Quick one
both players play with optimal frequencies and the get to the river.
If I get it right the eqilibrium point is when A bluffs 100 into 100 pot, he should have 33% bluff 66% value as player B gets 33% odds.
Should player B call with 50% of his combinations he gets to the river with or 33%? 
Just reverse the roles now and figure out odds for B. He has to call 100 to win 200 so 100/(100+200).
You have to include the 100 that B calls into equation because once he calls it becomes part of the pot. Left that out in my intial reply.
Originally posted by serverm07
Just reverse the roles now and figure out odds for B. He has to call 100 to win 200 so 100/(100+200).
that's direct odds, not game theory unexploitable frequency ken what I mean pal.
basically question is if my frequency should be on odds I'm getting or those he's getting on his bluff
2:1 value:bluff ratio from Player A's perspective is the balanced/game-theory optimal frequencies for his range.
33% pot odds for Player B is what he's getting. If he knows player A's exact range, he can make an exploitive call if he has >33% equity vs the river range.
If he has no idea what Player A's range is, he can use the optimal defense frequency of calling the top 50% of his range.
I misunderstood what you meant. Also wanted to point out that gto only focuses on own range you dont care about opponents strat.
yeah I got it right thanks mbml
it just felt weird that he has 33% bluff combos and I call 50% of my range
Calling 50% if the bettor is perfectly polarized.
Caller is getting 33% pot odds and winning 33% when calling.
If bettor is not perfectly polarized it's a bit more complicated: http://www.pokerstrategy.com/strategy/bss/2314/1/
Originally posted by LemOn36
yeah I got it right thanks mbml
it just felt weird that he has 33% bluff combos and I call 50% of my range
That's because of the dead money from earlier streets.
If there was no dead money and therefore villain bets 100 into 0, you'd need to call with 33% of your range against 33% bluff combos. Am I right?
nope.
If the pot is zero and villain bets 100, he is risking 100 to win 0, meaning that your optimal defense frequency is 0%
If we look at it from an exploitive standpoint, we need our range to be 50%+ against his betting range in order to make a call.
Originally posted by mbml
If the pot is zero and villain bets 100, he is risking 100 to win 0, meaning that your optimal defense frequency is 0%
assuming you never have nuts
For the caller there is a simple mnemonic to remember what is right:
If you get N to 1 to call then it should be 1 in N that you fold. Odds become "chance", which is not exactly the mirror image.
In the example you get 2 to 1 to call, and fold 1 in 2 = 50%. A half-pot bet would be 3 to 1 giving fold 1 in 3 = 33% (and not 25%).
Here is a slightly harder nut to crack:
On the river the pot is $100, you bet $100. You are raised, and it will cost you $200 to see showdown. You realize that your hand now is a bluff-catcher only. Your opponent is playing GTO. How often should you call if you want to be unexploitable?
Hint: Now the mnemonic above, in a way, both does, and does not work 
Edit: One more hint. This really is pretty tricky. Whether you call 100% or 0% or something in between for a while doesn't matter because the opponent makes this bluff with the correct frequency. But, in the long run, ... Think for a while about how much the opponent invests in a bluff of this sort.
Originally posted by YohanN7
For the caller there is a simple mnemonic to remember what is right:If you get N to 1 to call then it should be 1 in N that you fold. Odds become "chance", which is not exactly the mirror image.
In the example you get 2 to 1 to call, and fold 1 in 2 = 50%. A half-pot bet would be 3 to 1 giving fold 1 in 3 = 33% (and not 25%).
Not true.
Only true if your range is 100% bluffcatchers, meaning every hand in your range beats every bluff your opponent has. Or to put another way, only true if your opponent is truly polarize.d
Originally posted by lnternet
Originally posted by YohanN7
For the caller there is a simple mnemonic to remember what is right:If you get N to 1 to call then it should be 1 in N that you fold. Odds become "chance", which is not exactly the mirror image.
In the example you get 2 to 1 to call, and fold 1 in 2 = 50%. A half-pot bet would be 3 to 1 giving fold 1 in 3 = 33% (and not 25%).Not true.
Only true if your range is 100% bluffcatchers, meaning every hand in your range beats every bluff your opponent has. Or to put another way, only true if your opponent is truly polarize.d
Of course it is polarized range versus bluff-catchers. If you have a total mixture of hands, you can't even formulate GTO. Statements like "You should call with 50% of your range" are nonsense because a "range" could be pretty much anything. I thought this was obvious.
