more accurate ranges / ranges and speculative probability
Imagine you were putting yourself on a range. I.e., you take a step back from knowing what your cards actually are and asked yourself "what does my table image and my betting pattern represent to the rest of the table that I might have?". So you raised pre-flop from the button, and let's say you only do that with AA, KK, QQ, JJ, or AK. Then you call a bet of 3/4 of the pot on a board that is A-Q-5 with two clubs. A third club comes on the river, it checks to you, and you bet 1.3 times the pot. What have you told the rest of the players?
Of course, you know what you have told the rest of the players because you actually know your range for making decisions like this.
But what if someone does not know your range as well as you do? For instance, what about the guy who just sat down at the table. He might not know that you only raise pre-flop from the button with monster hands. So he puts 5-5 and Ace-x within your range when they actually are not. And he doesn't know what your calling and betting ranges are on the turn. So he may very well underestimate or overestimate your hand.
Then he plugs his numbers into pokerstove and makes an incorrect decision. Why? Not because what he is doing is wrong. But because tools like pokerstove only assign probabilities within the range based on the chances of the cards being dealt from the deck. In other words, in my example above with an Ace on the board, there are 12 ace-king possibilities and only three possible sets of pocket aces, so if you plug in the range AA AK into pokerstove, it will assume an 80 percent likelihood of Ace-King and a 20 percent likelihood of pocket aces. And your equity calculation will be based on that.
The problem with this is that this is only true if we are equally sure that the player will play pocket aces and ace-king the same way.
The easiest place to see this is with the pre-flop raiser. Unless you have seen enough hands to actually know what the person's raising range is, you have to go off (1) those hands that have been raised and went to showdown, and (2) the percentage of hands raised and re-raised. Now, with a very tight pre-flop raiser, which are we more certain of:
1. That he will raise with Aces?
2. That he will raise with Ace-King?
Obviously, for most players, the answer is (1). (Obviously, when you get a weirdo who slow-plays aces, you have to adjust to that.) So, in many situations, there is a core of the range that we are sure of and a perimeter of the range that we are less sure of.
So, instead of expressing the range this way:
AA-JJ, AK
We might express it this way:
AA-KK, and probably QQ-JJ AK.
Now, how does this effect equity? Straightforwardly, your equity calculations should be based on a weighted average of the possible hands in the range, with more probable and less probable hands being accounted for.
A weighted average is an average adjusted for the relative expected occurrence of various items within the sample. The best known example of this is the atomic weights of elements. Hydrogen's standard atomic weight is 1.008 because while most hydrogen atoms have no neutron and have an atomic weight of 1 (1 proton, 1 electron), in any sample of hydrogen there are likely to be enough deuterium and tritium atoms (atomic weights 2 and 3, respectively), to make the average weight per atom of hydrogen in the sample be 1.008 rather than 1.000.
Unfortunately, I see this all the time. Players will place, say, a hand like T8s within a particular player's range, and then use it to inflate the likely equity in the hand. But that ignores the fact that whenever a player is betting his hand and showing strength, we can be somewhat more certain that good hands are within his range and somewhat less certain that marginal hands are within his range, unless we know the player well enough to make a more accurate prediction.
Of course, if you throw out the marginal hands altogether, then the range estimate will tend to be too tight, as it gives a player who may be willing to play the marginal hand too much credit.
So, you need to weight the average. Assign a greater relative probability to hands that are more likely to be within the player's range, and a lesser relative probability to those hands that are more speculatively within the player's range.
