some EV calc. help. Mojo?

I just need to make sure im doing this right.

I want to find out EV of a 4bet shove preflop with a specific hand.

Pocket Tens UTG for this example.

villain is 28/22 , 3b = 9.5% , Fold to 4bet = 50% , and 3bet is premium (QQ+,AK) = 50%

first I need to find out what % of his range is calling our 4bet shove.

50% of his 9.5% 3betting range so 4.75%.
{99+,AJs+,KQs,AKo}

50% of that 3bet range is premiums.
{QQ+,AKo}

so that means when he calls our 4bet shove:
50% of the time we have 51.93% equity. ( TT vs. JJ-99,AQs-AJs,KQs )
50% of the time we have 36.41% equity. ( TT vs. QQ+,AK)

so our average equity when called is 44.17%.

villain will also fold to our 4bet shove 50% of the time.

summary so far:

50% of the time we win 100%. ( villain folds his 3bet to our 4bet)
50% of the time we win 44.17%. (villain calls our 4bet shove)

now we go to the $$ EV.

we open 4xBB UTG, Villain 3bets to 13 BBs, table folds, Hero 4bet shove 96BBs more.

50% of the time we win (4+13+1.5)18.5BBs 100%.
50% of the time we win 201.5BBs 44.17%.
(201.5 x .4417 = 89) (89 - 100 = -11BBs)

50% of the time we win 18.5BBs.
50% of the time we lose (-11BBs)

so the EV of this play is +7.5BBs.

damn I hate typing this out, I just wrote it all out on paper. my first 10k on WPN was rough and Im going through my database to make sure im not overadjusting to actually having stats on specific players now.

Cmonnnn guys..bump..bump bump it up!

I don't understand how "3-bet is premium" works as a stat but I'm skeptical of its usefulness.

Anyway if you assume he's folding exactly 50% of the time and having QQ+ AK the rest of the time (equity of 36.1%), the EV is as follows:

Outcome 1: He folds
Probability: 0.5
EV magnitude: +18.5bb
Product: +(0.5)*(18.5)

Outcome 2: He calls, we win
Probability: 0.5*0.364
EV magnitude: +105.5bb minus rake
Product: +(0.5)*(0.364)*(105.5 - rake)

Outcome 3: He calls, we lose
Probability: 0.5*(1-0.364)
EV magnitude: -96bb
Product: -(0.5)*(1-0.364)*(96)

Then you sum the outcomes:

EV = X = (0.5)*(18.5) + (0.5)*(0.364)*(105.5 - rake) - (0.5)*(1-0.364)*(96)

At your stakes I have no clue what the rake would be, I'll assume 3bb. Just note that the rake does matter quite a bit if the shove is marginal.

Plug 3 for rake and use www.wolframalpha.com solver:

X = -2.62bb

Now, we can use a variable to find the breakeven point for how often he would need to fold for the shove to be profitable. We set EV equal to zero and the 50% fold frequency equal to X, while setting the 50% call frequency to 1 minus x.

0 = (X)*(18.5) + (1-X)*(0.364)*(105.5 - 3) - (1-X)*(1-0.364)*(96)

X = 56.2%

I just noticed you were assuming he was calling with more than QQ+ AK. I think the way you're going about figuring that out is very confusing and probably inaccurate. I think it would be absurd to assume he calls your shove with a hand like AJs or KQs. I'll universalize the formula for you so you can plug different values:

EV = (fold%)*(18.5) + (1-fold%)*(equity)*(105.5 - rake) - (1-fold%)*(1-equity)*(96)

"3bet is premium is a new stat in HM2, that keeps track of the times villain 3bets AND goes to showdown AND holds QQ+,AKo,AKs. im skeptical aswell because so many 3bet pots don't make it to showdown that I don't think the number will be 100% dependable.

it would be absurb renton if it wasn't 2nl 6max. lol hes def. calling 99+,AQs,AKo,AKs.

so using your formula: ( which thank you so much for by the way!)

EV = (.5)*(18.5) + (.5) * (.4127)* (105.5 - 4) - (.5) * (.5873) * (96)

is that correct?

then it would step down to
EV = (9.25)+ (20.94) - (28.19)
EV = 2

this also doesn't take into account when hes restealing vs. a wide BTN opener. I really just wanted to see if my thought process was correct for figuring out the EV of a play using stats. which it is not, lol so thanks again renton for pointing me in the right direction.

