Hello,

can someone check my math for the following real quick? I want to calculate the EV for us 3-betting and calling a shove.

Hero and Villain both have 50bb. Villain open raises to 3bb, we 3-bet to 9bb, 1.5bb are added to the pot by SB and BB. Villain folds to our 3-bet 70% of the time and 30% of the time he calls and we win 60% of the time when we're all-in.

is it:

1) EV(3-bet/call)= (3bb+1.5bb)*0.7 + [(50bb+1.5bb)*0.6 - (41bb)*0.4]*0.3


OR


2) EV(3-bet/call)= (3bb+1.5bb)*0.7 + [(50bb+1.5bb + 9bb)*0.6 - (41bb)*0.4]*0.3



I always get a little confused here. Can someone tell me which one is right or are both wrong?


Thanks!

The EV has to be calculated with respect to some reference point. The proper reference point is the point before making the 3bet. The alternatives before that point are folding to the 3bb raise or calling it.

This is where the problem comes, that calling it is hard to estimate (even if possible if we use some heuristic). And folding is a no way in hell option if we have as you suggested a 60% all in hand when called.

This makes any EV calculation not material as a number that can illuminate anything because it has to be calculated vs something unreasonable such as folding whereas the best is to calculate it vs the calling option that is the only sensible alternative. That is complicated also by the fact that if you called maybe the blinds get involved too. But imagine if you called and some of the blinds pulled a bluff (especially if sb first called lol) and then the first guy pushed thinking you have little and then you called and the blinds folded their call/steal attempt 3bet and now you make a ton more on avg. How do you estimate this possibility? How do we dare ignore it in the calculation if we wish to do the calculation in order to appreciate better the 3bet option as better alternative? It wont necessarily mean a lot to know the exact EV of that choice if we cant compare it with a proper alternative which folding is not.

In any case if we insist to calculate EV from the fold level then the calculation is ;

EV(3-bet/call)= (3bb+1.5bb)*0.7 + [(50bb+1.5bb)*0.6 - (50bb)*0.4]*0.3

(the 9 doesnt come into it because it is the same as if we pushed right away and got called then - that 9 is never dead chips potentially, once the move to 9 is made 50 is all we can lose and it happens 40% of the time of that 30% )

There is the additional problem here that we ignore the probability the person just calls the 3bet and doesnt play as fold/push only. This is unreasonable profile in my opinion but i accept it if you insist on it.

thanks a lot for your response. that makes sense and yes ignoring the players behind will make the calculation less accurate, but i wanted to keep it simple because i was already struggling with the math as you can see

but since you adressed the issue, lets say we assume there are 5 players left to act, and if one of them finds a hand of a range of lets say: TT+,AKs,AKo, which is 3.47% (probably a little less, since we have blockers, but lets ignore that)

there is a 5*3.47% chance, that someone behind 4bets or jams. is that correct? if we folded to that, how would that change our equation? would it just lower our fold equity from 70% to 70% - 17.35% ? would the rest look the same?


EV(3-bet/call)= (3bb+1.5bb)*0.5265 + [(50bb+1.5bb)*0.6 - (50bb)*0.4]*0.3



i know there would be more cases that would change the outcome, like a player behind 4-betting and Villain 5-bet shoving, but that would certainly make it too complicated and i think ignoring a few rare scenarios is negligible. it doesnt need to be a 100% accurate, i think thats impossible anyway, since we never know about villains real tendencies.

These things become harder to analyze if we do not know the exact hand you had which can affect also the reaction of the others (the probabilities) and then our final decision too. We could however design a general approach about various hands and see what happens.

If you raised 9bb and folded then you need to include a -9bb loss in the EV of the 3bet in these cases (to be determined how often now based on what hand you had and their style and numbers and stacks and what the other guy did after them - give me some more idea about hand or stacks of players maybe) as a special branch in the EV and reduce the other 2 weights of his folding or pushing in the previous calculation. If you specify things more we can put some expression to see.

Also the chance at least one does the 4bet can be calculated again based on what your hand is and roughly using the 1-(1-0.0347)^5~16.1% (1 minus the chance none of the eg 5 had such top hand - this is not an exact calculation but its accurate enough for our purposes given the overall uncertainties of the problem anyway).

Its best to specify the exact hand you had to see exactly how all this happens as a calculation because this is exactly what you will be doing from your perspective every time to see if some option is better than another or if you have to call even with if one pushed after you as function of what the other guy did also. The branches can get a bit more complex but its doable easily if you analyze how often each scenario happens. The various reaction probabilities are significantly affected sometimes depending on your own hand.

