Pot is 1 on the river, hero bets 1(full pot) and has 2 value bets for every bluff hand as he should here.

Villain has only bluffcatchers that can beat heros bluff hands but loses to value hands.

MDF dictates 50% calling frequency agains pot size bet.

Villain calls 50% of the time randomly.

Does both players break even here?

if villain calls everytime, villain breaks even but hero wins?
if villain folds everytime, villain breaks even but hero wins?
if villain calls 50% of the time, both break even?

Lets have some game theory conversation TIA

Villain changing his calling frequency with bluff-catchers does not change the EVs, as long as hero keeps playing 2:1 value:bluff. Each call is exactly 0EV compared to folding, so calling 0%, 50% or 100% always results in exactly the same EV for both players.

the bluffcatcher essentially gets the call back(0ev), but the bettor earns pot+bet (+1 pot ev)

well what would be the right frequency of calls to make hero break even? can we calculate this someway? because there must be a response to combat heros strategy right? somekind of equilibrium?

Only way to gain ev for the caller is to bring more strong hands to the river.

At equilibrium in this 'toy game', when one player bets pot with a correctly balanced polarized range, and the other calls with pure bluff-catchers, the bettor's overall EV is pot, and the caller's EV is zero.
By calling at the optimal frequency, the caller can make the bettor's bluffs break even. He can't stop the value-bets from printing money.
Indeed, by calling at a certain frequency, the caller prevents the bettor from altering his strategy. i.e. If the guy with bluff-catchers stopped calling completely, the guy with the polarized range could forget about balancing his range and start betting ALL his bluffs. At equilibrium, he can only bet some of his air, because his bluffs will get snapped off some of the time.

great answer! this was pretty much what i was thinking of, i just couldnt figure out with my math if caller can make hero ev0 but now i understand its only for his bluffs. so villain calling randomly 50% in this case would be pretty good strategy even tho it wont change the outcome for him.

While changing the callers strategy won't change EVs as long as the bettor remains with the 2:1 value bluff ratio, calling with a frequency other than 50% will make the caller exploitable to adjustments by the bettor.

Calling <50%: The bettor can increase his EV by adding more bluffs.
Calling >50%: The bettor can increase his EV by removing bluffs.

The overall theory for one bluff and two values makes it okay to call any number of times. But it is another thing how to figure out the opponent's range.

If one has only bluff-catchers, often that is not the case.

Calling more or less than half the time here is not in theory nor in practice exploitable but possibly later.

One calls or not depending on the situation, and if there is a GTO bet, it is just the same what one does, and if the GTO player then adjusts, you can exploit him but if he thinks you will do so.

One tries to win money, so in practice, everyone is exploitable, or are you going to call (or fold to) that bet as the GTO suggests? Probably not, after you get more experience.