Assume a river spot where villain checks to hero.
The pot is 10, and stacks are 40.
Assume villain only has bluff-catchers in his range, and hero has a polarized range of the effective nuts and air.

It's optimal for hero to bet as large as possible.
My question is, where does the additional EV of betting 40 compared to say 20 'come from'?

In GTO, villain is indifferent between calling and folding no matter the bet-size (both have an EV of 0). Hero's bluffs are indifferent between bluffing and checking (both have an EV of 0).

Therefore, is EV_40 > EV_20 because hero's value hands now get more value? Or is it simply that hero gets to win the pot more often, and it's impossible to attribute the higher EV to a subpart of his range?

The additional EV just comes from our stronger hands balancing out our weaker hands, therefore our overall EV of our range of hands increase... like you mentioned when we bet a polarized range for as big as possible.

Here is a good thread
http://forumserver.twoplustwo.com/32...ht=overbetting

EV of your value hands goes up when you bet big your bluffs are 0EV every time they just telly you how big you can bet.If you dont have enough bluffs to balance your big sizing then you must bet bigger otherwise if you have 50 % bluffs then you should bet as big as you can.

If villain alters his calling frequency according to the bet-size and both players use optimal ranges to match the sizing, isn't hero's EV $10 (i.e. the size of the pot) every time?

e.g. Hero could bet $10 (a PSB) and balance his value bets and bluffs in a 2:1 ratio, villain calls 50% of the time, and hero's EV is $10.
If Hero bets $40 (4x pot) and balances value:bluff in a 5:4 ratio, villain calls 20% (?) of the time, and hero's EV is still $10.

I could be wrong here, but I thought hero could only increase his EV to more than pot if villain calls too often (more value for hero), or not often enough (hero's bluffs become profitable).

I believe the EV of hero's bluffs is the size of the pot, and the EV of hero's value is >pot.

When hero bets the river with a balanced range his EV is POT for that range. By betting larger he is laying villain worse odds to call, so he is allowed more bluffs in his range, and therefore wins POT with a greater percentage of his range. It doesn't matter how villain chooses to defend. If villain is balanced then Hero's bluffs will be breakeven and his value bets will gain EV as bet size increases.

For example, if Hero has 1 nut combo and 1 air combo on the river and he bets POT, in order to be balanced he will bet with a 2:1 value:bluff ratio, winning POT with the range he bets, so he can have .5 bluff combos and his EV prior to acting is .75 POT. If Hero decides to shove instead he should be betting with a 5:4 value:bluff ratio, and will now be betting .8 bluff combos with an EV of .9 POT prior to acting on the river.

The EV of our value bets against a balanced opponent is P+x*(P/(x+P)), where 'x' is our bet size and 'P' is POT. The EV of our bluffs is 0.

You mean he wins pot with a wider range, not with a greater % of his range, right?

Btw, in my previous post I wanted to say hero's bluffs are 0 EV, not EV of pot. Can't edit anymore.

That is value of his betting range but polarized range has checking range too which lowers overall EV overall EV.For example if hero has 5 value hands and 5 bluffs and 1 pot size stack then he bets 5 value hands and 2.5 bluffs so ev of betting is pot but he checks 2.5 so over all ev is
7.5*1+2.5*0=7.5 so 3/4 of the pot.What it is fun if hero got to the river with 2:1 value:bluff and bets the pot that strategy has higher EV then if he jams 3 times the pot and he get to the river with 1:1,that is why its important to get rid of some bluffs earlier in the hand.

So hero bluffs have 0EV but that dosnt mean it dosnt matter whether we bet because as soon as we check V wins the hand so his range has some +EV overall if we never check Vs EV is 0.You are right we only can win more then pot if V calls too much and we value bet to often,but then nobody is balanced.

I think this is correct. But to use a bigger sizing allows you to include more bluffs = maximize ev for your range imo

They are the same thing, aren't they?
Thanks to the other posters, I now understand this, I think.

Imagine hero and villain have 100 total combos each.
Hero has 50 combos of AA (value) and 50 combos of QQ (possible bluffs).
Villain has 100 combos of KK (bluffcatchers).*

If hero bets pot, he maximises EV by betting all 50 combos of AA and and 25 combos of QQ (the 2:1 ratio). With this sizing, he's checking back 25 combos of QQ. His betting range is 75% of all his combos and the EV of those is pot. His checking frequency comes from 25% of combos that never win, so overall EV of the range/strategy is (75% * pot) + (25% of 0) = 75% of pot.

If hero bets 4x pot, he maximises EV by betting all 50 combos of AA, and 40 combos of QQ (the 5:4 ratio). Now he's only checking back 10 combos of QQ (10% of his total range) and he's winning pot with 90 combos. Total EV is now 90% of pot, because he's betting a larger proportion of his range.

* Yes, I know it's impossible to have 50 combos of AA in real poker. Just imagine that AA represents "nuts", QQ represents "air", and "KK" reps "bluffcatcher".

They are. I was mistaking starting for betting range. I should read before posting You perfectly explained it btw. Looks like the extra EV when betting larger simply comes from the larger % of your starting range you can bet with EV of pot compared to EV of 0 (checking), and the fact you get more value with your value hands is just a side note.