Ranges, Ranges, Ranges
If you don’t like math (game theory I should say), stop reading.
If you’ve been paying attention, you already know that I’m a backgammon nerd. I play as often as I can and store massive spreadsheets worth of data. I minitab the hell out of it. The hours I spend on analysis have made me a much better player.
I play backgammon because I love strategy games. Another game that I love is poker. It’s complex, mathematical, and it’s a game of imperfect information. I get bored easily so the thought process keeps me very focused.
I want to post a hand and walk you through my thought process. Although I won the hand (actually stacked him), I’m not sure how I feel about my play. I am hoping someone out there can help me.
Because I’m a mathematical person, I rely strictly on formulas.
When examining a hand I use the following:
the bluff zone: (see comments for formula)
the profit zone: K=bu= ((a-b)b^2)/(a(a+3b))
where z is the hand strength, a is a large bet, and b is a small bet.
These formulas are theoretical and should only be used with hand ranges to determine betting patterns.
A hand range is every hand in which you would take the same action . If you are dealt AA and decide to raise, your hand is AA, but your range is a lot more than that. Your range can potentially be all pairs 1010-AA, AK(s)uited, AQs, and AJs. It could also look more like 7-6, AK, 10-8, J-10, J-9, K-J, and 5-4. It all depends on your position. If you are in early position (in the blinds or under the gun— first person after the big blind) you are generally going to play stronger starting hands. As you move into middle to late position, your starting hand values drop.
The ranges I state above may not seem too big, but they are! To give you an idea, there are 6 different combinations of pocket pairs, so the first range contains 42 different possible combinations (6 combos x 5 pocket pairs + 4 combos of suited cards x 3). In the second range, because suited isn’t a requirement, there are 16 different combinations of each starting hand, so we begin with 112 different possible combinations.
After each round of betting, your hand range changes according to your actions. Your goal is not to represent a hand, but rather to maximize the total possible combinations of hands that you can have.
Lets quickly assume that on the last round of betting (the river) there are 14 units in the pot and you bet 14. This means your opponent has to be right 1/3 of the time to make the call. If he calls and loses, he loses 14; calls and wins, he wins 28.
Lets put that math into two different hand range situations: the first situation is where we have narrowed down your range to 10 possible combinations of hands. However, his hand beats 5 of your possible combinations (5/10), so he is looking at 50%. That is more than the 1/3 required to call. The second situation is where you have 20 possible combinations of hands. If his hand only beats 5 of your combinations (5/20), he is looking at 25%, which is below the 1/3 requirement.
Now that you have a basic understanding of how the math works, I’m going to walk you through one of my hands using hand range logic, but there are some other quick things we need to know first:
There are 12 combinations of top pair top kicker
There are 3 combinations of a set (having a pocket pair and making 3 of a kind on any street)
I enjoy playing heads up (1 on 1) more than I do a full ring game, so that’s why there are only two players involved. The hero is yours truly, and the villain is my opponent.
Hero posts the small blind of 2 units
Villain posts the big blind of 4 units
Hero raises to 12 units
Villain raises to 36 units
Hero calls 24 units
The flop is [5h 9s 8c]
Villainbets 39 units
Hero has 15 seconds left to act
Hero raises to 104 units
Villain has 15 seconds left to act
Villain raises to 294 units and is all in
Hero calls 190 units
Villain shows [Kc 7h]
Hero shows [Td 9d]
The turn is [Jh]
The river is [3c]
Villain shows King Jack high
Hero shows a pair of Nines
Hero wins the pot 660 units with a pair of Nines
Now for the breakdown:
I am the dealer, which makes me the small blind. I made a standard 3xBB (big blind) raise. He re-reraised trying to represent a big hand. Here, I am giving him credit for pocket pairs 77-AA, AKs, AQs, AJs, Ax, 8-7, KJ, and 5-4. That’s actually a fairly tight range in a heads up match. Since there are only two players, all hand values are higher. I give him credit for stronger drawing hands in case of deception.
The flop comes 5, 9, 8 rainbow. The total pot size is 72 and the villain makes a bet of 39. This is an odd bet. It seems as if he is betting for value— typically, I’d expect to see a bet that is 80-90% of the pot here. This is a very, very tricky spot. I don’t think you value bet an overpair with a draw on the board, so I want to give him credit for 77, 88, AKs, AQs, AJs, Ax, KJ, and 8-7. In reality, I’m only giving him credit for .5(88). This is because there is an 8 on the board, and I think you check a set here half of the time for deception. Excluding Ax, which we will actually throw in as A9, which I beat anyways, there are 34.5 possible combinations of hands that he can have. Of those combinations, my hand, 10-9, beats 29 of his combinations.
It looks like I’m in good position, so I want to re-raise just to see if my range is accurate. After I raise, he comes over the top, which validates the fact that he has to overcards (but not a pair, or he would smooth call to maximize value), or is on a draw.
It turns out that I was right, well, he had Kx, but that’s not too bad.
Thoughts?
