[GTO] EV in over-folded Spots
Join Date: Sep 2012
Posts: 372
I don't know why, but I have some difficulties to understand the following result.
Assume villain folds with P(Fold) = Alpha + X, where Alpha is the probability that makes heros bluff break even and X is some kind of margin. If X is > 0, hero nets a profit and his ev becomes:
EV = (Alpha + X) x (1 + Bet%) - Bet%
Pot is standarized to 1 and heros bet size is given in percentage of pot.
We know that Alpha = Bet% / ( 1 + Bet%), therefore
EV = Bet% / ( 1 + Bet%) x (1 + Bet%) - Bet% + X x (1 + Bet%)
= X x (1 + Bet%)
What does this equation tell us? Heros ev linearly scales with the margin probability, that seems fine. What I don't understand is that it also scales with his bet size (Bet%). Why is that the case? For example compare the following nominal values:
EV(X = 0.1, Bet% = 0.3) = 0.13
EV(X = 0.1, Bet% = 0.55) = 0.155
Using a bigger bet size our ev is higher despite the fact that we risk more. What does that mean?
Join Date: Sep 2016
Posts: 7,411
There was a discussion on this.
EV=(overfold%)/(gto defending%)
If you bet 19xpot and opponent folds 96% insted of 95%. He actually folds 20% of his defending range.
Join Date: Jul 2012
Posts: 16,234
It might be easier to visualize if you graph it. That being said, since you're stating he is folding a specific value greater than alpha, then at some point where bet size dictates he folds =>90% you are capturing 100% of the pot (0.9+0.1). That's the easiest way for me to explain it. So, as bet size increases EV approaches 100% of pot with how you've created the equation.
Join Date: Jun 2017
Posts: 442
As said, because 10 percentage points overfold becomes bigger relative to MDF the bigger you bet.
Eventually, when you bet 9x pot, MDF is 10%, which means villain is folding everything, which means your EV reaches it's maximum: the whole pot.
You can also look closer at the EVs of getting called and getting the fold to understand:
If you bet 1x pot and villain defends MDF:
1*0.5 - 1*0.5 = 0.5 - 0.5 = 0
If you bet 1x pot and villain overfolds 10 percentage points:
1*0.6 - 1*0.4 = 0.6 - 0.4 = 0.2
So we are gaining 0.1 from when villain is folding and 0.1 from when villain is calling, relative to GTO.
If you bet 2x pot and villain defends MDF:
1*0.67 - 2*0.333 = 0.67 - 0.67 = 0
If you bet 2x pot and villain overfolds 10 percentage points:
1*0.77 - 2*0.233 = 0.77 - 0.47 = 0.3
So we are again gaining 0.1 from when villain folds, but now we are gaining 0.2 from when villain calls, compared to GTO.
So we are always gaining Overfold% from getting the fold and Overfold%*Bet from getting the call, compared to GTO.
Total EV is then: Overfold% * Pot + Overfold% * Bet = Overfold% * (Pot + Bet)
Or use the formula Haizemberg showed.