The way i would sum this up is that low cards lose the least equity against the button. That makes logical sense, because it fits the criteria that we were looking for:

[x] We don't share cards with many calling hands, thus we aren't dominated often

[x] We fold out the hands that dominate us

[x] We have more equity than we need for +EV (20%).

I think that you've made a mistake with the small pairs, in my sim they lose almost 10% of their equity. Apart from that, you can see that this approach generated a polarized 3betting range for you.

(This is how you could take realizable equity into account, even though i obviously pulled those numbers out of my a..)

Let me point out a (potential) mistake here:
You don't care about range vs range equity (although in this case, when only looking at your bluff range, this is fine). You care about the equity of every single hand in your range. Every single hand needs to have more than 20% equity.

Less? I think what we've figured out here (if anything) is that we wanna bluff more! We'd need 20% equity for +EV! T3o has more than 30% equity!

Nice bit of info right here. Yes, the boards that you hit depend largely on your kind of 3bet range, and it's not always that apparent which ones that are.
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Final Thoughts

What were we doing in the first place?

- We look at a single hand
- We assume that calling this hand is +EV (implicitly!)
- We utilize the EQ/FEQ trade-off in order to determine whether 3betting could generate more EV than calling
- We determine a potential bluff range

If you read that carefully, you'll see that we've made a couple of assumptions that i hadn't talked about before.
We assumed that both calling and raising were viable options! Think about that. This whole approach is useless in spots where you can't call much, or where you take 3betting hands out of your folding range.
It only yields decent results in spots where we want to consider 3betting bluffs out of our calling range.
And when do we want to do this?

Well, that's not such an easy question.
But i know one thing: We don't want to do that when the equity threshold for 3betting profitably is 20%.

55% * (SB+BB+2.5BB's) - 12% * 7.5BB's + 33% * (equity * 8.5BB's+SB+BB) + (1-equity) * -7.5BB's) = 0
-> equity = 20%

20%. 20%! Every single hand has more than 20% equity! Why would we take hands out of our calling range to 3bet? We won't! We'll take hands out of our folding range, and suddenly this entire approach is (almost) useless against this weak button (fwiw: Against this kind of player i would construct my 3betting range by taking bluffs out of my folding range and looking at board coverage).

So, all this for nothing? Well, I absolutely love our bottom line:

Some concepts, neat as they seem, are precisely useful against good players, or against sticky players, and in very specific spots. Against weak players they might just be fancy, meaning that using them would cause you to sacrifice EV i.e. make less money.

Back to you guys: What kind of equity threshold would merit utilizing this approach?

My next post will be about Richard and Leonard.

Part1: short confused excurse aka: Trapped in theory please send help (skip it, seriously)

Little confused rambling (a little scream of theroetical approach getting lost in the real thug life model so to say):

Is there really an equity threshold that would mark a point where our 3betting range gets more merged towards a linear structure that would supposedly be constructed to not take linear equity hands, but FEQ>diff(EQ(vsopeningrange),(EQ(vsdefendingrange)) hands in order to avoid reverse implied spots? It kind of gets so iffy with so many parameters coming into play (equity realization postflop, rake, postflop betsizing), that to narrow it to a certain threshold where we are in fact generating more EV using this concept, we would need BTN to be extremely predictable in his postflop play and ranges.
For the heck of it, lets just keep this simple and just ignore all that and just look at raw equity but keep in mind that we want to use every bit of preflop FEQ that is offered to us.
I mean we are currently in a scenario that does not value FEQ as that high commodity as it is in reality, meaning preflop FEQ is so much more valueable and let us turn trash into +EV hands while not losing EV on calling and 3betting hands and so overall extend our total range and probably expanding our ev roofing just because of the rake. This concept should always keep this in mind and that makes it even less applicable in a scenario where rake is just huge.

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Part2: The actual question and how I approached to find an answer

What is the threshold? Where would be a good point to start applying the concept due to equity constraints on our action range?

I. The assumption

In order to find a threshold that would utilize the concept profitably we need our opponent to not let us turn the trash of our folding range into +EV 3bet bluffs and instead start to see that we resteal linear, so we would be somewhere at >30-32% equity for all elements of our 3betting range.

Mathematically speaking we need BTNs parameters of raising, defend and fold frequency to match a certain plane that location initially gets orientation from the overall opening range. If for example BTN decides to only open 40% with 2.5BB, of course he would need to exactly 4Bet and defend with the same ranges, but you will quickly realize that we have much less FEQ. So the plane would be flatter and the equity threshold would increase.

