Scenario:
The game is Pot limit omaha. CO raise pot.
You hold 9864ds on the BTN and 3-bet. CO 4-bets.

• Let's say we have 100BBs.
  It is 25.5BBs to call the 4-bet and villain has 37.5% of his stack in the pot.
  
• We are almost certain villain got AA** which means we are not likely to be dominated by a better rundown

If we call another 25.5BBs PF with an equity disadvantage of

Hand Pot equity
AA ** 56.35%
9c8s6c4s 43.65%

56.3 to 43.6, how often do we need to hit the flop good enough for this to be EV + and do we hit the flop this often?

(37.5 * 2) * 0.4365 [Preflop] + (62.5 * 2) * X * Y [Post Flop] > = (100-12) [already invested 12]

Where X is the equity post flop and Y is how often we are able to continue.
This can not be calculated without some sort of integral calculus, but we can make a rough estimate.



Let's say we continue on all flops where we have odds to call his pot-sized shove on the flop.
We need around 30% equity cause the shove is a little less than pot.



Therefore we call on 68% of the flops. Now we calculate the average equity on these flops.

68%*30%+54%*10%+40%*10%+27%*10%
+16%*10%+8%*10%+4%*10% =

0.204+0.054+0.040+0.027+0.016+0.008+0.004 = 35.3%

And the triangle-shaped pieces

[(68%-54%)*10%+
(54%-40%)*10%+
(40%-27%)*10%+
(27%-16%)*10%+
(16%-8%)*10%+
(8%-4%)*10%+
4%*10%] /2

= [1.4% + 1.4% + 1.3% + 1.1% + 0.8% + 0.4%*2] / 2

= 3.4%

So overall, we have

35.3 +3.4 = 38.7% equity on average.

__________________________________________

Finally X = 38.7% and Y = 68%. The formual gives

(37.5 * 2) * 0.4365 [Preflop] + (62.5 * 2) * 0.387 * 0.68 [Post Flop]> =? (100-12) [already invested 12]

32.74 + 32.90 = 65.6 which is a lot less than the 88 we need

Conclusion: It is massively EV- to call the PF 4-bet

___________________________________________

Where are the misstakes?

er, uhhhhh...

This post does not belong in SMP (Science, Math, Philosophy) so I moved it here for mauling, castigation or ultimate destruction/deletion. Anyway, it certainly has more of a chance for useful answers involving poker/omaha here than in SMP.

-Zeno

Interesting - I've been calling here always, but mostly cause that's what DJ Sensei and RBK were doing in their vids

nice post,
dunno if it's relevant but
one point may be that there are some flops we easy can give up and so we save ~half of our stack.
also there are flops AA** is playing c/f

In your post flop equity calc of .387, you've calculated the average equity of the times you have the right pot odds to call, so by multiplying .68 aren't you effectively double-counting the percent of the flops you can continue? since you're average equity calculation was weighted by the percent flop, i think the 0.387 is a 'true' equity calculation assuming that 68% of the flops will give you the right odds.

also, why didn't you put in MMMM instead of 9864 since any middle rundowns are effectively the same.

I find it hard to respond to long posts. So I'll simply post this before I forget it.

You equate the (summation of Xi * Yi) to (N * Xavg * Y avg), which might not be true for the distribution of X and Y, especially at the edges.

Is there a way to get scientific notation to display properly on this site?

Also assuming that the 3-bet is inherently break-even, we have an equity overlay preflop as well which you fail to include in this computation. Or you could even argue that the 3-bet is inherently a -Ev proposition, ad you need to make up equity postflop.

You have not discussed your assumptions which seems that the flop is always a shove by the opponent, which is not always true, as there is an equity overlay because of bluffing/semi-bluffing involved.

ps: This is just from a quick read since you havent given many cliffs notes for your calcs.

______________
1) 88$ to win $200 @43.65% gives you a loss of 0.70$ as it seems. So are you askig if its better to shove or flat?

2) Why do you think that the area under the graph represents our average equity?
(more edits coming)

I don't have time to break this down in-depth right now, but I'm really suspicious of your little squares and triangles counting method that gives you 38% equity on average when you can correctly call a flop shove. In fact, I'm 100% sure that it's wrong. There's no way that your average equity on good flops can be lower than your preflop equity. I think you may actually need to divide that by .68 to get your average flop equity as GoChoo suggested.

I tried to make it as simple as possible.
This is hard enough for someone who hasnt read a single poker book

Something is wrong in the calcs for sure...was wondering what that might be and if its possible to fix it or maybe even the method is wrong?

Im asking about flatting his 4bet.

I assumed that the area represented the average equity but maybe it is wrong also.

The answer is to raise less when you 3-bet. Seriously.

But removing 68% seems wrong also...i think...

You got 38.7%(if my calcs are correct, which they probably arent...) on the flops you dont fold and if you dont include 68% its like you call on every flop with on average 38.7%, right?

scrap my last post.

Your presentation is spot on.

allow me this

The LHS of your equation is sklansky dollars of the call

The RHS is what you have left.

I dont see how you can equate the two.

you can compare the s$ of the call with the s$ of the fold( which I believe to be -24)

Allow me this:

You finish the hand Y times and win X *200
You do not finish the hand 1-Y times and lose 37.5

This has to be better than -12*2

X * Y * 200 - {(1-Y) *37.5 } >= --24

0.12 > -24

profit. Whats wrong ?

Now back to checking out the graph, although the area seems the right equity to be concerned about.

My guess is that you have to come up with a probability distribution function to accurately calculate the true equity, which is not a trivial task. Considering this problem appears to be non-linear...

Using your method, my interpretation is that you said that 68% of the time you have a 30% equity plus 54% of the time you gain 10% equity. If you're going to use that approximation, I think you also need to subtract the equity that is past 30% for the sake of consistency (undecided how precise it is yet).

I haven't excercise my math skills beyond excel for 10 yrs so I dunno how much more useful I can be in this discussion. Its a good start and I'm also interested in the results along with other situations.

I find it much easier to use to following formula:

EV(Post Flop) = CALL% [$Flop Pot Size * AvEq - $Amount to call on Flop]

CALL% = % we can call (around 30% as u stated in your example)
AvEq = Average all-in flop equity when calling

I cannot take credit for the structure of this formula as i read it somewhere else, the one i previously used was a little longer and similar to yours. however i cannot remember who/where it was posted (not on 2+2 i am sure), so apologies to the original source.

Ill make a very hastily and rough attempt at solving this. ill use an example of a .5/1 game so 1bb = $1 (and exactly 100bb stacks). after the initial 3-bet its $25.5 more to you ie; post flop EV must be >$25.50. thus on the flop you are calling $62.50 to win $137.50 = 2.2:1 = ~31% ev required (will use ur graph)

Im not posting my calcs as i simple cbf, but:

EV postflop = .68 * ($137.50 * .558 - $62.50)
=$9.60

which would imply that calling is indeed -EV
I used Quasars numbers for the most part, and also did this exceptionally fast, and therefore could be even more incorrect than the original

Think the .68 is OK, I did a similar calc in a post that a bit simpler not too long ago. The 38.7 is definately wrong though. Don't need advanced calculus, adding up area under curve by hand is correct, but you added wrong.

Looks like you're trying to do a Riemann sum (fancy term for adding up squares and triangles) but theres some weird 68% * 30% thing at front. Why are we adding in the equity for the times we fold? We're not calling 100% postflop.