For example, I open to $15, and get 3-bet by the SB to $50. I'm considering 4-bet bluffing to $115. I know my break-even % is 100/170, meaning I need vil to fold 58% of his 3-betting combos to break even. However, this doesn't consider future action and equities.

I know the range of this particular vil pretty well, and I know that at best he's only going to fold 33% of his combos, and when he does call my bluff hand is going to have 30% eq vs his range. Is there a way to factor this data into the inital bluff? Does this make sense? Is there a simpler way to go about all of this?

Put his calling range and your 4betting range in a solver using ~180 flop subset (or using all 1755 flops if you want exact number) and see what ev your hand has. Add that postflop ev to the ev when he folds preflop and subtract the ev when he 5bet jams and you get the total ev of the hand.

I don’t own pio or a solver. Is there a way to do this without?

I don't think you can get the exact answer in a reasonable amount of time with a reasonable amount of effort since you would just have to know the EV of all possible future actions in all scenarios and account for them.

You might be able to get away with some shortcuts if you have a database of your own play were you could use filters to get ev numbers and just generalize your EV returns on that but it's still only a guess as your stats/play can vary dramatically based on the games your playing, opponents, state of mind etc.

Edit: Plus depending on how many hands there are in your db you could have insignificant sample size to be confident your numbers were close to your "true" winrate for the scenario.

Would a slap-shot way of doing this be to find some bluffs that would have at least 40% eq v range?

You can use a semi-bluff shoving equation. You're not really shoving as the play will continue after he calls, however if this will be +EV then you can be pretty sure the whole play will also be +EV as you're in position.

EV = F($Pot) + C(%W*$W) – C(%L*$L)

F - times villain folds - 0.33
$Pot - the pot before we raise
C - times villain calls - 0.67
W - times we win when he calls - 0.3
$W - what we win when he calls
L - times we lose when he calls - 0.3
$L = what we lose when he calls

https://www.youtube.com/watch?v=ym1774AC5Ww
https://www.splitsuit.com/semi-bluff-shove-poker-math

The main question here is how do we calculate %W, %L, $W and $L.

Without that info the above equation is pretty useless.

There's no easy way to solve this.

You could use a solver to get an idea, but it also depends a lot on the skill levels and tendencies of the players.

Players don't always get to realize all of their equity (even if IP).

We open equilab let's say and put the equity of our hand against villain's calling range. It's not going to be perfect but we will get an idea.

Hot and cold equity would be pretty useless as an estimator preflop especially for 'bluffing' candidates in a 4bet range.

Actually, the assumption that you have 30% EV when called makes this a lot easier. What is missing is the 5-bet percentage. Here is my approximation:

When V folds (33%), H wins $53 (3-bet + BB)
When V calls, H has 30% equity in a $233 pot = $70. Less the $115 bet the EV is -$45.
When V 5-bets, the EV is -$115 (assuming a fold).

Just put in your probabilities, multiply through, and add up to get the total EV. If we assume 0% 5-bet probability, I get an EV of -12,73. Since folding to the 3-bet has an actual value of -15, this is slightly profitable. If you assume even a 10% 5-bet probability the EV is -19.72, so the play is unprofitable. With 20% 5-bet probability the EV is -26.71.

For the hell of it, flatting his 3-bet (unlikely) gives you 30% equity in a $103 pot, with $50 put in, for an EV of -19.1, so calling is slightly better than 4-betting if you assume a 10% 5-bet range... but still worse than folding.

All of this uses your "30% equity" as a proxy for your EV when you see a flop. As the aggressor you may feel like you have strong positive IO when he only calls. I don't think there is any reasonable quantitative approximation for post-flop play... at least not one where you have a clue as to the parameters.

My sense is that this only makes sense against a passive player who is unlikely to 5-bet.You know your V, but that kind of player would seem to have a pretty tight range to 3-bet OOP.

The problem I have with the calculation above is that it can give one the false sense of confidence in conclusions that are highly susceptible to fluctuations based on the assumptions you used to make the calculations.

This can cause you to cling to the numbers as proof of something rather than developing the skills to analyze a given situation and factoring in all information because your formula says it's correct.

By your logic, if the IP player had 100% equity with his hand in a certain flop spot, he should on average win exactly 100% pot (a bit more IP and a bit less OOP). That doesn't really make sense though does it? In reality he would win more like 300% of pot on average.

The same way if he only had 30% equity, he probably wouldn't win 30% of the pot on average, but should win way less.

This is not a problem with the calculation; it is a (potential) problem with the user. The thread title is "How do you calculate...". If OP doesn't know how to use the calculation he needs to start another thread . Seriously, proper use of model information is always an issue. Models provide insight; humans make decisions.

What you are trying to do is quantify the implied odds after the flop. Let's keep this simple. We know the outcome when V folds (+53) and when he 5-bets (-115). The question is assigning a value when he calls. We have a $233 pot with "30% EV". To me that means that if the cards run out with no more betting we win 30% of the time. That produces an EV of $70.

The betting after the flop creates implied odds. You seem to think that H has negative IO (he "should win way less"). Unclear to me. We are almost certainly at a huge range disadvantage, but we 4-bet and V just called. No sense making the play if we are going to shut down on the flop. Will he fold to a shove on the flop (simplest case)? If so we have realized positive IO; if he calls and we lose we have huge negative IO.

I don't think trying to calculate IO is productive or that any results will have much credibility. To JG's point, use the model to get useful info and then use your human brain to make a decision. In this case I would interpret the calculation results as: "This play is negative EV based on the likelihood V will fold and the value of the cards. I can only make this play if I am confident I have significant IO". We can assess whether we think that is the case, but at least we have the information we need to inform our decision based on our assumptions.

I will consider the question of calling the 30 raise. Facing a bet or raise can sometimes be best analyzed using implied odds. To simplify somewhat, with your bet and a raise to 30, I will assume the third player folds, not too unreasonable given the dry flop. You can show that calling the flop raise for the example hand is a -EV play using only the immediate odds.

Using an advanced implied odds model, I originally assumed if you hit the nut flush you will win 90% of the time if villain calls. The flush hit probability on the turn is 19.1% but pairing the ace gives you possibly a few more outs, so I raised the hit probability to 22% and reduced the win given hit prob. to 88%. If you don’t hit on the turn you will fold and lose the 20 you had to invest to see the raise plus the 10 call on the flop. (This may not be too realistic since the river card is still in play, but to include both turn and river action complicates the analysis quite a bit.)

Then I looked at various villain call probabilities for your turn bet assuming villain checks to determine the future bet required to give you the implied odds necessary to call the flop raise.

Villain Call% _ Future Bet …
50% 72 … 60% 62 … 70% 55 … 80% 50 … 90% 45 … 100% 41

So, under the conditions imposed with villain call probabilities ranging from 50% to 100%, the minimum future bet for +EV ranges from a bit more than the pot of 67.50 to about 0.6 Pot, depending on how likely villain calls.

Of course, one can argue some of the assumptions/values used but the point is that a calculation can be done. As stated by several others, doing this type of calculation offers insight for making an informed decision and usually should not be the lone guidance.

This is very true and well worded.

Also I should note I think just doing these types calculations for mental stimulation, learning, etc are all worth while endeavors in and of themselves even if I still somewhat object to using the model to justify a strategy.

Thanks. A few decades of experience preaching it to everyone who ran a regression and thought they knew something because they calculated an R-squared

Haha that would do it. Thanks again for the contributions!