I do like the content of video, however I see a flaw in your math. You concluded that in the top 54.4% of flops your equity would average 60%. Multiply the 2 and your equity is around 33.2% on just those flops. The conclusion preflop was that your equity vs AAxx would hover around 32%. In the bottom 45.6 % of flops your equity should be at least 11%, that would add at least 5% to total equity.

Subtracting this 5% from total equity of 32% leaves you with only 27% equity from top 54.4% of flops, which is slightly worst than a flip.

If we plug these numbers in .45 (0)+ .55(.50x8300 - .50 x 2000) = $1732.50

This means we recoupe only $1732 of $2100 preflop call. The numbers can be tweeked a bit on postflop calling frequencies, but still looks like a losing call preflop.

Hey Zach, nice video :)

If you want to do integration from the graph, you may need to break the rectangles down into smaller intervals. You're essentially doing a Middle Riemann Sum, although with some other estimation methods mixed in. Given small enough intervals, it will eventually give you the exact result though.

However, you may want to know that PPT OO can do the integration for you. Just plug this in, and run equity stats:

When I run that on your example I get an average equity of around 52%, which unfortunately makes the preflop call a losing play.

PokerJuice also does the same thing:

Unfortunately, still a negative EV preflop call :(

This was an outstanding video with valuable, practical applications. Well done, good sir!

@ ~16:00, I was hoping to get some clarification from anyone on Zac's EV analysis, as I think this is important. As I understand, the practical definition of EV is all outcomes times their associated probabilities. But to determine the full, net expectation, you must considering the betting and deduct what you risk. So, here is application from hand 1 of Zac's video:

EV(flop,call) = .55(.60*10300) -2000 = 1,399
Assuming villain stacks off on every flop:
EV(prf,call) = 1,399-2100 = (700)

This is a significantly different result, so it would be lovely if someone could verify the correct method, as I am not 100% confident in my own.

edit: correct EV calc = .45(0) + .55(.60*10300-2000) = 2,299
which is number Zac got, just different format. I stand corrected.

U only lose the 2000 55% of the time not 100%

Cliff:
- Op reminds us why calculus is cool ie talking about area under the curve and some straight up math wizard shit.
- Ev of calling pf with kkxx vs aaxx is ev of (flop play) - (pf call)
- Very low spr vs AA is better than medium spr, because we don’t need much equity to stack off.
- Very high spr is still the best, because we “leveraged harder”
- Lakers sucks because Jeremy Lin spend more time boat up in PLO instead of shooting in the gym
- Poker juice is really cool

Thank you for making this video. I really like the detailed analysis you have on each spot.

I am going to go through and address these comments 1 by 1. Thanks for the feedback and I would not be surprised if I made some errors; I apologize if I have.

I do like the content of video, however I see a flaw in your math. You concluded that in the top 54.4% of flops your equity would average 60%. Multiply the 2 and your equity is around 33.2% on just those flops. The conclusion preflop was that your equity vs AAxx would hover around 32%. In the bottom 45.6 % of flops your equity should be at least 11%, that would add at least 5% to total equity.
Subtracting this 5% from total equity of 32% leaves you with only 27% equity from top 54.4% of flops, which is slightly worst than a flip.

I don't fully understand your wording but I do see that 60% seems too high given that having 60% on 55% of flops yields more equity than we have overall on the full spectrum of flops. I should have caught this as a sanity check.

Hey Zach, nice video :)
If you want to do integration from the graph, you may need to break the rectangles down into smaller intervals. You're essentially doing a Middle Riemann Sum, although with some other estimation methods mixed in. Given small enough intervals, it will eventually give you the exact result though.
However, you may want to know that PPT OO can do the integration for you. Just plug this in, and run equity stats:

When I run that on your example I get an average equity of around 52%,
which unfortunately makes the preflop call a losing play. PokerJuice
also does the same thing:

Thank you very much for showing me a more accurate and quicker way to approximate my equity in situations like these. I will go play around with it. IIRC this isn't the first time you helped me with something similar.