Yeah, that stat is worthless. But you did the formula right and I wouldn't be surprised if you had 41% equity when called at 2NL. I would, however, be surprised if you got him to fold 50% of the time at 2nl.

Also, don't do your own arithmetic, just use www.wolframalpha.com.

Originally Posted by Renton

Yeah, that stat is worthless. But you did the formula right and I wouldn't be surprised if you had 41% equity when called at 2NL. I would, however, be surprised if you got him to fold 50% of the time at 2nl.

Also, don't do your own arithmetic, just use www.wolframalpha.com.

yeah I forgot to put 2nl in the first post. at higher stakes this is not the case.

I have 1.5k hands on villain so I think stats are O-K.

ty for the site. will do, I have 9 more hands to do now. lol too my top 10 losing pots and making sure my play was correct. I will also do my top 10 winning but I start with my losers.

alright, Im sort of stuck now.

I have 2.5k hands on villain.

he opens pretty tight, 16.7/12.4 . where I want to exploit him though is his 3bet stats. he calls a 3bet 52%, F23B 39% , 4bet 9.1%.
his fold to cbet is 72%. his fold to turn cbet is 0%. if he sees a turn, he either calls or raises, (50/60% respectively)

I did the preflop EV like you did above. my problem is when he calls, how do we write that out?

Outcome 1 = we 3bet , he folds.
probability = .39
EV mag = +13.5
product = +(.39) * (13.5)

Outcome 2 = we 3bet, he 4bets, we fold ( for simplicity I guess, we always fold to 4bet since he only 4bets AKs,AA,KK)
probability = .091
EV mag = -13.5
product = -(.091) * (9)

Outcome 3 = we 3bet , he calls
probability = .52
EV mag = this is where im stuck. how do we breakdown the times he calls and folds to our cbet and the times we cbet and give up OTT. again for simplicitys sake were never putting money in on the turn.

in the end I want to know how often I can 3bet him with *bluff hands* and take the pot OTF profitably. theres a glaring weakness and I know how to exploit it but I don't know the math to double check my frequencies. thanks

*$25NL 6max too btw*

* edit I think I solved it*

outcome 3 = we 3bet, he calls, we cbet 13 into 19.5, he folds.
probability = (.52) * (.72) (.52=times he calls 3b, 72% is how often he folds to a cbet)
EV mag = +32.5
product = +(.52)*(.72) * (32.5)

outcome 4= we 3bet, he calls, we cbet 13 into 19.5, he raises or calls and we surrender.
probability = (.52) * (.28)
EV mag = -21 ( I am wondering about this number being correct, we raise 9 pre and cbet 13. we lose the 21 we invested, or the 32.5 in the pot?
product = -(.52) * (.28) * (21)

EV = (.39) * (13.5) - (.091) * (9) + (.52)*(.72)*(32.5) - (.52) * (.28) * (21)
EV = +13.5BBs right?

Before I show you how to simplify that I want to make sure you're aware of the basic method of calculating expected value for any event where the outcomes are known.

EV(action) = P(outcome 1)*EV(outcome 1) + P(outcome 2)*EV(outcome 2) + P(outcome 3)*EV(outcome 3) + .... for all outcomes of the action.

P is the probability of the outcome occurring, and EV is the value from that outcome. All P values should add up to 1, or 100%.

There are three immediate outcomes to 3-betting a hand preflop:

EV(3-bet) = P(fold)*EV(fold) + P(4-bet)*EV(4-bet) + P(call)*EV(call)

If we assume he opens the button for 4bb and you 3-bet from the BB to 12bb, he folds to the 3-bet 39% and 4-bets 9.1%:

EV(3-bet) = (0.39)*(5.5) + (0.091)*(-11) + (1-0.39-0.091)*EV(call)

I want to clear up that your EV of 13.5 when he folds to the 3-bet makes no sense, all you win is his raise plus the blinds, 5.5bb in my example.

As you can see, we have no value for EV(call). That value is an entirely separate EV problem that is actually quite complex, as we don't know what the flop will be, what our action will be, what his response to that action will be, and what turn and river runouts will come, not to mention the different action sequences of those runouts.