Just try a specific hand and we can put some numbers into it and see what it looks like based on the reads you have for how they roughly play. It all boils down to creating a more complex tree of possible outcomes and correctly evaluating the EV in each branch and the probability with which it happens.

thanks again, very helpful! okay then, can you show me how to implement the 16.1% and the 9BB we lose, once this occurs? let assume everyone has 50BB.

Villain open raises to 3BB from EP, we are MP and 3-bet to 9BB holding the AQo. Of the 5 players sitting behind, theres a 16.1% chance (might be less now since AQo blocks some of that 3.47% range?) that one of them 4-bet jams a range of TT+AKo+. If that happens we'll fold not getting the right odds therefore lose our 9BB. If all of them fold, villain folds 70% of the time and we win his 3BB + 1.5BB of SB and BB. The remaining 30% of the time when all 5 behind folded, villain 4-bet jams and we always call having 60% equity against his range.

This might not be entirely realistic but if you can you show me how to put it all together in one equation, I'd be very happy!

One last thing before i show you how i think about these situations using estimated probabilities and equities.

What do you use as starting range for the first guy that you assume pushes only 30% of the time over your 3bet? What is that pushing range that you estimated as happening 30% of the time (we need these details instead of defacto calling it 30-70% because we will derive the 30-70% or whatever it is from them instead more accurately).

Was this a 9 handed table and you had 5 acting after you or were you all 7? I mean i need to know the exact position of the opening guy to put a rough range estimate on them. Usually utg is 9-12% for example 9 handed for a size of 3bb, rarely 15% but maybe this guy was looser than that so you tell me.

Also we continue to assume that he never calls the 3bet to simplify things? If he had 50bb and he called 9bb, another 6 say above his raise and went to flop with 41bb left and the pot was 19.5bb then its not exactly a bad play always depending on his hand. I mean if he has AA,KK he can sometimes do that slowplay and get a cbet from you rather than a fold if he pushed. He can still take down QQ,JJ,TT,KK also usually on most flops or get them pot committed after they cbet and others that connected at flop or later. If he has AQ or TT or 99 etc it may make sense to call still if your range is not very tight because he pays 6 to see a 19.5bb pot and will connect or have a friendly flop often enough to justify it if he is careful.

As an example. If he had 88 and he called paying 6bb and assuming that he can manipulate you to put everything in later given you are both just 41bb behind and the pot would be already ~20bb at flop he can get a set 12% of the time. So he paid 6bb to try to win with close to 87% chance 41+20 and lose 41 13% of the time out of the total 12% of the time he has set (so 10.4% vs 1.56% overall). Then there are also some 4*4*4*4/C(50,3)~1.3% non set open ended straight flops that he has 40%-60% (depending on cards) vs your range. He also has about C(6*4,3)/C(50,3) ~10.3% flops without set and no overcard where he has ~50% equity vs your range. C(n,m) is simply n!/m!/(n-m)!


All those numbers suggest that paying 6bb for a 20bb flop with 41bb back is not exactly a bad idea if you can play well post flop. Push is not as great though for some of these hands. For example pushing 88 when you know you will be called always even if opponent had a rather wide 9% 3bet range, leaves only 43% equity or

0.43*(50+50+1.5)-47(what he had after his opening raise)=-3.355bb

Even if he assumed you fold 30% of the time it doesnt get good enough because he will be facing then some better range offering only 38% equity and leading to

0.38*(50+50+1.5)-47=-8.43bb when called and 9+1.5+3=+13.5 when you folded for a net result of

-8.43*0.7+13.5*0.3=-1.85 bb overall loss for pushing (fold is taken as 0 and i can argue call and careful postflop play even out of position is higher than that for many of the non pushing hands)

If your 3bet range was even tighter it gets even worse.

My point is there will be hands that play better as calls than folds/all ins for him to be unreasonable to break down his decision vs the 3bet to a fold or push. Some of the hands you think he would be folding because they are bad as push he could use to just call and see a flop putting both on edge of course but still better than folding.



You gave the other guys TT-AA,AK+ as 4bet after your 3bet. Do you for example give the original guy some similar a bit wider 99-AA,AQ range for push?

The problem i have with some of these pieces of information is a bit of self consistency.

For example it doesnt seem right to say you have 60% vs a pushing range when you hold only AQo even AQs or even AKs.

Here is why.

I use old pokerstove or you can use http://www.propokertools.com/simulations to see these equities.

AQo has 60% only vs some very wide for EP range like 25% such as 66+,A2s+,K6s+,Q8s+,J8s+,T8s+,A7o+,K9o+,QTo+,JTo

But notice that then not only this range is unreasonable for him to push vs your 3bet range for 50bb but its also unreasonable as opening range as well (or suggests if he were to fold 70% of the time that he opens like some 80% of hands if then you get 25% to push (or ~30% of 80)).

