My question is this

If we need to call lets say 50 percent to make to prevent our opponent from bluffing with any two cards.... where does are profit come from? Should we actually be calling a little more than what the equation tells us.

(I think I know this but I'm kinda going two different ways with it so I would love some clarity)

It's not about making profit it's about preventing additional loss.

Remember we're out to make the most +EV decisions we can and sometimes that means trying to chose between a 0 ev decision and a -EV decision.

And no you should not be calling more. The indifference calling frequency balances paying off villain's valuebets and snapping off her bluffs.

why is villain now a her?

Haha I try to be gender neutral in conversation even though I know there is the feminine noun villainesse. I use villain for convenience just like I might use actor to refer to an actress. I'm not perfect! Haha

My understanding is...

If you called epsilon % more than your mixed strategy optimal calling percent, the nemesis will automatically adjust to never bluffing as now EV Bluff < EV Not Bluff and you will only be paying off value bets

If you called epsilon % less than your mixed strategy optimal calling percent your opponent will always bluff and although you will always pick him off, the villain will more than compensate by picking up the pot enough when your hand is worse than your (now smaller) calling frequency

Now of course you would observe this and adjust yourself leaving the game with oscillating pure strategies. It's been shown that this actually isn't optimal

Bill chen's book (MOP) offers god insight into this

Good luck at the tables
Hutch

Now adding to the above post....... by doing this we can prevent ourselves from being exploited since they don't have incentive to bluff more or less frequent.

Now the second part would be.......also in the process of doing this we can gain if our opponent bluffs too often as we will win more often in showdown and if he bluffs too infrequently we will then profit by being able to releaze our equity more often and we can now bet and profit if they fold too often (and maybe gain as well if they call too often since we then win at showdown more??)

I guess I want some clarity on the second part as well.

Thank you so much for everyone that has read and responded!!!

If a villain bluffs more frequently then equilibrium you should call 100% of the time you hold a bluff catcher. If a villain bluffs less frequently than equilibrium you should fold 100% of the time you hold a bluff catcher.

You can think of it almost like getting the right odds to call on a draw.

I follow that. As far as we can exploit them once we know they are deviating from equilbrium..... but we would still profit from there unbalanced stragey, in those other respects I stated???

That´s a great way way to explain it.

It's about calling frequencies with our bluffcatchers, not about betting and raising.

Opponent bluffs too often -> we call our bluff catchers
Opponent bluffs too little -> we fold our bluff catchers

Since our opponent has a bluff, he will either bet it or check/fold or check/raise but never check/call. Since our bluffcatchers win the pot 100% of the time against his bluffs (at least in theory) at showdown and he will never check/call his bluffs, there is no reason for us to bet the river with a bluffcatcher. At least if we don't go two steps ahead and assume that we can bet our bluffcatcher because he will check/raise his bluffs too often and we can call profitably.

The exploitation is that we don't pay additional money when their range is full of value hands and get more money when their range is full of bluffs.

The assumption here is that you have a bluffcatcher. This means we are on the river and we know villain is only betting with hands that win 100% vs our bluffcatcher and hands that lose 100% vs our bluff catcher.

first step - being unexploited
as soon as you spot inbalance you start exploiting him.
If he checks you can bet all the hands he would bet and some top part of his checking hands. You put it right....so cant see what the quest is.

I guess I'm asking if we just defend the whole way using those percentages and don't deviate, where do we make money on this strategy?

If they bluff too much are we making money cause we win at show down much more? If they bluff to infrequenty do we make money because now we have a chance to bet, or can realize equity we wouldn't.. ect. ect.

You asked about making the other player indifferent.

If your strategy wins money and therefore his strategy loses money, he has an incentive to change his strategy. If he has that incentive, he's not indifferent.

Thanks for the response. I think I was confusing some aspects of flop and turn defense frequencies and River play (betting to make villains bluff catchers indifferent or calling to make him indifferent to bluffing.