Let's say we bet pot $100 into pot $100 on the river.

If V call every time we bluff (33%), and fold every time we value bet (66.7%).

How is that indifferent? even V make all the right decision every time we still won't lose money? How?

how does he know when we're bluffing and when we're valuebetting?

If it is a heads up pot and you have both put in 50 dollars at the river. With those numbers pretty sure you are losing money, it isn't equal.

if he calls a valuebet you win $200, and if he calls a bluff you lose $100. $200*2/3-$100*1/3=$100. if he folds you always win $100 regardless of whether you're valuebetting or bluffing

I think you misunderstood indifference. It's not about making money if V makes all the correct decisions. As you made it out to be, V is folding when you are Value betting and calling when you are bluffing. Unless you have a gaping tell, that won't happen.

Indifference is achieved when you bluff at a rate that, makes the SAME action not effect the fact that V is losing money ie if he always calls he loses and if he always folds he loses. Here's how it works:

You bet $100 into a $100 pot. 1 out of 3 times you are bluffing and 2 out of 3 times you are value betting. We assume that V always wins if he calls and you bluff, and you always win if he calls and you are value betting. SO:

If he always calls:
1st time, you bet $100 as a bluff, he calls, wins $200, you lose $100
2nd time, you Value bet, he calls, loses $100, you win $200
3rd time, you Value bet, he calls, loses $100, you win $200

Now if he always folds
1st time, you bet as a bluff, he folds, loses nothing, you win $100
2nd time, you Value bet, he folds, loses nothing, you win $100
3rd time, you Value bet, he folds, loses nothing, you win $100

In both cases, you won $300, so no matter what action he takes, you gain, and thus we say he is indifferent to either action. If you bluff less, let's say, 1 out of 4 times, then he clearly should fold more and not pay your value bet 3 out of 4 times, and lose out on value himself only 1 out of 4 times. If you bluff more, say, half the time, then he should call more, snap off your bluffs half the time and win $200 and lose $100 half the time when you have value. So by bluffing the correct amount, you put him in a spot, where he can literally flip a coin, try to find a read, or ask a magic 8-ball, and he still loses money.

In reality, if they have ways to know when you are bluffing and when you are value-betting (if you have a tell with sizing, or just a physical tell, or when your story doesn't make sense) then you can't really achieve indifference, but if you can play your value and bluffs the same way, with the same size, and bluff the correct amount, not too much and not too infrequently, then your opponent might as well fold every time or call every time and it won't matter. You gain money.

So, you may ask, why bluff at all? Can't I just play my Value hands? Well, you can, and you still win money, but your opponents can catch up and not pay up, but more importantly, if you play X number of hands for value, and Y number of hands as bluffs (let's say X= 2y so you are bluffing a third of the time) then you win money with X+Y hands, as opposed to just winning money with X hands, thus, you win more than your fair share of pots.

Hope this helped

Because he's not psychic, and we assume Hero isn't giving off tells.

Villain is Russ Hamilton, apparently.

OP, your opponents are not superusers. They become indifferent when they break even in the long run whether they always call, always fold, or randomly pick any percentage of either option.

e.g. If someone always calls, they win 2x pot a third of the time (when you're bluffing) and lose 1x pot (the cost of calling) two thirds of the time (when you're value-betting). Their total EV is (1/3 * 2p) - (2/3 * 1p) = 0 = breakeven.
If they always fold, they never win the pot, but they also never lose a pot-sized bet in making a bad call, so the EV in that situation is 0 too.
If they call with their bluffcatcher 50% of the time, they still only win 2x pot a third of the time, but lose 1x pot the other two thirds.

No matter what strategy the opponent chooses, he cannot do better than break even. Indifference occurs when there is nothing to be gained by changing strategy. When your strategy is perfectly balanced and optimal, the opponent kind of has to shrug and say "No matter what I do, I never make money. It's pointless trying to find an exploit, because there isn't one."

I might be missing something here that might be embarrassing but OP is up $100 in this scenario where the V makes the perfect decision.

OP takes down 2 @ 100 without a showdown and loses 1 @ 100 when called and bluffing. The OP didn't lose the 100 in the pot when he was bluffing. The chips in the pot don't count here IMO since they aren't 'at risk' once they are in the pot.

As I read further we are looking at the 'indifference' from the V side of things, not the Hero side of things ... GL

I don’t see where anyone said a player loses the $100 pot. But, a player can certainly win it.Three posters have shown that hero’s EV (per hand) with a pot size bet and 1/3 bluff fraction is $100 just as you have posted.

I think the more pertinent question is when and why do we want to be indifferent

Sent from my MotoG3 using 2+2 Forums

when you don't know whether villain is overfolding or overcalling

We don't.
I don't know about you, but I prefer making easy folds, or fist-pump calls, not sigh-calls.
Generally speaking, we want to be the ones giving our opponents the "impossible" decisions, not the other way around. That's why it always feels great to shove the river (especially when you're nutted), but it's often horrible to be faced with a shove on the river when you have a bluff-catcher. Putting the last bet in is a very powerful tactic, so if we can do it with a balanced range that makes villain sigh, then we're winning.

You don't lose $100 in the pot when bluffing. You lose your bluff bet, however when you win, the chips in the pot certainly count, but your $100 bet doesn't anymore.