cashgame nl500 6max :

UTG & MP fold
CO raise KcQd
BTN & SB fold
BB call

Flop : Qc9c5s (pot = 1)
BB check
CO bet 0.5
BB call 0.5

Turn : 7c (pot = 2)
BB check
CO bet 1
BB call 1

River : 2c (pot = 4)
BB check
CO bet 2
BB raise 4x2=8
CO?

So the pot size is 14 and I have to call 6. I hold 2nd nuts.

Is there a mathematical way to solve that?

I am not looking for an exploitative logic based on reads, Villain's profile, etc. Let's say we face a balanced strategy : I'd like to know if there's a mathematical solution.

If you're holding the second nuts here at 500NL, call xD

on a serious note:

6/(14+6)= 30% is the amount that we should be winning when we call, for our call to be 0 EV. Therefore that's how much villain should be bluffing (when he's counting in the effect of our bluffcatcher's blockers).

If we include the 5% rake for example, it would be 6/(20*0.95)= 31,6%.

If we include the fact that we beat/block some of his value range here, our call becomes +EV. The exact math depends on how often he has Kc/Qd in his value range, or Qd in his bluffing range, but this should be one of our best bluff catchers.

We should be bluffing river here 2/8 = 25% of the time.
When we're check raised by the GTO strategy, we should be folding 8/14 = 57% of the time to be balanced.

ty ZKesic for your answers

where do you get this from? I don't understand the idea here. You are talking about Villain, right? or Hero?

I don't get this one either sorry.

OK, so Villain should do this move with a range including 30% (or 31.6%) of bluffs, right?

yes

When we bet 2 into a pot of 4 on the river, opponent has to call 2 to get 6 right? So our bluff:value ratio should be 2:6. Therefore, we should be bluffing 2/(6+2) = 25%, for opponent's blufcatchers to be breakeven.

When opponent check raises river to 8, he's risking 8 to get a pot of 6.
Meaning, his bluff has to work 8/(8+6)= 57% of the time.
That's why we're folding 57%. If we folded more, his bluffs would be +EV, if we folded less, he shouldn't ever be bluffing.

I don't think think anyone can come up with some math for this situation. You dont have a range here, you can say it's polarised,ie he is raising with the nuts or nothing.

But think logicaly, if he was bluffing ,would he have checked the river when the fourth club came, if you don't have a club you probably check back. He would have Lead into you, if he was bluffing.It makes sense when there is 3 to a flush and he has Ac as a blocker to make this move.In your particular case Iam surprised if he don't have the nuts.

CO bet size is what determines the answer, imo.

CO bet half pot for value, and got checkraised full pot. Seems like a clear fold in any scenario where we allow the BB to bluff.

Even if BB is bluffing, CO preserves EV by folding. The math involved here simply resolves back to bet size in these scenarios. Better is to understand bet size as a signal, and that BB was more balanced in bet size than CO.

If CO wanted the option to call, modeled by accounting for the ability of BB to bluff, then CO would have bet full pot.

So, CO was tilted towards value with his bet size, and BB is perfectly balanced. So, CO should concede the pot and his equity there based on the logic of his bet size as CO.

If you want value by calling bluffs, you should bet the size of the pot to seize this value.

Of course, if BB is a very predictable player and we model him as one that never bluffs, then we SHOULD bet less than pot, but ALWAYS fold if checkraised.

You just have to work out the chance of him holding that nut A. You then compare that probability to the odds given to you by the price.

In this example we're getting a price which is a little better than 1-2, so, if you think he holds the A more often than 2 in 3 times you should fold.