Ranges & Geometry - something I'm thinking about

Several threads and post (ISF's blog post from this week) have been about ranges and hand reading. I haven't had much say because I've been working on something - trying to get my own mind around what few things I understand. Maybe you can help me now that I've finally arrived at something I can put into words.

I see my own hand on a continuum from very weak to very strong (x-axis) and my opponent's a similar continuum (y-axis). The diagram looks like this:



The 45-degree (or the line y = x, for fellow math geeks) are where we're tied - we have the same hand. Above the line, villain is ahead. Below the 45-degree, we are. Near the line in the football-shaped region are the big money pots - both hero and villain have high quality hands, but someone's going broke.

I learned this concept reading Steve Badger's website: you're closest to winning a huge pot in poker when you're closest to losing. Two pair wins big against TPTK, but is vulnerable. You can be ahead and get all your chips in the middle with a set but lose to a straight or flush. So hands where both players have big hands, but one is slightly behind, are where the big pots reside.

Geometrically, every time we bet we face a rectangle of possibilities defined by the action so far. The range we're representing is the "width" of the rectangle, and the range we can put our opponent(s) on is the "height." When we can put villain on a "narrow" or "thin" range, the range rectangle is "short." When the range we are representing is narrow, the rectangle is "thin."

Given that the players open roughly the same types of hands and ranges, the player whose range is largest on later streets has an advantage.

Okay, enough theory. An example. Two TAGG's, both playing 18/14/3 styles, face off. MP TAGG limps, Button TAGG raises to 5xBB, and MP calls. Who has the advantage? Well, depends upon the flop, but generally the button has the advantage because his range is wide while the limp/caller's range is narrow. This is the situation pictured in the graphic above: the yellow rectangle represents the relationship of the two ranges.

A positionally aware TAGG could have just opened nearly any Ace, a good King, all pp's most broadways and some sc's and 1 gappers. What can the limp/caller have? Well, probably pp's TT and lower, though JJ and QQ are possible. A very few other hands like AJs, ATs or maybe KQs are possible - non very likely.

When the flop comes down K J 3, who's ahead? Parts of hero's very wide range connected solidly with the flop, and very little in villain's narrow range did. So both players (should) know that hero's range is way ahead. Of course, if villain had 33, his just hit the dream flop, and his cards are way ahead of hero's range. But his range is still far behind, and herein lies his problem.

If villain shows strength, he either just hit his set or is turning his underpair into a bluff with a cbet. And hero knows it. Villain can stand very little pressure with most of his range, so hero can cbet relentlessly. If villain show any aggression or "plays back," hero is instantly wary because there are very few possible hands. And hero can tell almost down to the card what villain's holding is. He's knows when he's got it beat.

Villain has a problem when hero acts because his range is so wide: a small bet might mean he's hoping for a raise so he can shove; a big bet might be real or an attempt to steal the pot; a check might indicate weakness or a desire to induce a bluff on the turn/river. Hero's wide range makes it difficult to know his holding, so more of his aggressive actions have to be respected - generally.

Absent a wide range, hero likes a polarized range, which is like the above yellow rectangle with the "middle" missing. A polarized range has most of the same advantages of a wide range: deception, value for big hands, room to bluff, and so on. We want to avoid narrow ranges against aware opponents - it's hard to earn any money with a narrow range, and it's fairly easy for our opponents to know when they've got us beat.

To finish the example: positional aggression is generally better than oop passive play. The raiser will typically have a much wider range and more maneuvering room. The typical calling range is much narrower and constricts movement later in the hand.

To relate this geometry to the ISF Low Stakes theorem: "Versus passive lines, you should be inclined to play weaker ranges, and versus aggressive lines you should be inclined to play stronger ranges." In our example, hero can play a much wider range by using position and villain's passive play. Against a TAGG's MP raise, however, hero will need to confine his play to much narrower and stronger group of hands.

I know this isn't groundbreaking, but I'd like to work this out in terms of calculating ranges and finding places where villain's betting patterns don't match his range. Let me know what you think so far, and I'll update the thread with some range estimating tricks and tips I'm working on.

This is interesting Robb. I'll watch into this more detailed.

I already wanted to add that in my perception ranges have basically 2 attributes, being it's:
1- strength
2- wideness

Ranges can both be wide and weak, wide and strong, tight and weak and tight and strong. Of course there are in betweens, but I think this is a good start.

It's interesting to see in what situations what ranges are present and how we can exploit it.
I.e. any weak range can be exploited by bluffing into, especially tight and weak ranges (like a 18/13 calling range). Any strong range can be exploited by playing for implied odds.
UTG opening ranges and 3-bet ranges are usually tight and strong. BTN and SB opening ranges are usually wide (and therefor weak). That's all preflop tho.

When someone is able to apply this thinking postflop (which I'm am still far from able to do correctly in most situations) than I think this is very profitable and a big step forwards for almost any player. It's basically ISF theorem I guess

Something else you might want to look at is one of the graph options in PokerEV that pits your Sklansky bucks against how you played the hand or something. I don't have time to read all of this but the initial theory sounds good.

