Quote:
Originally Posted by yaqh
Hey Emus, glad to see you found your mistake. I have a couple comments, though, that might give you a more intuitive understanding of the result or at least help everyone else put the algebra in context.
So, we're looking at a situation where Hero is mostly polar, and Villain mostly holds bluff-catchers, but Villain has a few slowplayed or trapping hands as well. In particular, trapping hands make up a fraction T of his range, and T is fairly small -- 5% maybe. After a bit of work, we see that Villain's bluff-catching frequency must make Hero indifferent to bluffing. That is he must continue often enough when facing a bet so that for Hero's air hands, the EV of bluffing and his EV of giving up are equal.
Then, it turns out that the particular amount that Villain must continue versus a bet with hands that can beat a bluff to make those EVs equal is P/(S+P). (Here, P is the pot size, and in this example, Hero's bet is all-in, so S is the stack size and the bet size.) So, P/(S+P) of Villain's range must be continuing versus a bet. We know that T of Villain's range is nut trapping hands, and these are obviously continuing versus a bet. So, the other (P/(S+P) - T) of his range which is continuing must be made up of his bluff-catchers. Basically, Villain continues with all of his traps, and then he adds in enough bluff-catchers to his continuing range to reach the total required continuing frequency P/(S+P). Thus, we see that the fraction of Villain's range which is bluff-catchers which will call Hero's bet is (P/(S+P) - T).
Yes, once I figured out the P/(S+P) relation the rest sunk in pretty easy.
The thing that is confusing for me is this:
according to section 5.2 you create an eq distribution graph as follows
X h
Y EQ(h)
H0 (meaning closest to intersection) => highest EQ
H1 (after normalization) => lowest EQ
graph starts around (0,1) or upperleft & ends around (1,0) belowright
Now I look to graph at page 239,
X (1-Hb) & normalized
Y EQ(hb)
because of section 5.2 relation;
(1-Hb)0 or closest to intersection => has to have lowest EQ
(1-Hb)1 => has to have highest EQ
And therefore, the graph is to be exected to start around (0,0) belowleft & end around (1,1) upperright
BUT you started again the graph around (0,1) or upperleft & ended it around (1,0) belowright which means you first started to add the lowest EQ hands and then the highest EQ hands.
This is of course according to you page 240 Hc definition & this is not important for the math (if I got it right) but it is rather confusing once you are getting used to eq distributions.
To phrase it otherwise, by changing the definition of H in graph page 239 (by just using a different ranking method) then you did in section 5.2 you got instantly the idea that a different kind of graph was used but actually graph 239 is exactly the same as an equity graph in 5.2 the moment you realise
(1-hb) page 239 = h 5.2 OR
hb1=(1-hb)0=highest EQ = h0 5.2 =(1-h1) 5.2
(1-hb1)=hb0=lowest EQ =h1 5.2 = (1-h0) 5.2
Did I got it right?
Last edited by Emus; 01-10-2013 at 06:38 AM.