i'm having some issues with a GTO toy game and it doesnt seem to add up. Hoping somebody can help me:

Assuming both players are playing GTO

So when Villian bets $100 into $100 OTR and gets called by Hero, both villian and hero should be Zero EV correct?

Villian should be betting 2 value combos for every 1 bluff combo, so when Hero calls say 100% of the time, his EV is:
-100 + -100 + 200 = 0ev
(lose first call, lose second call, win third call)

So ofc Hero is indifferent to calling or folding, because his EV is zero regardless.

But what about Villains Ev?
Villian is betting PSB, so hero should call 50% of the time.
So Villian's EV is then
.5(200+200-100) + .5(100) = 200 positive Ev?!

That doesn't seem right.

Incorrect. Villain's bluffs should be 0 EV if both players are playing GTO, but villain's value bets should be +EV.

It is right, though. If stacks are equal to the pot size, hero holds nothing but bluff-catchers, and villain holds 2/3 value bets and 1/3 bluffs, then villain can bet 100% of the time and simply claim the pot. Whether you calculate his EV as equal to the pot size (i.e. what he expects to win) or twice the pot (i.e. his expected stack size at the end of the hand) makes no difference.

Edit: Actually, I wrote the above without looking at the formula you used. Your formula is wrong, because villain's EV when called is not 300; it's 100. And villain's EV per hand is 100, not 200 (you can see that it can't be 200, because that could only be the case if hero called and lost 100% of the time). So the formula should actually be

.5((200/3)+(200/3)-(100/3)) + .5(100) = 100

Hope this helps!

But the toy game you’re referring to has one player with either the nuts or bluffs and the other with only bluff-catchers. When you switch villain and hero roles, they both have 0 EV as overall strategies.

Note that on a particular hand, no matter what the holdings, the EV sum has to be the pot size before final bets are made. Looking at an overall strategy however, like GTO, you can talk about both players having “long run average EV” of 0.

That is a very simple toy game that you made up. Study more sophisticated game theory, and then it will make sense. But simple Nash equilibrium calculations is the most basic of game theory examples, and games like rocks paper scissors is 0EV for players playing gto. But real gto in poker is +EV, and can't be evaluated through small toy game models.

There is a logical tree that leads to the river situation. It is not just about GTO even if one can make efforts to have enough bluffs in one's range.

The bluffs might be independent vs GTO. But if they are to balance one's value bets, that are not absolute, one has something to think about.

There are better and worse spots to do the bluffing at.

As a GTO river call, you call based on the pot odds the bluffer gets, and with a balanced bet range (easier said than done), you make the same money no matter how often or rarely it is called. If the opponent never bluffs and one calls it according to GTO, the non bluffer makes no more money than he would make with a balanced bluff range.

The only thing you need to do is to make balanced bets and balanced calls, as far as this part of the game goes.

Bluff are always meant to balance value bets, and you don't balance calls in gto, you call with a wider range than your opponent bets, to make him indifferent to bluffing.

Irregardless, value bets are the primary driver in gto, and bluffs are only meant to make your opponent indifferent to calling between his indifference points.

thanks, but Why are the terms being divided by 3?


I thought each time hero calls villian would win 2x($100 (pot) + $100 (call)) = $200

mnus the 33% of the time villian was bluffing and loses $100

As you said there's already $100 in the pot. If hero and villain both had 0EV what happened to the $100 in the pot? I know rakes are getting higher but I don't think the situation is that bad yet.

Good stuff in this thread guys.

I like how this scenario shows us just how bad it is to have a range of (bluffcatchers) vs a range with (nuts + bluffs). The bluffcatcher is simply getting his call back on average. He's breaking even while the bettor is earning (pot) every time. If instead, we allow the bluffcatcher's range to include some strong hands that beat some of the bettor's value range, then we'll see both strategies profiting from the pot. The bettor will no longer be able to earn (pot), which will benefit the bluffcatcher.

Hint when studying gto, fold equity doesn't exist, and bluff catchers are only in the AKQ game. But bluff catchers don't really exist in real life gto poker, because you are calling with your value hands to make your opponent to bluffing, which means your bluff catcher are really good hands, that are not strong enough to bet or raise with. So bluff catcher don't really exist in the way that most people think they do, like you don't call down to the river with ace high just to catch a bluff.

In these spots the value of the bluffcatcher comes from the times villain doesn't bluff and you get to show down the best hand and win. If villain only has a few nut combos and therefore rarely bets, this value can be very substantial.

Of course if villain has us beat 50% of the time, villain will bet 75% of his range (assuming PSB), giving us 25% of the pot as game value. But if villain has us beat 50% of the time, we have quite a ****ty bluffcatcher.

BTW, adding a few nut combos to our range hardly changes the value of our bluffcatchers. It will limit the amount villain can bet. But even betting infinity dollers only gives 1 bluff for 1 value hand. So assuming being beat 20% of the time, if we add enough slowplays where villain is forced to decrease his bet size from infinity to a PSB we go from facing a bet 40% of the time to 30% of the time. Ergo a change in game value from 0.6 to 0.7. Not insignificant but not exactly groundbreaking either.

My wording was poor. When I said this:

I meant that it will benefit the player doing the bluffcatching, not the hand itself.

The terms are divided by three because you've added the EV of three different hands together and are trying to find the EV of a single hand. If you fail to divide the terms by three, you'll triple villain's EV when hero calls.

Right, so after three hands, villain has won $300 total, which is $100/hand.

If hero calls 100% of the time, villain's EV is $100/hand.

(200+200-100)/3 = 100

If hero folds 100% of the time, villain's EV is $100/hand, because the pot is $100 and villain simply claims it.

If hero calls 50% of the time and folds 50% of the time, villain's EV is $100/hand.

EV when hero calls: .5((200/3)+(200/3)-(100/3)) = 50 (note that if you fail to divide the terms by three here, the answer becomes 150, which is triple the correct answer, as previously mentioned)

+

EV when hero folds: .5(100) = 50

= $100/hand.