Originally Posted by minSim

When someone is able to apply this thinking postflop (which I'm am still far from able to do correctly in most situations) than I think this is very profitable and a big step forwards for almost any player. It's basically ISF theorem I guess

Yeah, this is where I'm headed. I play a lot against villains who are 40/5/1.2 or something similar. And I want to think about things like that - what common opening stats VP$P/PFR lead to what kind of ranges when villain is opening, limping, or calling. And would a 40/5/1.2 3bet anything but AA and KK? I don't really see it happening. I will get back to this later this evening or tomorrow and try to move it forward.

Quickly assessing ranges based on VP$P/PFR stats is difficult for me - and I'm a math geek. I have tried and failed with various attempts: printing out lists of hands from PokerStove, rote memorization, keeping a chart by the computer. Lots of crap that didn't work. To go with my x-y axis range thinking above, I've been working on a way of "estimating a villain's range" in terms of blocks of hands, a quick way to throw in about 5% of the total range - mix and match on the fly, basically.

The basic question we have to answer is this: villain opens 25/12 and knows nothing of position - which hands is he likely to be playing? Well, a different group of hands depending upon whether he raised, called or limped. So I needed some flexible way to move hands in and out of his range.

Mathematically, all we need to know is a 4-hand group of unpaired cards is 5% of the total range of hands (and all pp's = 6%).

So here's my list of "hand groupings" that I'm using at the table to estimate ranges. I sort according to playing type so that I can pull them out of the range or shove them in as needed for calling ranges, limping ranges and/or opening ranges. (Note: AT means both suited and unsuited, ATs or ATo will be specified.)

Robb's Top Third of NLH Starting Hands
6% are pp's (3% are 99+)
5% are premium broadways (AJ+, KQ)
5% are weak broadways (AT, KJ, QJ, QTs, JTs)
10% are A9 or worse (2.5% Axs, 7.5% Ax)
5% are sc's + suited 1 gappers (T9s...23s, J9s...24s)

This is only approximately the Top Third of poker hands. We would have to throw a couple hands (JTo, K9s, etc) to make it the real-deal Top Third. And Ax is WAY over-represented. Based on my experience at the micros, I feel it's an accurate description of how my opponents tend to play. But the point was to get most everything typically played in a nice, neat group to insert or pull out of a range estimate.

Finally, let's talk about VP$P - I've been giving this a lot of thought, something I should have done long ago. Villains who play 20/10 obviously open 10% of their hands for a preflop raise, but what are they doing the other 10% of the time? Well, a combination of limping and calling others' raises. Limp/calling with small pp's might account for 2-3%, so this person is either limping weak aces, weak broadways or sc's/1 gappers, or or some combination thereof. I would love to hear other's views on this - really, I'm just a newbie thinking out loud here.

Examples
1. Villain is 25/8/1.5 and opens from MP. What might he have? Probably pp's 99+, a big Ace or KQ.
2. Same villain, limps behind. We have 17% of his range in the limp/call section of VP$P, so small pp's, lots of Aces, weak broadways and sc's + 1 gappers. This overestimates his range a bit, so a specific read on which group of hands he overplays might be helpful (my standard guess at the micros is Ax - jeez, that hand gets played too often!).
3. Villain is 12/8/2, and calls EP raise from cutoff. At PFR ~ 8%, he's opening most pp's and premium broadways. With 4% in his limp/call range, it's difficult to know what he's adding to pairs 99 and lower: medium aces, sooted broadway stuff is most likely.

The math goes like this, just for those interested in developing your own system of grouping hands and quickly calculating what percentage of the total range they are.

There are 1326 different combinations of starting hands possible in NLH, so 5% of them would be about 64 - 66 combinations.

Any hand made from two unpaired cards can be made in 16 ways, so if we think in terms of 4-hand groups like "premium broadways" (AK, AQ, AJ and KQ), we have 4 x 16 = 64 total ways to make those hands - or 5% of all possible. And pp's can be made in 6 ways, so all 13 pp's is 6%.

The reason I think in terms of broadway cards (T+) and "others" is because there are 8 hands possible between A9 and A2, or K9 and K2, so 10% for group. A quarter will be suited, so we can grab 3-4% and call them "suited + med Kings," for example, when we see a villain playing that garbage. We can see villain show down A3s and throw in 5% for sooted/weak Aces, minimum. As soon as we see A6o getting played, we know 9-10% of villain's range includes bad aces.

Enjoying the analysis Robb...

There are 1326 different combinations of starting hands possible in NLH

Are you going into specific suits here? That may be overkill. Just stick with 169 starting hands - suited and not.

Also - since you are being specific in your analysis - you should break down the vpip more. pfr + limp + call. Easy enough to show on your HUD if that will help you with the range.

Keep at it and lets see where the research ends up. I am most interested seeing how your analysis stacks up against the evidence. You should be able to look at poker tracker against a player you have 1000+ hands on and see if your assumptions are right vs the hands he shows down...

Originally Posted by paulwright

There are 1326 different combinations of starting hands possible in NLH

Are you going into specific suits here? That may be overkill. Just stick with 169 starting hands - suited and not.

I'm just trying to count quickly so that I know if 4 hands: say, A9, A8, A7 and A6 are in his opening range, then that's 5%.

Also, what HUD do you use? I don't think mine is capable of displaying limp vs. call - just VP$P and PFR. But let me know if there's a slick and easy way in PA HUD to do that.