EV(call) = Sum of P(a million different things)*EV(a million different things)

...basically.

Theorists use the "R value" to simplify this problem. R stands for realization of equity. In this case I will refer to it as the percentage of the pot that you will win on the flop if your 3-bet is called. We know the pot size will be 24.5bb minus rake, if you assume a 4bb open and 12bb 3-bet. Your R value will be the percentage of that 24.5bb pot you expect to win, on average. This counts all the times you will make profitable bluffs, all of the times that you will get to check down the best hand, but also all the times you'll be able to get value from worse. It's expressed as a percentage of the 24.5bb pot, but that percentage can be over 100, and would be if for example your entire 3-bet range was AA, a hand with heavy implied odds against worse hands.

EV(call) = (24.5 - rake)*R - 11

So this expresses that you will win R% of the pot on the flop, deducting that you already lost 11 when your 3-bet was called.

The new full equation is as follows:

EV(3-bet) = (0.39)*(5.5) + (0.091)*(-11) + (1-0.39-0.091)*((24.5 - rake)*R - 11)

if you assume the rake will be 1.5bb,

EV(3-bet) = (0.39)*(5.5) + (0.091)*(-11) + (1-0.39-0.091)*(23*R - 11)

Now, we can set EV(3-bet) equal to zero to find out what your R will need to be when called to justify the 3-bet:

0 = (0.39)*(5.5) + (0.091)*(-11) + (1-0.39-0.091)*(23*R - 11)

R = 38.2%

first let me say, wow. lol I had to write it out on paper to fully understand what you were saying.
im still a little confused about why we are setting EV(3bet) to zero. what is that percentage? I mean I understand how why you need to know it to fully solve my original question, but what does that number represent? our break-even equity on the flop? our equity by the river? or is that how often I will win regardless of my holdings? thanks for taking the time to go through this with me btw.

Originally Posted by acg123

first let me say, wow. lol I had to write it out on paper to fully understand what you were saying.
im still a little confused about why we are setting EV(3bet) to zero. what is that percentage? I mean I understand how why you need to know it to fully solve my original question, but what does that number represent? our break-even equity on the flop? our equity by the river? or is that how often I will win regardless of my holdings? thanks for taking the time to go through this with me btw.

EV(3-bet) is the expected value of your strategic option of 3-betting with a given hand. It is what you ultimately want to solve for. I set it equal to zero because I want to discover the minimum hand necessary for 3-betting here. Thus, I'll know that this hand and all stronger hands will be +EV to 3-bet.

You could set EV(3-bet) to any value you want and solve for R to find the minimum necessary postflop value (R value) hand to 3-bet and have an EV of at least that. This is sometimes useful, as in the case BB vs BU facing a raise you have a third option: you can call.

If you face a 4bb open BB vs BU, you have three EV values:

1) EV(3-bet to 12), the expected value of 3-betting your hand to 12bb
2) EV(fold pre), this is always zero.
3) EV(flat pre), this is the EV of flatting

In order to maximize your value, you're going to VPIP with all hands that have a EV(3-bet to 12) or EV(flat pre) of greater than zero. Additionally, you're going to 3-bet to 12 with all hands that have an EV(3-bet to 12) of greater than EV(flat pre) or EV(fold). This will typically have you 3-betting your strongest hands for value in addition to some marginal hands as semibluffs. The semibluffs will have a lower EV(flat pre) or might be minus EV to flat at all.

So really, instead of solving for EV(3-bet pre) = 0, you might want to solve for EV(3-bet pre) > EV(flat pre), but this gets complicated. I'd stick to using zero for a while until you get the hang of this.

ok, so the 38% is the minimum equity needed vs. villains 3bet calling range.

I can see where EV(3bet pre) > EV (flat pre) will come in very handy later on as my stats converge. it seems most of the villains at my stake tend to play a "set" style or they play predictably based on their stats.
so let me try it then.

same stats, 16.7/12.4,calls a 3bet 52%, F23B 39% , 4bet 9.1%.his fold to cbet is 72%
villain opens 4bb from BTN.

