EV in over-folded Spots
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?
5 Replies
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.
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.
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.
thanks, your answers are very helpful.
seems like over-folding is worth a lot more with big bets.
can anything be derived from this for practical use?
i guess that the over-folding margin does shrink for bigger bets, because it gets easier for villain to defend at the right frequency aka mdf.
It can. Your opponent must defend very close to perfect vs big bets, even "small" over fold or over call can be exploited stealing a lot of ev.
