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Old 01-19-2008, 10:17 PM   #1
The Bryce
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Join Date: Oct 2005
Location: Edmonton
Posts: 4,245
Maximizing EV to Find a GTO Betsize

Half-street Game:

- Our opponent can only check and call or check and fold
- Villains calls always lose to our non-bluffs, and always beat our bluffs
- Out opponent's entire range is bluff-catchers
- The bet size and pot size are known

In this game our GTO bluffing frequency is determined by looking at the price we are laying when we bet. For example, if we are laying 2:1 we would bluff 33% of the time, as any deviation of that would be exploitable.

a=s/(p+2s)

a: the amount our opponent needs to win the pot based on the price we are laying in order for our distribution to be GTOl*
s: bet size
p: pot size

Note that this isn't the same "a" that Chen uses in the AKQ betsizing game, though the equation looks similar.

In any case, I wanted to jazz this up a bit and say that n% of my opponent's range beats my non-bluff range, but my bluff range beats his whole range o% of the time (this is a clunky attempt to simulate the equity of a semi-bluff). We know that the amount we're bluffing and lose plus the amount we're not bluffing and lose must equal a%, so:

o:The % our opponent wins when he calls and we're bluffing.
n: The % of our opponent's range that beats our non-bluffing range.

b*o+(1-b)n=a
b*o+(1-b)n=s/(p+2s)

My lack of a formal math education catches up with me here, as I do know how to simplify that to b= (or perhaps it can't be done).

The other thing I was curious about, was that if the bet size was unknown and you could simplify that expression to b= whether you would be able to write up an EV calc that took into account all the permutations of what you would win or lose with bet size "s" and then maximize "s" for the highest possible EV output (which you normally can't do, I think, since the amount we bluff (b) and the amount we bet (s) are co-dependent, but if we can substitute the other side of b= for "b" then it should work).

Anyways, to those who actually understand how to do this stuff: thanks for humoring me
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