1= EV(3bet to 12)
2 = EV(fold pre)
3= EV (flat pre)
im going to use the EV=0 formula

1= 38.2% ( not going to retype it.)
2 = 0%
3 = EV(flat) = hmm..this seems complicated as well. how would this work, just use range vs. range? ex. villain opens 30% OTB.
ive wrote like 5 lines and keep deleting them. how would this work renton?

No, you're not understanding it completely. 38% is the minimum amount of the pot you need to win when called. It might correlate with how much equity you have, but they are not the same. If you flop a 90% equity hand, such as a set, you figure to realize much more than 90% of the pot on the flop, because you will put bets in against hands that you dominate. If you flop a 30% equity hand, such as bottom pair, you will usually realize much less than 30% of the pot. Certain hands that you 3-bet will realize equity better than others, because they will flop more profitable hands such as TPGK or flush draws, oesds, and gutters, while others will realize equity poorly because they will rarely flop strong and often flop medium strength hands. This is why we 3-bet with 97s and call with KTo.

To answer your example question:

Villain opens button for 4bb and you're in the BB. You have three choices:

1) folding
2) 3-betting to 12bb
3) flatting for 3bb

1) EV(folding) = 0bb

This one is easy, since we know folding is always 0EV.

2) EV(3-betting to 12) = (0.39)*(5.5) + (0.091)*(-11) + (1-0.39-0.091)*(23*R - 11)

Copied from above. Again, R is the percentage of the postflop pot you will win, on average, when called.

3) EV(flatting for 3) = 8.5*R - 3

When you flat, the only outcome is that there will be postflop play in a pot of 8.5bb (minus rake). As before, R is the percentage of the pot you figure to win on average, and it is important to understand that this is not the same R as for 3-betting and getting called. That is a different pot vs a different range, and a different R altogether.

One more thing, the EV values will always be in big blinds or dollars, while the probabilities and equities will always be percentages (numbers between 0 and 1).

how would you solve for R in the flatting scenario?

It's very difficult, if not impossible, to solve R. R represents the postflop value of your hand, all of the profitable bluffs, value bets, and semibluffs you'll be making, subtracting all of the mistakes you make. It will be higher if your hand makes strong, dominating pairs, or if its suited or connected, or if your range is very strong otherwise and you'll thus be able to make a lot of profitable bluffs. The best you can do is estimate R, and the more you play the more you'll get a feel for how profitable your hands are to play.

I imagine the point of 3BPremium is to identify the players who use some of those premium hands to balance their Calling ranges. this guy seems to be playing straightforwardly with that stat (appropriate at this level).

Originally Posted by Timlagor

I imagine the point of 3BPremium is to identify the players who use some of those premium hands to balance their Calling ranges. this guy seems to be playing straightforwardly with that stat (appropriate at this level).

that was I was using it for, but quickly realized the sample size has to be pretty big to be useful.

Originally Posted by Renton

It's very difficult, if not impossible, to solve R. R represents the postflop value of your hand, all of the profitable bluffs, value bets, and semibluffs you'll be making, subtracting all of the mistakes you make. It will be higher if your hand makes strong, dominating pairs, or if its suited or connected, or if your range is very strong otherwise and you'll thus be able to make a lot of profitable bluffs. The best you can do is estimate R, and the more you play the more you'll get a feel for how profitable your hands are to play.

yeah ive done some research on google after I read this reply the first time and it seems impossible to truly find the value for R. I don't want to get into that, lol I just want to make sure im making correct plays in tough spots. I appreciate your help Renton you have definitely pointed me in the right direction and handed me a compass to find my way. lol

Sorry to have left you hanging, acg. I was spending all my time in the "Werewolf Village" forum here on FTR.
My team lost 'cause I made a couple of really bad choices.

***
Renton has lead you well, and the only critique I have for him is semantic.
He used "EV" when he should have use "value" a couple of times.
Here's what he wrote:
EV(3-bet) = P(fold)*EV(fold) + P(4-bet)*EV(4-bet) + P(call)*EV(call)

It should be:
EV(3-bet) = P(fold)*V(fold) + P(4-bet)*V(4-bet) + P(call)*V(call)

I just took the E's out of the RHS (right-hand side) of the equation. This is because we know the value of that play, and it's not a random variable.

I.e. If we bet, and Villain folds, we will win a non-random amount. That is a value.
When we multiply that value by a rate at which Villain folds, that's an expected value.