This was an outstanding video with valuable, practical applications. Well done, good sir!
@ ~16:00, I was hoping to get some clarification from anyone on Zac's EV analysis, as I think this is important. As I understand, the practical definition of EV is all outcomes times their associated probabilities. But to determine the full, net expectation, you must considering the betting and deduct what you risk. So, here is application from hand 1 of Zac's video:
EV(flop,call) = .55(.60*10300) -2000 = 1,399
Assuming villain stacks off on every flop:
EV(prf,call) = 1,399-2100 = (700)
This is a significantly different result, so it would be lovely if someone could verify the correct method, as I am not 100% confident in my own.

Pokertroy77 answered for me. Thanks.

I am going through the math and analysis again and will report and update what I find.

Thanks

Hey my first post here. I just couldn't resist to comment..
I enjoyed the video and I did some math on my own because some of your conclusions were suspicious.
I'm talking about conclusions about low SPR, med SPR and high SPR ( vs AA) and that medium SPR is the worst because we have to fold a lot of equity while low and high are better.
That sounded a little suspicious at first glance( I studied math), so I did the math for the case that on the flop you have SPR = 1 and that we continue 38% of the time and our average equity was about 70%.
Calculated EV is still higher than the cost of preflop call.

Your conclusion was that you fold a lot more equity with lower SPR. That is true, but you also have higher average equity and you have to include both variables into calculation.

So is there something that I am missing here?
To me it seems that right logic is(in theory), the higher the SPR , the better it is for us, since AA is always favorite against our hand preflop, so we would like to put as little money as possible to see the flop. In practice it is little more complex.
For example , at 1 SPR I see people folding naked AA on lets say flush flops, or 89T flop...

Anyway great video, I learned something ;)

I wanted to add that I think Zac is one of the most informative instructors RIO has to offer. His videos focus on analytical methods and not just the tired, old HH reviews or session replays. I find this to be much more valuable.

This discussion is positive and Zac's openness to relook the spot is really cool as well. Lets keep discussion going!

Thanks again to all of you for pointing out the error in results. The box counting (approximating integration) was done incorrectly. Regardless, the tools that Jonna pointed out to me are clearly superior for estimating equities on subsets of flops. The results he showed look correct and will make the PF peel a losing one. This doesn't surprise me. It was the very reason I chose to analyze this spot. (I wasn't sure if I should call or fold PF).

Hey my first post here. I just couldn't resist to comment.. I enjoyed
the video and I did some math on my own because some of your
conclusions were suspicious. I'm talking about conclusions about low
SPR, med SPR and high SPR ( vs AA) and that medium SPR is the worst
because we have to fold a lot of equity while low and high are better.
Yes I maintain that when peeling vs AA order of preference is High SPR, Low SPR, Medium SPR. The thresholds that define high medium and low will differ for each hand.

That sounded a little suspicious at first glance( I studied math), so
I did the math for the case that on the flop you have SPR = 1 and that
we continue 38% of the time and our average equity was about 70%.
Calculated EV is still higher than the cost of preflop call.
Yes I arrive at approx the same result.

Your conclusion was that you fold a lot more equity with lower SPR.
That is true, but you also have higher average equity and you have to
include both variables into calculation.

No, my point was the opposite. At low SPR we fold the least often and even though we are getting in money in on the flop as a bigger average dog, the amount we get in is less.
That said, I suspect that there is no low SPR that we can profitably peel vs AA holding a hand that is bigger than a 2 to 1 dog AIPF. For that reason let me use a different hand vs AA to show my point. Below is analysis of T872$ss vs AA.
You can see that EV starts declining as SPR increases until an inflection point where it starts increasing dramatically.

So is there something that I am missing here? To me it seems that
right logic is(in theory), the higher the SPR , the better it is for
us, since AA is always favorite against our hand preflop, so we would
like to put as little money as possible to see the flop. In practice
it is little more complex. For example , at 1 SPR I see people
folding naked AA on lets say flush flops, or 89T flop...

