm just balancing my play" when they ****up, and balancing your range helps a lot in sports (I.e. a Striker in football going for far post is 'value' (Its a cooler goal, and hard to cover) and near post is 'bluff' etc. goalie ends up having to 'guess' or just 'hope you miss the far post.

Table tennis etc. Definitely gets you a small edge (even if you are doing it wrong/exploitably, any thinking is better than none anyways, it gets you focused and against strategy>nostrat generally)

Perfect balance allows us to make our opponent indifferent between his decisions points. If we don't value bet and bluff in optimal ratios (ie perfect balance), then our opponent can take a different line of action, increase his equity in the pot by exploiting our unbalanced ranges. In the words of John Von Neumann, we must use mixed strategies (balanced ranges) or else our opponent can find out our strategy and change his strategy to minimize our gain and maximize his own gain. So perfect balance is using mixed strategies to make our play unexploitable.

And yes, GTO is +EV vs all other strategies except for playing against someone else also playing gto, in which case- gto vs gto is both 0EV for both players, excluding rake.

Nice to know that tips. But I still can't see the point in betting for example 12% of size pot OTR, when we have a polarized range of 90% value, 10% bluff to be balanced when our oponent has only bluff catchers. (A classic river scenario very used to talk about balacing betting strategys)

If calling and folding has the same EV for villain on this scenario when balanced, why would be chose that ridiculous small size vs a estandar size to be balanced? What is the advantage we get vs a std size?

If we bet std, we at least give the villain the chance to make a mistake by a bigger amount (Something really good for us if our range is so heavily towards value)

What if our river range were 100% value, what would be the optimal bet size?

Your misconception is that gto correlates to expected value, which is wrong. We are in no way trying to maximize expected value in poker when we play gto. Rather, in poker, GTO is the minimax solution to the game, so we must use mixed strategies to make our opponent indifferent between his decision points. Thus expected value goes out the window, and we bet in optimal ratios and bet sizes, such that villian cannot unilaterally change his strategy and improve his equity in the pot. Nash equilibrium states if one player can change the decisions of his strategies and improve expectation, then the players are not playing optimally. So your error in reasoning is that gto is centered around value, but rather when we play gto we need to make sure villian cannot improve his equity by changing strategies, and thus once we have achieved this goal, our strategy is optimal. In other games, equity connotes win percentage, we try to maximize or minimize each player's win percentage through the use of optimal strategies. Again, gto is not centered around expected value.

It's likely that if ranges are that skewed on the river non-GTO actions where taken earlier in the hand, so talking about an equilibrium solution on the river in that scenario is pointless.

In other words, from my last post, the heart of gto poker is decisions, not value.

The "make our opponent indifferent" idea confuses people a lot because it usually involves a single-street scenario in which ranges are fixed. In other words what you're doing is creating a toy game that isn't part of the GTO solution to the actual game, so in many ways is worthless.

Anyway, with a lot of the fixed-range-on-river toy games there are multiple solutions. If we have 90% value, and our opponent is 100% bluff catching, as long as we bet more than X% of the pot our opponent is folding. So betting X+20 or, X+50, or overbetting allin are all GTO solutions, neither we nor our opponent can individually divert our strategy to gain EV.

This. I think I got what you said last.

The confusion for me was that, the definition of a GTO solution was that while our betting range is balanced with our size, villain would be indiferent to call or fold on the last mentioned spot. And definition also stablishes that villain can't not increase his EV by changing to any other strategy against us, so we could think that always we bet in a non-GTO way, villain could improve his EV against us changing his strategy.

BUT in the last theoretical example where we hold 90% of value and 10% bluffs, if we size our bet in a balanced way is true that villain can't get more value against us by chaning his strategies, but if we bet Pot for example, villain should fold all the time, but that would have the same EV against us that if we bet in a balanced way and villain Fold/call. So altought we are not balancing our betting range with our size, villain still can't do anything to improve his EV against us by changing his strategy unilaterally.

So there are certain spots where we can play Non-GTO and villain still can't improve his equity against us by changing his strategy unilaterally?

I hope I explained my concerns well.

No, all sub optimal play is exploitable, so any deviations from gto and villian can change his strategy and unilaterally improve his equity.

That's not really the definition, its more just a means of calculating a GTO solution.

This part is in fact the definition.

No, again, betting pot, betting 1/3 pot, half/pot etc.. are all in fact GTO solutions (for this particular toy game). That our opponent isn't indifferent about calling/folding doesn't disqualify them from being GTO.

So in the last example, the only non-GTO solutions would be to bet less than the indiferent point for calling/folding of villain right (Less than around 12%)?

And in the case we'd have 100% value on the river, the only non-gto solution would be not betting at all, right?

Really thanks for the patience! I think I got this perfectly now!