Originally Posted by Renton

I just noticed you were assuming he was calling with more than QQ+ AK. I think the way you're going about figuring that out is very confusing and probably inaccurate. I think it would be absurd to assume he calls your shove with a hand like AJs or KQs. I'll universalize the formula for you so you can plug different values:

EV = (fold%)*(18.5) + (1-fold%)*(equity)*(105.5 - rake) - (1-fold%)*(1-equity)*(96)

The equation may make more sense if presented like this:

EV = (fold%)*(18.5) + (1-fold%)*(something)

something = (equity%)*(105.5 - rake) - (1-equity%)*(96)

EV = (fold%)*(18.5) + (1-fold%)*( (equity%)*(105.5 - rake) - (1-equity%)*(96) )

EV = (fold%)*(18.5) + (1-fold%)*(equity%)*(105.5 - rake) - (1-fold%)*(1-equity%)*(96)


All I did was show why in the hell the (1 - fold%) term shows up twice.

Originally Posted by acg123

how would you solve for R in the flatting scenario?

You spend a decade making ever closer guesses and being wrong a lot. (OK, that's just cynicism.)

The number of variables in trying to determine the total value of a post-flop scenario from a PRE-flop knowledge base is potentially huge.

Some simple scenarios may be easy to solve for, though. If Villain is short stacked and only has a PSB left in his stack, we can examine post flop scenarios in which only one bet happens (a shove from Hero or Villain).
Still... we need to solve that for a LOT of different flop textures, then try to move on to turn and river cards where the money didn't go in on the flop.

It's a very big question and mostly Villain specific (I'm guessing). I have never spent too much time thinking about this other than in the broadest terms of feels good / bad.

Originally Posted by MadMojoMonkey

Sorry to have left you hanging, acg.
no worries sir lol

I was spending all my time in the "Werewolf Village" forum here on FTR.
My team lost 'cause I made a couple of really bad choices.

hate when that happens..lol

***
Renton has lead you well, and the only critique I have for him is semantic.
He used "EV" when he should have use "value" a couple of times.
Here's what he wrote:
EV(3-bet) = P(fold)*EV(fold) + P(4-bet)*EV(4-bet) + P(call)*EV(call)

It should be:
EV(3-bet) = P(fold)*V(fold) + P(4-bet)*V(4-bet) + P(call)*V(call)

I just took the E's out of the RHS (right-hand side) of the equation. This is because we know the value of that play, and it's not a random variable.

I.e. If we bet, and Villain folds, we will win a non-random amount. That is a value.
When we multiply that value by a rate at which Villain folds, that's an expected value.
makes sense. also defines EV which will be helpful. ty sir

k

Originally Posted by MadMojoMonkey

The equation may make more sense if presented like this:

EV = (fold%)*(18.5) + (1-fold%)*(something)

something = (equity%)*(105.5 - rake) - (1-equity%)*(96)

EV = (fold%)*(18.5) + (1-fold%)*( (equity%)*(105.5 - rake) - (1-equity%)*(96) )

EV = (fold%)*(18.5) + (1-fold%)*(equity%)*(105.5 - rake) - (1-fold%)*(1-equity%)*(96)


All I did was show why in the hell the (1 - fold%) term shows up twice.

I see it showed up twice, and this might be a dumb question but why does it show up twice?

I think R is fairly easy to approximate for hands which flop polarized equity. Unfortunately, in NLHE the only hands that flop truly polarized equity are 22-55. Those hands realize 90% of their equity from sets, and its just matter of how many bets you expect will go in when you hit one. I think finding the R for hands like those has something to teach us about other less-polarized equity distribution hands, though, as the amount they will win when flopping trips, straight, whatever is comparable to the value of a flopped set. You then have a significant piece of the R puzzle figured out for all hands.

This is all extremely Theoryland shit that probably has no business in a beginner's circle thread though.

Originally Posted by MadMojoMonkey

You spend a decade making ever closer guesses and being wrong a lot. (OK, that's just cynicism.)

The number of variables in trying to determine the total value of a post-flop scenario from a PRE-flop knowledge base is potentially huge.

