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01-20-2008, 05:07 PM   #4
schubes
journeyman

Join Date: Jul 2004
Posts: 229
Re: Maximizing EV to Find a GTO Betsize

Quote:
 Originally Posted by The Bryce I was a little stuck in trying to figure out why were weren't taking into account how often we won the pot by bluffing, but then I realized that if we're indifferent to bluffing the EV on our bluffs is 0. Am I correct in my line of thinking there?
Yes.

Quote:
 Originally Posted by The Bryce So if were going to add equity into this what that does is change our point of indifference when bluffing. Say we are betting 1 unit into a pot of 2 units, in a nuts or bluff scenario our point of indifference when bluffing is if our opponent calls 33% of the time. If we only lose the pot 85% of the time when bluffing (15% equity), however, our point of indifference becomes 0.33/.85=0.388. So we could represent this in the equation: 1/(s+1)/o Where o is the % our opponent wins when we're bluffing.
Well, it's an easy mistake to make but the opponent would call 2/3 of the time not 1/3. More importantly, I think you're jumping ahead too fast here. 1/(s+1) came from making us indifferent to betting with our bluffing hands. The semibluff aspect means that our bluffs lose less when they are called, but it also means that we win less when our opponent folds compared to checking, since we can check and still have some equity in the pot. You really should start with the indifference equation.

I'll say o is the chance our semibluff draw misses, c is chance opponent calls.

ex-showdown EV of bluff = c*o*-s + c*(1-o)*s + (1-c)*o*1

The last term is the probability opponent does not call, times chance we wouldn't win a showdown, times the pot size 1.

Setting this equal to zero and solving for c we get:

c = 1/((1-w)s + 1)

where w is the fraction of hits to misses with our semibluff. (w = (1-o)/o)

You can also set up an indifference equation for villain calling and folding with his bluff-catchers to find out how often hero bluffs. You'll find he'll bluff more often when his bluffs have equity, as you'd expect.