[***EDIT***]
I didn't realize that it's actually specified in the OP that stack sizes are 1 PSB. This post was made thinking there was no limit on the bet size. It is all still correct I believe since I included stipulations on the limit of the bet size.
[/EDIT]
Quote:
Originally Posted by robert_utk
Another way to look at it, is that Player B ACHIEVES a loss of 1 pot by defending optimally.
This is kind of a weird view. Player B doesn't need to do anything but fold 100% in order to achieve the maximum loss.
Quote:
Originally Posted by Maroel
I think that any strategy between calling 0% and calling 50% against a Pot Size Bet is optimal for player B because player A can't unilaterally improve against them in this specific scenario
Quote:
Originally Posted by ZKesic
This is correct.
However in NLH, spots like this shouldn't really exist. You pretty much always have wider bluff range than value range.
It's not correct, as I'll show at the end of the post, unless stack sizes are not greater than 1. It's true if the bet is limited to the size of the pot.
Quote:
Originally Posted by robert_utk
Ok, no problem. We don’t have to continue going back and forth.
However, I don’t want anyone else to read:
Originally Posted by Maroel:
Having some hands that can win is not a justificacion for using a calling strategy in a gto situation.
and think that such a statement could ever be correct. It is not.
Good luck,
-Rob
He's right and you're the one misunderstanding something here, or we are misunderstanding your point. The game we're talking about is an example of what Maroel is saying.
Quote:
Originally Posted by Kingkong352
It s confusing bc you guys seem to have changed the rules. But maroel is right. Polarized 70-30, bluffcatcher has to fold more than 3/4 bet size and call less. And call like 57% of the time vs 3/4 size.
I think what Robert wants to say is, say player A is perfectly balanced and polarized, he makes your bluffs indifferent. So if player B has a hand that beat some value hands, this one is logically nOt indifferent and call is +ev.
But i dont know if we re still talking about the original toy game.
The only equilibrium strategy is for player B is to fold 100% of the time to any sized bet. This is pretty clear when trying to find another equilibrium strategy. For example if player B calls sometimes to a half pot bet, A's value hands will choose that size and win more than pot. A's bluffs can still win the whole pot by betting 2x pot if B never defends at that size, so clearly B has to defend some at 2x pot for there to be an equilibrium, however we already know that A can range bet for 2x pot and profit more than pot if B calls sometimes. Clearly B can not defend against any bet size by calling or A can profit.
I also tried to prove it using a pure math approach, although it was a little tougher for me than the logical approach.
This equation is the EV of player A's value bets following the maximally exploitative strategy against player B:
EV = max(P+(1-f
b)*b) = P+x
1, where x
1 is the greatest gain over pot that B's strategy allows A's value bets.
P >= 2/3*(P+x
1)+1/3*max(f
b*P-(1-f
b)*B)
1/3P-2/3x
1 >= 1/3*max(f
b*P-(1-f
b)*B)
1/3P-2/3x
1 >= 1/3*max((1-x
2 /B)P-x
2 ), letting x
2 be [0, x
1].
1/3P-2/3x
1 >= 1/3*max(P-x
2(P+B)/B)
-2/3x
1 >= 1/3max(-x
2(P+B)/B)
-2/3x
1 >= -1/3x
2, taking the limit as B -> infinity to find the functions maximum
The final equation is only satisfied at x
1 = 0, therefore A's strategy can only gain EV if B allows A's value bets to profit more than pot.
B can defend some hands by calling without being exploited if the maximum bet is limited to the size of the pot or less, as the final step in the proof requires unlimited bet sizing to be allowed.
Last edited by browni3141; 11-26-2017 at 06:52 PM.