Some simple scenarios may be easy to solve for, though. If Villain is short stacked and only has a PSB left in his stack, we can examine post flop scenarios in which only one bet happens (a shove from Hero or Villain).
Still... we need to solve that for a LOT of different flop textures, then try to move on to turn and river cards where the money didn't go in on the flop.

It's a very big question and mostly Villain specific (I'm guessing). I have never spent too much time thinking about this other than in the broadest terms of feels good / bad.

yeah, after renton explained the "R" in the call equation I realized im getting too technical, the point of this entire post is for me to able to check my plays vs. regs to make sure im not exploiting myself but doing a certain action in a certain spot too often or not enough.

* on the plus side, going through my HHs to find hands for equations I found bet sizing tells on 4 of the regs I play with very often. they are very reliable as well as most players have a tendency to bet certain ways with certain equities. after a large enough sample I was able to figure out what those represent which means more monies in my account.

Originally Posted by Renton

I think R is fairly easy to approximate for hands which flop polarized equity. Unfortunately, in NLHE the only hands that flop truly polarized equity are 22-55. Those hands realize 90% of their equity from sets, and its just matter of how many bets you expect will go in when you hit one. I think finding the R for hands like those has something to teach us about more equity distribution hands, though, as the amount they will win when flopping trips, straight, whatever is comparable to the value of a flopped set. You then have a significant piece of the R puzzle figured out for all hands.

This is all extremely Theoryland shit that probably has no business in a beginner's circle thread though.

but it helps me understand, which could help someone else another day. which is +EV. lol

Originally Posted by MadMojoMonkey

The equation may make more sense if presented like this:

EV = (fold%)*(18.5) + (1-fold%)*(something)

something = (equity%)*(105.5 - rake) - (1-equity%)*(96)

EV = (fold%)*(18.5) + (1-fold%)*( (equity%)*(105.5 - rake) - (1-equity%)*(96) )

EV = (fold%)*(18.5) + (1-fold%)*(equity%)*(105.5 - rake) - (1-fold%)*(1-equity%)*(96)


All I did was show why in the hell the (1 - fold%) term shows up twice.

Originally Posted by acg123

I see it showed up twice, and this might be a dumb question but why does it show up twice?

Start here:
EV = (fold%)*(18.5) + (1-fold%)*(something)

Either he folds, or he "not-folds." In this case, since he's responding to a shove, there is only 1 "not-fold" option, so we don't need to separate call and raise.

It would normally start like this
EV = (fold%)(value_fold) + (call%)(value_call) + (raise%)(value_raise)

Any of those variables in parentheses can be 0, eliminating that term from relevance.
In this case, Her has gone all-in, so Villain can not raise. This makes our equation look like this:
EV = (fold%)(value_fold) + (call%)(value_call) + (0)(value_raise)
EV = (fold%)(value_fold) + (call%)(value_call)

We know that our equation is incomplete if the %-ages don't add up to 100%, or 1.
So we know that
fold% = (1 - call%)
and also that
call% = (1 - fold%)
Renton picked fold% to work with and eliminated one variable for his choice.

So now the EV euqation (when Villain is responding to a shove) looks like this:
EV = (fold%)(value_fold) + (1 - fold%)(value_call)

So now we know the value of a fold (the dead money in the pot before Hero shoved) and we know (or hypothesize) Villain's fold%.
The thing we need to solve for now is
value_call = something

Well... We can work out the something... since there are no more bets allowed at after this action, that makes it purely an equity analysis.
something = EV when called = (equity%)*(value_win) - (1-equity%)*(value_lose)

We use the same principle above and assume there will be no ties (which gives the mathematically same answer as if there ARE ties, so we're not even approximating, here).
That allows us to shorthand equity%_Hero in to just equity%, since we know that either Hero wins or Hero loses, so only Hero's equity is important. We could get the same results by using Villain's equity, but we're looking at it from Hero's perspective, so we'll choose Hero's equity.

OK, taking it back. Now we have this:
EV = (fold%)(value_fold) + (1 - fold%)(value_call)
and this:
value_call = (equity%)*(value_win) - (1-equity%)*(value_lose)

So we substitute the 2nd equation into the 1st.
EV = (fold%)(value_fold) + (1 - fold%)( (equity%)*(value_win) - (1-equity%)*(value_lose) )

The (1-fold%) term distributes through the 2 terms in the (value_call) substitution.