So in order for 30% equity giving the scenario 50% range, 2.5BB openraise, the frequencies have to correlate to form this plane.

x= frequency we pick up the pot from BTN folding to our 3bet
y = frequency we lose to 4bet
z = frequency we get called and realize all our equity on the flop

z = 1.78x - 3.33y or z= -.445(7.5y -4x)

I solved it for z, as z is the parameter that is the most questionable (explained in my above ramblings why equity realization rake yada yada, postflop tendencies etc)


II.
So what does this mean?
As we can see we need a BTN that does not like to fold, but also does not like to extend his calling range as this would just give us more equity with all our hands! We need a BTN that actually starts to 4bet more frequently taking that from both his defend and folding range, so that the bottom part of his defending range is so good that we would get really screwed with hands with <25% equity against the range, that would also play poorly initially, meaning we cannot bolster ranges with hands out of our preflop folding range against 50% opening range.

What if BTN just starts to raise 60% or more to begin with?

Extending his preflop opening range while also extending his defending range, meaning the equation is simply projecting a congruent plane IF also our ranges increase accordingly. At that point we just also reach the threshold of ALL our hands having 30% and in that case the concept also applies as we never underfold, never overfold and generate FEQ.

I am pretty confident my assumption of 30-32% equity as a threshold for introducing linear, "combinatorically smart" 3betting ranges that keep us from turning folding hands into polarized bluffing hands is the threshold and we seriously have to start thinking about realistic ranges to combat villains 4bets as well.

-------

So I hope Richard and Leonard are cool dudes that know a lot about 4betting.

Meanwhile for sure:

Richard and Leonard are remarkable in every way. I admire them for how they treat each other, and for how they treat me, and I wish that everybody had friends like them.

Richard, amazing as he may be, can barely write. To be honest, he doesn't do much with his life nowadays, even though he used to be an excellent piano player.
If anything, he focuses on manual labour, which is what he does for a living.
His job is to deliver stuff to places. I have to admit that I don't know much more about it than that, but it fits his personality extremely well. He has trouble handling delicate things (but not delicate matters), so he mostly doesn't. Except when asked to: Then he he handles his task with extreme care. He never declines or complains, and he gives it his full attention.

Leonard, on the other hand, fills his days with things that he's good at. He's an amazing writer, and he's quite skilled at dealing with delicate things. Funnily enough, he doesn't play the piano quite as well as Richard, but still well enough, and definitely better than most.

As you can see, Richard and Leonard are quite different from each other. You would expect this to be a problem sometimes, especially because in addition to being good friends, they also work together.
Working with them goes absolutely seamlessly. Leonard, this has to be said, is a lot more dominant, and a lot more skilled. But Richard never holds this – or anything else – against him. They split their work with maximum efficiency, meaning that Leonard generally deals with difficult matters, while Richards does what's needed in order to get those jobs done. Don't get me wrong: Both of their jobs are very important, and even though that's the case they never let pride or ambition get in the way.
In addition, with Leonard handling things of more importance, accidents do happen. He has injured Richard several times while working on a job. But their mutual respect and trust keeps them from ever assuming that there are any bad intentions.

They treat me the same way: I love cooking, and it's an even greater pleasure when Richard and Leonard give me a hand. When I don't have much time they go and grab the ingredients for me, and Leonard generally prepares them (as you can tell by now, this agrees with his skills and personality).
When I play the piano both of them love playing with me, and Leonard turns the pages.

Finally, they are admirably great at something that many people struggle with nowadays:
If they need to get something done, they give it their full attention. They work on it until it is finished, and only take breaks when told to do so. They don't complain, and they understand that the jobs given to them are of great importance.
And they are equally great at taking breaks. None of what happened before lingers in their minds, and they can relax as if they had no cares in the world. I am trying to learn this from them, as I am starting to realize what a great skill it is. I would go so far as to say that watching them take a break is inspiring.

After writing all this, I realize how much I wish that I was more like them.

Past Simon just played a session for my benefit.
When he was done, i immediately focused on the one mistake that he made. Like, literally, he quits the session (i didn't even wanna let him close the tables) and there i am attacking him, urging him to analyze how big his mistake was.
In my world, there wasn't a single well played hand, i didn't pause to give him credit for the session that he played, or to think about whether his play was good. It was all about this mistake.

So, we got over that, and then we sat down together and past Simon made me focus on the good stuff. Wow. I didn't know that he had those aggressive lines in him. You have to understand, certain types of aggressive lines make me anxious, and it's very hard to stay clear-headed when you're emotionally involved. Reviewing the session, i almost got the feeling that past Simon doesn't have those problems. He made some really ballsy and good moves, and he never had a hand that didn't make sense for it.

