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:
http://xs125.xs.to/xs125/08096/rangegraph836.jpg
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.
