I want to know if my expected value calculation for the upcoming scenario is correct. Assume that there is some villain who underfolds on the turn and overfolds on the river. We hold a pure bluff and want to exploit villain, however, we are unsure how often he has to fold in order to net us a profit.

Only considering the river in vacuum, we know that

p(fold river) > bet(river) / (bet(river) + pot(river)),

but what about the money we've invested on the turn already? This should not be part of our winnings.

For simplicity, assume that p(fold turn) = 0 (villain never folds aka underfolds). Our expected value is

ev(bluff river, p(fold turn) = 0) = p(fold river) x (pot(turn) + bet(turn)) - (1-p(fold river)) x (bet(turn) + bet(river)) > 0

<=>

p(fold river) > (bet(turn) + bet(river))/(pot(turn) + 2x bet(turn) + bet(river)).

Compared to the first equation, we need a lot more folds from villain on the river to make our double barrel bluff profitable. Am I right?

If so, my conclusion is that only because someone is slightly overfolding the river (compared to gto), it does not mean we should force this spot at any cost. The overfolding leak needs to be significant such that we can deviate from our baseline strategy.

Of course not at any cost, but if they are folding ~alpha ott, then going into future node where its profitable makes a lot of sense. It's really not going to happen where villains are folding way less than alpha ott, but way more otr... usually their defense is going to be within a few % of alpha ott with a big discrepancy for river.

For the situation you've described where probability of folding turn p(fold turn) is 0, then if hero uses the bet size of 100% for turn and river, then:

EV(river) = p(fold river) * (pot river) - p(call river) * (risk)

Here we want to solve for p(fold river) and since villain folded turn 0%, then we need to make up for the bet we made on turn of 1.00, so we can set EV(river) = 1.00, or substract 1.00 from the right side of the equation.

EV(river) = (% fold river)*(3) - (1 - % fold river)*(3) - 1.00 = 0

risk = 3 because in this example we bet turn for 1.00 into 1.00 and river size is 100% and river pot is 3.00 units. The - 1.00 is just adding the extra bet we always lost ott.

Let x = (% fold river)

1.00 = 3*x - (1 - x)*3 = 3x - 3 +3x

4 = 6x

x = 4/6 = 2/3 = 66%

so if villain *always* called turn then we would need him to fold 67% otr assuming our bet size was 100% of pot and same ott and otr. In reality if his folding freq ott is ~alpha, then any increase in alpha for river should result in decent EV

This would actually be a good thing for tombos to cover in one of his GTOwiz videos, I'll send him a message.

My understanding is that we will gain EV if villain underfolds turn and then calls "GTO" (false GTO) on the river.

Say that in GTO villain starts with 20 combos on the turn and calls 50% on the turn (10 combos) and then 50% on the river (5 combos).
So calling the river with 5 combos is "GTO".

Say that villain only folds 9 combos on the turn: 9 / 20 = 45%

Say that villain now calls the 5 "GTO" combos on the river, which means he folds: 6 / 11 = 54.5%

Turn EV: $100 * 45% - $100 * 55% = -$10
River EV: 55% * ($300 * 54.5% - $300 * 45.5%) = $15
Total EV: +$5

*******

Or in the OP example where villain folds 0% on the turn, so he ends up folding 15/20 = 75% on the river if he only calls the 5 "GTO" combos.

Turn EV: $100 * 0% - $100*100% = -$100
River EV: 100% * ($300 * 75% - $300*25%) = $150
Total EV: $50

********

River size does not seem to matter?

If we bet 1x pot on the turn and then 1.5x pot on the river, GTO would be to call river 40% = 4 combos
He arrives at the river with 20 combos, that is 4/20 = 20% call = 80% fold

River EV: 100% * (300*80% - 450*20%) = $150 = same as when we bet river 1x pot

Thank you for clarification. Given a 100% pot sized bet on the river, we need villain to fold 50% from a vacuum perspective. Taking the turn into account this % goes up to 66, so we need 66%-50% = 16% more folds, when we double barrel. That's not too much, but it's still quite a bit.