Logic tells me otherwise and I think I have shown that. Think of it like this: When our average allin equity is <50% we want SPR as small as possible. Once SPR is high enough such that our average allin equity is >50% we want SPR as high as possible.

Do you think AA26o will shove KQJ flops, where your range was appropriate for potting OOP v the full table?

very good video,

just curious that in hand 2 on the River you didn't consider him bluffing with any A:hh since he will peel all of them on the turn and they have significantly less SDV then AK . It may still be a correct fold depending on reads and gameflow but I assume that would alter the distributions quite a bit.

Zach,

You don't have to count boxes. You can use the 'require that' conditional on OO to find the mean equity of KKQ9 the times you get it in. You set the 'require that' equal to the min required equity, in this case 19%.

http://gyazo.com/58ef86a5af927d6fe07a3d8e35a23bdf

edit: looks I'm late to the party

Seems like your method of integration is realy mess up. You basicly count the high of each restangle in 2 diffrent ways(looks that your numbers are misteriusly similar) and than multiplay insted of multiplay the hight and the width (here 0,1 for each i belive).

Question is : Is that graph including the calculation that if we get on the flop with low SPR, then that implies we had to pay more preflop (well not necessarily, but if we are doing the calc with same stack size, it is true) hence, we have to deduct this higher cost from flop EV ?
BTW:It would be interesting to see how double pairs (no suit, suited or double suited, with no connectivity , some connectivity or connectors) do against AAxx if we get 4bet.
I wonder where is the borderline.

Porshy,
My calcs are inclusive of PF action, yes. However I don't know what you mean by we had to pay more PF. As long as we have any money left behind to play the flop, our PF call size will be constant. The formula for EV in the graph is:
EV= (freqallin*(equityallin(1+SPR)-(1-equity_allin)SPR))
EV_adjusted= EV-(1/3)
** This is because PF we always call a raise size of 1/3 of the flop SPR.

As for the borderline, it takes repetition and intuition to draw the lines. I don't know this to be true but I doubt we can ever call a pot sized 3bet with low SPR with a hand that is < 33.33% allin PF vs villains range. I doubt equities will ever be concentrated so abruptly that we can recoup enough post flop equity.

I know this is an extreme example, but what percentage of flops do you think he will fold on, and what is your equity on those flops? - I think factoring this in realistically would effect your g.i.i. equity by a significant amount.
Excellent video by the way - got me to start playing around with some spots in a similar way.
Cheers

By the way, how would you estimate a strong players' range for getting to the flop in that way with that amount behind? If you tell me that I'll write something that will model him stacking off accurately v your range and u v his and I'll try to calculate which hands within the range you give are really profitable within a range played this way versus a thinking opponent.

John,
Are asking about this specific spot? Or are you asking about what he should be doing?
Well, firstly AA62o has 12% equity on KQJr and his required equity is 19% so as you can see even considering the worst case scenario he isn't making a big mistake shoving blind. Basically, at this SPR he will ~never have <19% equity on the flop. For example, On JT9hhh AA62o has 29% vs a range of 8%!(aa,kk:xyzw)
Certainly as SPR increases he will have to fold some flops, but we also gain some on those flops by semi bluffing. This exercise is to look at spots where villain will jam nearly every flop. And at HU SPR <1 that is fairly reasonable.
As for what range would a good player call PF with little behind? Any hand that has > 33% equity and has decent visibility.

Haven't had time to watch the full video yet but is this the same calculation being done in PLO from Scratch Part 3 - http://en.donkr.com/articles/plo-from-scratch---part-3-219 ?

loved this video, keep up the good work.

really great video. This is very helpful analysis to be seeing done on hands. I am wondering how much you like your turn call based on your analysis of the river spot with QTT9. Do you still feel like you want to be playing the hand the way you did on earlier streets?