Originally Posted by MadMojoMonkey

EV = (fold%)*(18.5) + (1-fold%)*(equity%)*(105.5 - rake) - (1-fold%)*(1-equity%)*(96)


All I did was show why in the hell the (1 - fold%) term shows up twice.

It can be contracted even further:

EV = (fold%)*(18.5) + (1-fold%)*((equity%)*(201.5 - rake) - 96)

I just chose the long form because it more obviously displays the different outcomes and how they relate.

yesss I get it now.
the (1-%) is just giving us the remaining equity for the calculation.
(value_call) is when a flop happens, we need to calculate how often we will win/lose(equity) and how much (BB or $) to get a single (+ or - value) to plug into the full equation.

example: hero facing a 4bet shove.(villain opens UTG4x ,we3bet 12x,allfold,he 4bet shoves, 100bbstk)
villains range is TT+,AK,AQs
hero 3bet =88+, ATs+, KQs, AQo+, (86 combos) 100%
call4b=JJ+, AKs, AKo (40 combos) 46.5%
fold=TT-88, AQs-ATs, KQs, AQo (46 combos) 53.5%


EV= (fold%)*(-12) + (1-fold%)*(value_call)

value_call= (equity%)*(value_win) - (1-equity%)*(value_lose)


EV=(.535)*(-12) + (1-.535)*(value_call)

value_call= (.5735)*(95) - (1-.5735)*(95)

EV=(.535)*(-12) + (1-.535)*((.5735)*(95) - (1-.5735)*(95))

EV=.07

neutral EV then correct?

now I want to manipulate the ranges a little, say I call with only JJ+
fold = 72.1%

EV=(.721)*(-12) + (1-.721)*(value_call)

value_call= (.6198)*(95) - (1-.6198)*(95)

EV= (.721)*(-12) + (1-.721)*((.6198)*(95) - (1-.6198)*(95))

EV=-2.3

that didn't work. try TT+,AQs+,AKo 50 combos /86 = .581 (1-.581) = .419

EV=(.419)*(-12) + (1-.419)*(value_call)

value_call= (.5348)*(95) - (1-.5348)*(95)

EV= (.419)*(-12) + (1-.419)*((.5348)*(95) - (1-.5348)*(95))

EV= -1.18

fuck I expected to able to do it right the first time. ahhh cut down the 3bet %. duh . lol
3bet = hero 3bet =88+, AQs+,AKo 62 combos
fold = 88-TT,AQs 22 combos .354
call = JJ+,AKs,AKo 40combos

EV=(.354)*(-12) + (1-.354)*(value_call)

value_call= (.5735)*(95) - (1-.5735)*(95)

EV= (.354)*(-12) + (1-.354)*((.5735)*(95) - (1-.5735)*(95))

EV= +4.77

yesss

If you only want to know with what hands to call the 4-bet shove, then we're doing this the wrong way.

Villain's range is { TT+,AK,AQs }.
Hero faces a bet of 88 into a pot of (113.5 - rake)
bet/(bet+pot) = 88/(201.5 - rake) ~= 44%

Hero calls with any hand which has at least 44% equity against Villan's range.
{ JJ+, AKs }

Knowing the rake on a pot this size would help narrow down that 44%.

***
If you want to know what range to 3-bet, then we also need to know:
What's Villain's range to open UTG?

Originally Posted by MadMojoMonkey

If you only want to know with what hands to call the 4-bet shove, then we're doing this the wrong way.

Villain's range is { TT+,AK,AQs }.
Hero faces a bet of 88 into a pot of (113.5 - rake)
bet/(bet+pot) = 88/(201.5 - rake) ~= 44%

Hero calls with any hand which has at least 44% equity against Villan's range.
{ JJ+, AKs }

Knowing the rake on a pot this size would help narrow down that 44%.

***
If you want to know what range to 3-bet, then we also need to know:
What's Villain's range to open UTG?

{22+, ATs+, KQs, AJo+, KQo}


I like where this is going.

This is effectively a heads-up situation, which I just want to point out before we get into this. This is a vital fact when we analyze Hero's 3-bet. We are assuming that all people left to act behind Hero will fold, and this is certainly not always the case (unless Hero is BB).