This is effect might be reinforced when we take more streets into consideration, e.g. the maximum distance from preflop to river. Then we need significantly more folds. My main takeaway is that river leaks are hard to exploit. When we reach a particular river spot, we can bluff with all of our low sdv hands, however, the number of bluffs is bounded by our baseline strategy (gto). Increasing the number of river bluffs by playing more hands preflop (say any2) is basically impossible to implement, because the required fold probability is far too high.

It is quite a bit... but it is taking the most extreme scenario-- where villain folds 0% ott. Him folding turn 30-40% which is still quite a bit below 50% would make our required river folds < 66%, still >50%, but quite a bit lower than 66%.

And I disagree with the statement, "river leaks are hard to exploit." River is going to be the biggest the pot ever is, so small deviations from equilibrium can result in large delta for bb/100.

Hard to exploit in a frequency sense. For one particular river spot bb/100 might be high due to the size of the pot and villains overfolding leak, however, this does not take into consideration how often you reach this node. Until now I thought that one wants to push as hard as possible to reach this lucrative river spot right from the start (preflop). Now I think that one basically can not adjust his preflop baseline (gto-ish) strategy in order to exploit someones river overfolding leak. I still think this is a somewhat strange result, because it seems rather counter-intuitive, at least to me.

Well, you definitely cannot just start opening any two cards, but you would certainly be able to open hands that are ~0EV if people have multiple exploits post flop with all else being equal. There also could be circumstances such that Hero should be tighter than equilibrium preflop, but since we're specifically talking about overfolding then we would be wider. The individual doing the exploiting is also encouraged to reach future profitable nodes at higher than equilibrium frequencies, but obviously you can't just do it with everything. Hands that are mixed and equal EV in theory between betting/checking on turn where in the check node villain continues to play theoretically correct so there is no EV gain going into this node, Hero would certainly be incentivized to continue betting to get to the future profitable node. Depending on the EV to be gained going into the future node would determine how wide/with what hands Hero would want to push forward into it.

The easiest way to solve these types of problems is just to keep track of contributions to the pot.

For example, consider a triple barrel bluff, pot-pot-pot sized bets with a starting pot of 1.

Villain always calls flop and turn. How often do we need them to fold the river?

• Starting pot = 1
• Villain's contribution so far = 1 + 3 = 4
• Hero contribution if we bluff river = 1 + 3 + 9 = 13

So on the river we are risking 13 to win 5 (villain's contribution so far plus the starting pot)

alpha = risk / (risk + reward) = 13 / (13 + 5) = 72.2%

So if they always call flop and turn, we need them to fold at least 13/18 = 72.2% on the river for the triple barrel bluff to be profitable.



Let's try another example where villain folds 60% on the river:

We can calculate that triple barreling is not the most profitable line, losing 2.2 pots total. You'd be better off just giving up on the flop. However, you must pull the trigger on the third barrel if you have already double-barreled.



The intuition here is that your river bet is profitable; they're overfolding 10% on the river. But it's not profitable enough to recuperate your flop and turn bet.

Here's a spreadsheet calculator you can use the play with this concept.

You can input different bet sizes and fold% for all streets and find the EV of different lines.



https://docs.google.com/spreadsheets...it?usp=sharing

I like this way of thinking about it and thanks for the calculator!

If your turn bet is losing xbb, you can just add that to river size bet to get required fold frequency. (I didn't check if this is truen,i think it is )

In general if river is over folded by any significant amount, and there is some fold equity and/or your hand has equity when called barreling will be very +ev.

It is true, and this is a much better way of conceptualizing it imo.

Also vs weak opponents you size up smaller ott if you are planning to bluff most rivers. That way they get to river with wider range whcih in most cases means higher fold equity.