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 Originally Posted by IowaSkinsFan
Seems like you have a pretty good understanding of ISF theorem.
The other thing your talking about is aejones, when he talked about "merging" your range, which is essentially just really thin value betting. Its not that hard to explain, but when you valuebet you don't have to necessarily be ahead everytime, just a certain % of the time depending on the situation. River is the easiest example, here let me try to explain.
Look at a typical scenario where our range is ahead of our opponents range. Let "L" Be our hands, and "B" be our opponents hands. The hand is ahead of the other hand when it is above it.
Example:
L
L
B
B
In this case, both L hands are ahead of the B hands.
Now look at a more difficult range example. Assume we are on the river.
L
L
L
B
L*
B
L
L
B
L
L
B-------
L
B
L
B
B
Assuming opp is calling with all B hands, what L hands should we be value betting?
I put a line like ------ beside of [what I think is if I understand this correctly] the midpoint of our opponent's calling range, and it would seem like we need to bet with all of the L hands above that line (so that we get a positive return on our bet). This is assuming each B hand has the same number of possible combinations.
What happens if we start betting with more hands than this? Will our opponent start calling us with more hands? If so, then don't we start getting more value with the L hand that I marked with an asterisk since we will win more often when we bet with it?
L
L
L
B
L*
B
L
L
B
L
L
B-------
L
B
L
B
B
Okay I'm just going to go through this the best I can and see how it turns out. Right now there are 10 L hands and 7 B hands. If we assume that x is the size of our river bet, then each L has the following EV associated with it if each B is equally as likely:
L = x
L = x
L = x
B
L = 6x/7 - x/7 = 5x/7
B
L = 5x/7 - 2x/7 = 3x/7
L = 5x/7 - 2x/7 = 3x/7
B
L = 4x/7 - 3x/7 = x/7
L = 4x/7 - 3x/7 = x/7
B-------
L = 3x/7 - 4x/7 = -x/7
B
L = 2x/7 - 5x/7 = -3x/7
B
B
So the bottom two L hands have a negative EV for a value bet of size x, which seems fairly obvious without this calculation. If we just bet the top 8 hands, then we would end up with an average EV of 0.61x on those 8 hands, or an EV of 0.488x average over each of the possible 10 L hands, assuming each L happens with equal frequency.
I guess what I'm asking is, if we started betting with all L hands (even the ones that are apparently -EV), would villain start calling with a slightly wider range of B's to compensate? Since I don't really know if this change would happen, and even if it did I don't know how big the change would be in our opponent's calling range, let's just suppose there is only one more B added for the sake of example. Then we would end up with:
L = x
L = x
L = x
B
L = 7x/8 - x/8 = 6x/8
B
L = 6x/8 - 2x/8 = 4x/8
L = 6x/8 - 2x/8 = 4x/8
B
L = 5x/8 - 3x/8 = 2x/8
L = 5x/8 - 3x/8 = 2x/8
B
L-------
B
L = 3x/8 - 5x/8 = -2x/8
B
B
B
The L with the line beside of it would now have a neutral EV, but the average EV the L hands that we bet is now 0.5x.
In the first set of hands, we're betting 8 hands for an average EV of 0.61x. So for the 10 possible L hands, we land an average EV of 4.88x/10 = 0.488x. In the second set of hands, we're betting all 10 hands for an average of 0.5x, so now out of the 10 possible L hands we're landing an average EV of 0.5x. I hope this makes sense.
The example I'm trying to give is probably a bit screwed, but I'm really just trying to apply Shania to value betting. I imagine that even if I'm correct, it would be something so subtle that it's practical application would be difficult.
Edit about a half an hour later: The opposite occurred to me as well. If we bet less L hands, suppose only the top 5 or 6 in the original example, then maybe our opponent stops calling with the worst of his B hands to compensate which could cause a net gain or loss for Shania (much like when we start betting with more hands, it could cause a net gain or less for Shania, but often it's going to be hard to decide which). Then it would make sense that we should switch between betting a few more hands and a few less hands than what we deem optimal so that our opponents are always calling either too tight or too loose against us -- I think gabe said something similar about oscillating ranges in some other thread, though I doubt it was in the context of river value betting (not that it matters).
I'm really f'ed up on cold medicines right now so if some of this doesn't qualify as English then I'll come back and revise later.
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