The worst part is, him feeling no pressure and being able to be clear headed in big pots is what i want, but i often behave like i'd rather hold him back. After all, as i already mentioned, he plays these sessions for me. If it wasn't for me, he wouldn't need to play a single hand of poker.
But he does. And so far, he hasn't even complained about the way i was treating him. He just swallowed it, felt more pressure, made a couple of bad decisions because of that and then got bashed even more. Putting it like this makes me feel sick of myself.

In addition to all this, i am a hypocrite as well.
I make people believe that all i care for is past Simons well being and then... well, let's not dwell on it.

I am so glad that we straightened things out today. Focusing on the positive aspects of the session felt amazing. I know what i'll have to do in the future!

Man this is awesome.

My confidence in my assumption on equity threshold dwindles with every day past Simon did not chose to adress the depressurized crazyness that was my last post. I get it though. I am still happy I wrote it. It was fun.

Envying and admiring people for their trades is one reason I follow this blog. So I see where you are coming from, though I am not big on "only live in the now" kind of mentality that completely disregards past and future events. Maybe one of my traits that prehibits some potential, but preserves my sanity and decision making. I could use some dedicated deterministic asskicking hardworker trade too, you think we germans get that in our DNA. Well. We lied world. We lied...

manual labour must become the piano

You'll get an answer.
Yes we did. Have a nice weekend y'all!

Interesting to hear that you are German. At first I wanted to reply to your post right away but then I shied away because I am not qualified enough to criticize the mathematical equation of it. First of all he never talked about linear/ polarized ranges. It's a logical independent concept that he explained and you shouldn't mix it with with those old terms.
Not gonna say too much about the equation but for me it's not clear enough how you set it up.
The number 30% for the original question might be accurate, but your conclusion seems weird, because you are thinking about linear ranges and polarized ranges again, where you are getting of the track, more thinking about your own question than the one from Simon.

Maybe Ser Midas can describe the issue more clearly Or maybe I am completely wrong.

You might be right that I approached it again with a point of view that was flawed. Maybe it is just what I view as polarized, because the term is just not true and so open to interpretation.
As linear should be logically always considered the hands with the most equity against the range. Again flawed, because we know that the agressive actions forcing a rangesplit in certain spots will mean what is linear against all elements of the range, will no longer be linear against the partial quantity set of elements that remains.
So the whole linear/polarized concept is flawed because it is just based on equity against a range, not against parts of it depending on the reaction and for sure it is blind towards approximational factors like realization factors, future action etc etc.

So while not thinking about that I approached the given dilemma as blunt and basic as I could and that was taking in account those flawed concepts to not deviate, which in the end I did anyway. Just to somewhat explain my logic and how it seems off from the start just through the point of my view. Also my 30% was just an assumption based on what the equation offered as a possible value that still seem to be representative for a certain style found in today's games.

I'll keep ignoring you guys for just a bit longer.

Situation:
My results this year haven't been amazing. Far from it. In addition, i've made some reasonably poor investments.

What's gonna happen:
For my last 3 weeks in Thailand, i'll play just 4 tables of zoom(500 or 1k). I'll hit a 5bb ev winrate.

Which means:
I will write more poker related posts. I will go back to describing my sessions. I will outline differences between 4 tables and more tables. I might even post some strat.

Ah, great! I look forward to this

Just 4 tables of zoom.

Click. Boredom. Click. Click. Flop. Miss. Click. Fold. Booredome.
Click. Click. Click. My heartbeat is faster than the click.....click.....click. 4 zoom tables - one - decision - every - 10 - minutes.
1st session was 3 hours.
3 hours of playing, meditating, swimming, playing. Walking around.
Trying to enjoy the sun.
2nd session 2.5 hours, which i just finished. Same, just dark outside.
Small break, but mostly this weird meditative state of clicking and then occasionallly thinking a tiny bit and making one more aggressive click and then going back to the passive clicks and the boring clicks and the loooong and sloooow and conscious clicks and all the clicks and is online poker anything except boredom and clicking and really not thinking cause i don't want to and can't and know all the spots anyway and ranges are so tight ahhhh 6max.