For example a hand like KTo for him has only 34% (even vs a rather wide in my opinion 11% 3bet range facing an ep raise with 5 left to act for as much as 9bb). Basically such hands are instant folds. Not only that but hands like JTo, A7o, K6s etc do not belong to say utg range vs 6+ others for 3bb size because they are often dominated by whoever calls or 3bets and the chance all folded is very small (you risk 3 to take very infrequently 1.5 and be behind when called or forced to fold to a 3bet? why?). You can use these of course from later positions.

If i were to imagine ranges (but you played there and know better of course) consistent with TT-AA,AK for the others as 4bets i would give the first guy some range no worse than 15% to open from that position even if loose. And then if he really were to fold 70% of the time you would be up against roughly a 4.5% range. Vs such range AOo has only 35.6%. Even if he opened 20% AQo has no more than 40% vs a 30% fraction of pushes. So you cant have 60% as imagined when you call him.

Did you mean to say the opponent would have the 60% equity then and not you? Because that sounds more reasonable.

Let me see some feedback further and we can post more when we fix these numbers to be more reasonable.

It may come out that AQo is not a better play as 3bet for 9bb even with 5 left to act. It is good that Q,A are helping as blockers but it may play better either with a smaller raise or a call. You might still wanted to raise anyway AOo if he was wide to open because of putting some fear in he others and to play a bigger pot in position vs him also. Yous ee AOo gets in toa tough spot if you call and then someone 3bets and the first guy pushes etc. Even vs 2 takes some effort to play. So maybe a call has a few problems but it depends on how tight his opening range is and what the others tended to do if they saw a raise and call for 3bb each.

From your perspective for example in order to call after doing that 3bet, you need to have over 41/(50+50+1.5)=40.3% vs his pushing range.

You can do that if he pushes say 6.2% like 88+,AJs+,KQs,AQo+

Just give me some feedback about his perceived open and push range and the exact position he had (how many left 6 or more?) plus prior image and how he saw you and we can see more.

hey thanks man, interesting to follow your thought process. are you a highstakes player? and well I guess you're right, it's unlikely someone ingame opens that wide and 4-bet shoves loose enough to give AQo 60% equity, i guess some whales might . maybe i shouldve clarified that this hand was never played out that way, its all hypothetical. i was just using numbers that would make it clear that we can call his shove easily while at the same time we'd always have to fold to a cold 4-bet shove of some of the players behind. and well yes, i guess there are many more things that can happen, like he can call our 3-bet, or some of the people behind could call, or min-3bet ect.
for now i'd be happy to know how to calculate our EV using these hypothetical numbers. can you show me how to implement the 16.1% times we fold to cold 4-bet shoves by the 5 players left to act (with the 9bb we lose) into the original equation?

Ok i will assume what is more reasonable to me even for a loose player that he raised 15% from the utg+2 position in a 9 handed table with 6 left to act and you are the next after him.

When i get the time i will try to create a model for how it all should work out.

Shall i assume still he never calls even if i showed you that to be better for some of the hands that face such 3bet?

Shall i create my own proper push ranges for him assuming you are like a 9% 3bet? I will use for others TT-AA,AK as you said to push after the 3bet. I will also show you how the A,Q you have (blockers) makes the reaction probability for them even smaller.

I will try to do that some time soon.


But in general in order to understand the logic here is how we evaluate the outcome of something in EV.

Lets say we have found out that there are 3 things that can happen 1,2,3 and they happen each with probability p1,p2,p3 adding up to 100%. We then evaluate what the effect on our stack is on avg in each of these cases. We do that by multiplying eg the resulting pot times our equity minus the cost of the action we took. If we folded we see this as 0 but if we did something that cost x and we then later folded in one of those scenaria then we have to count that branch as having -x as EV.

Lets say we have established that each branch results in some avg results like x1,x2,x3.

You then have the total EV that is:

EV=p1*x1+p2*x2+p3*x3


So all you need in such problems is to find those 6 (here) numbers, the probability of each branch to happen and the corresponding avg result when this happens. This is the total EV of an action that can be branched out in several outcomes.


You always calculate how the stack changes from the position you took the action. In this case your original stack was 50 so the EV for every branch will be the difference between the avg outcome of each branch for your stack minus 50.

For the guy that has used 3bb already the basis is now the remaining stack of 47 for what to do afterwards. After your 3bet if someone else pushed and its back to you your decision has to be evaluated assuming you have 41bb left. If you had decided never to fold it would be still in that branch from the basis of 50bb stack as you were evaluating the initial choice to 3bet. If you were considering folding if a specific thing happened (eg multiple pushes) then in that branch it happened your avg result would have been -9 for example.

