I'm having some confusion as to how overbetting a balanced range increases EV in comparison to a smaller sizing. I understand the basic principals of when its effective, but having a disconnect with the math. I was given the layman's explanation of, "it allows us to bluff more combo's while staying balanced, so we can face them with a bet more often and "claim" the whole pot. This doesn't quite make sense to me, because aren't our bluffs in theory neutral EV/break even? And if they are calling with a GTO frequency, we will net the same amount when called with our value hands? Thanks for any help.

That's a pretty good question. Here is a tentative answer.

If Villain is calling with GTO frequency (i.e. is properly responsive to our bet-size) in some spot, then there is no question of exploiting him, though, of course, there still is the question what value bet size is optimal. I think the key thing to remember is that when both players optimize bet size and betting/calling frequencies, this makes Villain indifferent to folding or calling with his bluff-catchers -- though not, of course, with trash hands that don't beat a bluff or with hands that beat most of Hero's value hands. Now, except in rare cases, river calling ranges are elastic. That is, whether one hand belongs to the (indifferent) bluffcatching sub-range or to any one of the two other sub-ranges ('trash' and 'value-call' hands) is dependent on Hero's bet size. So, it isn't generally the case that betting bigger is better. The optimal bet size, at equilibrium (when both players play GTO) is dependent on the specific shape of both player's ranges in that spot.

The "bigger is better" maxim, I think, mainly applies to cases where we want to exploit hapless Villains who have unbalanced capped ranges and/or don't adapt their calling frequencies to bet sizing very well. They are exploitable when they get themselves in tricky river spots through poor play on earlier streets or when they might get there correctly but then can't defend their capped river ranges with proper calling frequencies. In that case, if villain is weak, we can bet bigger and, while we lose some value through making him fold 'too much' we are more than compensated through being able to bluff him much more frequently. In the case where Villain rather is a river-calling-station, then we lose more when our bluffs are called, but are more than compensated with the value part of our betting range (though we would win even more though not increasing our bluffing frequency, of course).

So, the neat thing is that when we ourselves are boosting our betting size against Villain's (unbalanced) capped ranges then, in case we don't know if Villain is weak-tight or a calling-station, it's still possible to exploit him with theoretically correct betting frequencies just by dint of the fact that Villain's range is capped. Of course, when we have a proper read about Villain's calling tendencies, we may exploit him even harder through bluffing (almost) any two cards or (almost) never bluffing. But even while maintaining a balanced value/bluff ratio we can still bet bigger in both cases and increase EV so long as Villain stays in any direction from correct calling frequencies.

The bigger you bet, the less often villain needs to call to avoid exploitation, therefore, if for example you are betting a range that wins 40% of the time when called, you need to bet bigger than when you are betting a range that wins 60% when called.

This sentence needs a lot of qualifiers; overbetting doesn't always increase your EV. Your post is alluding to the nuts-air (AQ vs. K) toy game or some similar variant, where our value bets have 100% equity when called regardless of the width of villain's defense range.

Thus the answer to this question is no:

Consider the following situations: (a) we jam one PSB and villain calls with 1/2 of his kings, and (b) we jam nine PSBs and villain calls with 1/10th of his kings. Both calling frequencies are the "GTO frequencies" in this toy game for their respective bet sizes.

EV of jam a: (1/2)*(1+(1+1) - 1) + (1/2)*(1) = 1.5

EV of jam b: (1/10)*(1+(9+9) - 9) + (9/10)*(1) = 1.9

Our value shoves gain more when we can bet larger.

I think it should be relatively obvious why, in practice, people don't just indiscriminately overbet jam rivers with their entire betting range--but if you want an explanation, let me know.

The inverse of this sentence (from our perspective instead of villain's) is "It allows us to make a bet that has an EV of +(pot) more often, despite the fact that a higher % of our betting range consists of bluffs (i.e., more neutral EV bets). If the EV of our bet remains +(pot) even as the proportion of neutral EV bets in our range increases, it must be that our value bets are becoming more +EV."

We can simplify that further to "It allows us to bet for value more profitably, which increases our EV," since we're value betting the same % of our overall range regardless of our bet size.

The layman's explanation is confusing because "claiming the whole pot more often" isn't actually the reason our bet is more +EV; it's more like a side effect.

Only threshold bluffs are 0ev; on the river all bluffs are threshold bluffs when considered a part of an optimal strategy. Though on the earlier streets semibluffs will show a profit.

I don't know of any formula that takes into account the money won from a value range. Before the river this would be hard to compute because of all of the possible river cards. Once the river has fallen though, it's easier to compute the money won from a value range as a percentage of the pot determined by the number of combos the opponent will call with.

If your opponent only holds bluffcatchers and you have nuts or air, then your value hands own 100% of the pot. If your opponent has some value hands and some bluffcatchers, then your value hands own a smaller percentage of the pot depending on the overlapping parts of your ranges; some of your value hands will lose to the top of your opponents calling range.

Regarding overbetting specifically:

I can think of a few reasons to overbet.

1) Your opponent sucks at poker and will call with hands that should be clear folds.

2) You're trying to set up a smaller spr on the next street.

3) The stack to pot ratio has been mismanaged by someone on a previous street.

I meant "strays" (deviates), not "stays".

If there's 1 in the pot, the EV of your jam can't exceed 1. Nemesis is not putting a cent in the pot if it would make your EV 1.9.

I guess he's computing ev of final hero's stack (hero and villain start with a stack of 1 and a pot of 1).
Although this is not the most used convention it's surely the best one in a lot of spots.

Those are the EVs for our value shoves, not for our entire range, as context indicates.

I'm not.

Ah ok, makes more sense then. All the 9's inside the parentheses took me by surprise.

Well, when you look at it from villain's perspective when we get to bet big: he doesn't care whether it was the extra we make with our value bets, or if it's the extra Q's we get to bet, he was just robbed some EV because he faces 0EV spot more often.

I'd rather call "value bets gaining more" the side effect (I'm not usually particularly interested in how much each part of my range makes, but how much all of my range makes), but really, I don't think that matters as long as the math works. If you find this the easier way of explaining the effect, it sounds fine.