_______________________________________
Upcoming posts:

Waffle - Last reply regarding the 3betting toy game
My next poker session - I will post at least 3 hands
Decisions - Why you can't make them

Love the post, but keep in mind : have you ever worked a real job? A 9-5, factory work, kitchen work? When I put my past jobs in perspective, I love clicking buttons for a living

did you see the post in high stakes PLO about the bots that are playing and sharing cards that have taken 3.5 million dollars out of the plo economy? There is something like 25 of them and 8 were still active this week.

It's kind of a big deal. It's why being a plo pro is hard as F.

themoreyouknow.jpg

its borderline friend. It's going like **** for me personally for 2 years but still not quite **** enough to go back to a regular job lol but still man pretty tough to stay ahead of the curve

I really enjoy reading this thread, please do continue!

Hey can you post some graphs? Be fun to sweat.

subbed, enjoying the strat content =)

OK. I will post a graph. 4 tabling so far:



The shape of the graph is due to the nature of pot limit omaha and the way i run at it (and/or the way i play it. Please bash me).

Just finished the 4th out of those 4 sessions, and it's spew monkey day (It often is when i 4 table).

I promised 3(or 4?) hands, so here we go:



Ace is (obviously) a good card cause it blocks his value range. The fact that my calldown was very very right against his hand is good (in case you were wondering).



Probably close, but that guy just offers me those additional $65... I've always had a weak spot for raising small pairs when i shouldn't.



Well. I think turn is interesting, i just suck at these spots. We clearly don't barrel turn for balance, so why do we barrel turn?
I feel that checking turn has to be better in a vacuum (if there was no river play), but there IS river play. as played obviously lol call and lol lose (i am practising 2p2 lingo in case you didn't notice).
hm, is this where people post some racist stuff like lol russians? I'll just not do that.

4th hand just in case:



Thing is, we have pocket JJ (or JJ, or pocket Jacks. Whatever you prefer).
Jack-Jack also works. AND he opens 26% in that position. It's likely that calling makes more money, but i just couldn't help myself (the thing is that we can go all in for protection and hope that this guy watched the wrong sauce video lately and just four bet ace-queen suited cause yolo).

Take-aways:
- Do you think my play sucks in some of these spots? well, pad yourself on the back then. You play those spots better than me. People focus too much on the negative stuff in poker.
- I am considering banning myself from PLO for this month. It's just not going well at all.
- Because of the way i've run this year, I am as happy as i could ever hope to be. So thanks everybody for hitting great hands, sucking out or taking the perfect line against my actual hand!

even the gods get struck by variance

I'll post graphs, but not always. Today, for instance, i won't. (Edit: Hm. I might not post graphs. See below.)

“Things are going downhill with you!' he said to himself, and laughed about it, and as he was saying it, he happened to glance at the river, and he also saw the river going downhill, always moving on downhill, and singing and being happy through it all.”

I mind-banned myself from plo. Instead, i just play no limit. I've started checking results more often, i am not sure why. Probably because i realized that when i post graphs, i'll see results anyway. That's a problem, i will have to think about how to deal with that, cause it's not good.

Sessions 5 and 6 were fairly uneventful. I've changed many small things in my mind, i am not beating up past Simon at the moment.
Sessions feel very relaxing.

Poker:
Well, i did play poker during those sessions, but it's not really about that. I started using an RNG, and that's cool. The poker just fades away over all the things that i need to deal with, that just concern myself and my self. And me.

Let's take a minute to consider the state of 500 zoom:
The strategy at zoom 500 right now is to have 10 sizes in every spot, and hope that everybody's too confused to exploit any of them. There's 2 camps right now:

Camp 1:
Guys in camp 1 are thoroughly confused. They seem to try to decide what every size means and which one is weak or strong, but their brains can't handle it, so they black out every 10 hands or so.
Naturally, this is the wrong strategy, and the only guys who can employ it are the ones who are blessed with exactly the hand that they need against each size in each spot.

Camp 2:
Camp 2 probably started out having the same problem as camp 1, but neither worked it out nor was favored by variance, and got sick of trying to guess the right button every 2 minutes. So they found a remarkable way of dealing with it:
They simply ignore the sizes, treat them all the same, and click their buttons with grim determination.

Camp 3:
Camp 3 applies the century-old strategy of information betting and information raising. This way, they avoid spots where their opponents bet, thus avoiding any confusing bet sizes. It's quite an elegant solution, in fact.
The fact that they end up playing 85% of their own hands in a non-optimal way must be considered a minor nuisance in the grand scheme of things, and is probably a sacrifice that needs to be made.

Which camp do i fit in?
You'll get the same response from every 500 zoom reg who reads this:
Neither.

because they do not know which camp they belong to, or?