Edit: You can also assume that your range is "perfectly balanced". Then you explicitly have to point that out. If you hold what you think is a bluff-catcher, then it's "place" in what you think is your total range in this spot can act as a randomization device.
Better, and probably simpler in practice is to user your watch as an RNG. The position of the second hand could tell you what to do. This is much simpler than, in a live situation, figuring out where in your range your hand is. It eliminates the risk (the fact) that you aren't playing perfect GTO poker coming into the river.
On the river the pot is $100, you bet $100. You are raised, and it will cost you $200 to see showdown. You realize that your hand now is a bluff-catcher only. Your opponent is playing GTO. How often should you call if you want to be unexploitable?Hint: Now the mnemonic above, in a way, both does, and does not work confused fish confused confused
Ok let me try. 
So we are getting 5/2 odds so we should fold 2/5 = 40%.
If we fold 40% then opponents EV of bluffing is: 0.6*200-0.4*300=0
Hmmm I thought there would be a trick somewhere... 
Where did I make a mistake???
EDIT: Ah yes I made a mistake!
If we fold 40% then opponents EV of bluffing is: 0.4*200-0.6*300=-100
So folding 40% is not the GTO.
Great mnemonic rule, YohanN7.
If you have a total mixture of hands, you can't even formulate GTO.
Doesn't feel right. Could you please comment on this, lnternet?
I meant that it becomes insanely difficult to formulate GTO strategies unless ranges are assumed perfectly balanced or totally polarized.
I'll be back with the answer to the question, I had it nicely formulated, but now I'm a bit dizzy ?(.
I'll give it a try below. I hope by Jupiter that this is correct because I'm really dizzy, and these matters can be confusing as hell, even if you (think you) know them.
If you call more than 40%, then you'd be exploitable; the bluffer should never bluff (he loses $300 every time you call, but wins only $200 when you fold).
On the other hand, if you call less than 40%, then the bluffer could bluff 100% of his hands and show a profit.
So if we fix the calling frequency at 40%, what happens if the bluffer bluffs 100% of the time? He'll break even (0.6*$200 - 0.4*$300 = 0). Likewise, if he never bluffs, he trivially breaks even.
This independence of the result of his bluffing frequency shows that 40% is the GTO calling frequency. Note that the pot odds tells a different story, the "mnemonic" above (valid for bluff bet and call) would say call 60%. The difference is that it is much more expensive for the bluffer to bluff with a raise than a bet.
How often should the bluffer bluff? If he bluffs too often, you could exploit him by calling more, and if he bluffs too little by calling less. He is giving you 5:2 to call, so it should (as usual) be 5:2 against that he is bluffing. Let us fix the bluffing frequency at 2 in 7.
Now what happens if you call 100%? 2 times of 7 you will win 500 and 5 times of 7 you will EDIT:win -> lose 200. This is zero zip nada. The same as if you call 0%. Again, this independence of the outcome of the frequency (this time of the caller) shows that 2 in 7 (or 5 to 2 against) is GTO.
Hence, if you bet $100 into a $100 pot and are raised (all in) so that it costs you $200 to call, what are the GTO bluffing and calling frequencies?
The answer:
The bluffer should bluff 2 times for every 5 value hands, i.e. 28.6% (5 to 2 against or 2 in 7).
The caller should call 40% of the time (2 in 5).
Neither can deviate from these numbers without becoming exploitable. But both can deviate provided the other player does not.
These results are perhaps surprising, and the corresponding numbers for check-raise-bluffing are even more surprising. There it is easy to look too much at the pot odds for a call.
Originally posted by YohanN7
Of course it is polarized range versus bluff-catchers. If you have a total mixture of hands, you can't even formulate GTO. Statements like "You should call with 50% of your range" are nonsense because a "range" could be pretty much anything. I thought this was obvious.
I've posted a link earlier how to determine a close to GTO calldown for an arbitrary range: http://www.pokerstrategy.com/strategy/bss/2314/1/
Originally posted by YohanN7
Better, and probably simpler in practice is to user your watch as an RNG. The position of the second hand could tell you what to do.
Blocker effects rank your bluffcatchers. In most poker situations this helps define a pretty clear threshold for which bluffcatchers call and which fold. Only rarely are all your bluffcatchers exactly equal.