As an example before the analysis of the prior case that offers to opportunity to see many more things try this simpler example;

Say you are the SB and have only BB to act. BB will either fold or push say in that simplified model because he doesnt want to call and then face a push by the other guy and not look strong given that he has smaller stack too.

Say that based on some analysis that we do not have to go into detail here (as we will do it for the other example) we have decided that this is what will happen next with our 9bb 3 bet with some hand that is not exactly perfectly fine to call anything (like say QQ-AA,AK would have been here). So it might have been AQ or AJs or 99. Say its 99.

We have decided that if the big blind pushes who is only 35 bb after this 3bet its a legitimate big hand that beat us and we can call only if the first guy folded if we had vs that pushing range (35-9)/(35+35+3)=35.6%. Our hand lets say has 37.5% vs such strong range and we can call then.

We have also decided that if the big blind folds and the the first guy pushes we need 41/(50+50+1.5)=40.4%. Say we do not have it vs that guy and we fold then.

Now if both go all in we decide to also fold obviously.


The big blind will go all in say 6% of the time and the first guy 30% if the big blind has folded. But if the big blind has called he will go all in only 20% of the time because he is afraid 2 people now.

Here is therefore what can happen after we 3bet to 9bb re-raising the original guy that had raised to 3bb.


1) Big blind folds and first guy folds; This happens with probability 94% times 70%=65.8%. The avg result then is a gain of 1+3=4bb

2) Big blind folds and first guy pushes; This happens with probability 94% times 30%=28.2%. We then fold as described. The avg result is a loss of 9bb

3) Big blind pushes and the first guy folds. This happens now with probability 6% times 80% =4.8%. We then call the big blind and have 37.5% in a pot of 35+35+3=73bb. The avg result is 27.375 that is higher than if we had folded (35-9=26). We have lost in that case 27.375-35 =-7.625 (its 35 because this is the maximum risk we have vs the big blind when we place at risk our 9 so we called the other 35-9=26). This is better than folding and losing -9.

4) Big Blind pushes and the first guy also pushes. This happens with probability 6% times 20%=1.2%. We then fold and the avg result is a loss of 9


Overall therefore the total EV of the choice to 3bet to 9 is;

4*65.8%+(-9)*28.2%+(-7.625)*4.8%+(-9)*1.2%=-0.38bb

Notice also that as expected 65.8+28.2+4.8+1.2=100.

The decision to 3bet to 9 in that case proved a slightly negative EV idea with that hand and these ranges assumed.

sure, i'd really like to know. obviously its best to use reasonable numbers. I didnt really think about it, I just wanted a formula that worked so I could fill in whatever numbers, although it would be interesting to see how a reasonable villain would construct his ranges. would also love to see how you
calculate the effect of card removal given our own holding.



so i guess in my scenario, its

1) villain raises, we 3bet, everyone folds (70% - 16.1%)
2) villain raises, we 3bet, players behind fold, villain shoves and we call (30%)
3) villain raises, we 3bet, get cold-4bet shoved, we fold and lose 9bb (16.1%)

or in other words:

EV(3-bet/call)= (3bb+1.5bb)*0.539 + [(50bb+1.5bb)*0.6 - (50bb)*0.4]*0.3 + (-9bb)*0.161



correct?

In your case we need to rederive these number to be accurate with the situation. But if we assumed a model of ;

Original guy folds 70% if it gets to him. The others push 16% (that is different because of AQ blockers but we can re-derive it later keep it 16 for now). The first guy then calls 20% of the time and folds 80%.

When the others push they have strong range, stronger than what the first guy has when he pushes so we fold then.


When the others fold and the first guy pushes we have 43% then when we call which is more than the needed (in this example but not the other before i gave) and we call.

We have ignored 3bet calling cases here which is generally a weak assumption for the first guy but lets ignore it here to learn how to do simple cases first.

Here is what happens then;

1) The others fold 100-16=84% of the time and the first guy folds 70% of the time therefore the total has a chance of the product 0.84*0.7=58.8%. We pick up 3+1.5=4.5bb then.


2) The others fold 84% of the time and the first guy calls 30% of the time therefore the chance is 0.84*0.3=25.2%. We have 43% equity then and the avg result is 0.43*(50+50+1.5)-50=-6.355 which is better than if we had folded and lost 9. This is why we called even if we knew we were behind.

3) The others push 16% of the time and then always we fold and lose 9 regardless of what the first guy did.


16+58.8+25.2=100

The EV is ;

0.588*4.5-0.252*6.355-9*0.16=-0.395

So the 3bet to 9 with that hand in this model is a negative idea.


If we did the math a bit better to find the proper probabilities using the removal effects of A,Q (or when i do it later if i get the time) things would get a bit better for the EV and it may turn positive.

i read your 2nd post too late, it was sent while i was typing. anyway, BIG THANKS!