Villain raises 4BB w/ 146 hands { 22+, ATs+, KQs, AJo+, KQo }

Hero raises to 12 BB w/ x hands { XX? }

If Villain does not fold, Villain will shove all-in. <- This is a major assumption.

Villain folds 96 hands @66% { 99-22,AJs-ATs,KQs,AQo-AJo,KQo } and shoves for 96 BB with 50 hands @ 34% { TT+,AQs+,AKo }.

Hero faces a bet with alpha of 44%, and calls 88BB with 28 hands { JJ+, AKs }.

How many hands does hero fold, so that he's not exploited by the shove?

***
When Villain shoves 96BB into a pot of 17.5BB, Hero wants that to be at most 0EV for Villain.
Villains has { TT+,AQs+,AKo } and bets into Hero.
Villain's EV to shove = (fold%)(17.5BB) + (1 - fold%)(something)

something = (42%)(105.5BB) - (58%)(96BB) = -11.4BB
(Using Equilab for pre-flop equities with those ranges)

substitute back
Villain's EV to shove = (fold%)(17.5BB) + (1 - fold%)(-11.4BB)

Set it less than or equal to 0 and solve for fold%. (We want to remember that right now we're looking at Villain's perspective.)
(fold%)(17.5BB) + (1 - fold%)(-11.4BB) <= 0

distribute the (1 - fold%)
(fold%)(17.5BB) + (-11.4BB) - (fold%)(-11.4BB) <= 0

clean that up
(fold%)(17.5BB) - 11.4BB + (fold%)(11.4BB) <= 0

collect terms with (fold%)
(fold%)(17.5BB + 11.4BB) - 11.4BB <= 0

move the known values to the RHS
(fold%)(17.5BB + 11.4BB) <= 11.4BB
fold% <= 11.4BB/(17.5BB + 11.4BB)

Now just solve it
fold% <= 39%

So now we know that in order to make Villain's 4-bet shove -EV, Hero needs to fold LESS than 39% of the time.

***
Hero will call with 28 hands, from the x hands in { XX? } that Hero 3-bet with.
So x = (y + 28)
where y is the number of hands Hero 3-bets, but folds to a 4-bet shove.

fold% = y/x = y/(y + 28)
and we want that less than 39%
y/(y + 28) <= 39%
y <= (39%)*(y + 28)
y <= (39%)*y + (39%)*28
y <= (39%)*y + 11
y - (39%)*y <= 11
(1 - 39%)y <= 11
y <= 11/(1 - 39%)

y <= 18.2

So Hero wants to fold less than 18.2 hands... let's use 18

Woo hooo. Don't it get you all chubby right around now?

29 + 18 = 47 hands to 3-bet for Hero.

{ 99+, AQs+ } is 44 hands.


(Open to revision... I haven't done this type of calc in over a year.)

Hero raises to 12 BB w/ x hands { XX? }
x <= 47
{ XX? } = { 99+, AQs+ }
(which is 44 hands, which is less than 47 hands, so ensuring Villain's shove is -EV, so hopefully the rake doesn't screw you both.)

I didn't actually use Villain's opening range in any of that.

It turns out your 3-bet range is based solely on villains habit to fold or shove, the range of Villain's shove, and the ESS.
(in this model)

Spoonitnow may have to remind me of some things.

I think this model is to prevent hero from being exploited, as opposed to a model which maximizes exploitation of Villain.

Did you go and sob into your pillow over that post?

Was it informative?

I'd hate to think I did that for nothing.
Most of that is just algebra. All I did was start at the end and work my way back to answer your question. All of the actual equations are EV calcs, which you should be starting to be comfortable recognizing and even writing out for yourself.

Originally Posted by MadMojoMonkey

Did you go and sob into your pillow over that post?

Was it informative?

I'd hate to think I did that for nothing.
Most of that is just algebra. All I did was start at the end and work my way back to answer your question. All of the actual equations are EV calcs, which you should be starting to be comfortable recognizing and even writing out for yourself.

lmao! nah i was playing..and life. sorry "sir mojo". lol

so sick man, i wish i could remember all this stuff. i definitely write everything down, i keep a binder for poker calcs, player profiles, exploits etc. this will be added pronto! and i will repost the same calcs with different ranges to make sure i fully grasp the concept. thanks for the help MMM.