Hello! This question might be stupid, but its been on my mind for a while, so I figured Id ask:

Assuming on the River villain bets, we know are good 40% of the time and 60% of the time we lose, that means, if villain bets 2 times the pot, we are offered 1:1.5 and we are still calling break even.

Assuming he bets 2.5 times the pot, we are not getting the right odds to call. Is there actually still a way to profitably call this, for example if we call the river x% of the time and fold y%?

You maximally exploit him by folding 100% of the time. If you'd rather not maximally exploit him (so that maybe he doesn't adjust as quickly and you save $ in the long run), you can exploit him by folding more often than the percentage that makes him indifferent.

Can you figure out the % that makes him indifferent?

Hm sadly I dont know. Im also not sure what exactly indifferent means. Could you please explain?

The pot is 1. He bets 2.5

40% of the time you win the pot plus his bet, a total of 3.5
60% of the time you lose your bet, a total of 2.5

EV = .4*3.5 + .6*-2.5 = -0.1
hopefully, you understand this part?

Now let's say instead that you fold some of the time and call some of the time. Let's say you fold with probability F and call with 1-F. The calling part of the EV is the same as before, but the folding part of the EV is 0. If you fold sometimes then you lose less than if you call all the time, but unless you fold every time, your EV < 0.

EV = F*0 + (1-F)(.4*3.5 + .6*-2.5)
you see how this is the same as before except broken into the folding/not folding cases?
EV = 0 + (1-F)(.4*3.5 + .6*-2.5) = (1-F)(.4*3.5 + .6*-2.5)

So your EV if you fold, say, half the time is
(1-0.5)(.4*3.5 + .6*-2.5)
(0.5)(.4*3.5 + .6*-2.5) = -0.05

If you want to find the breakeven point, it's where your EV is 0. So,
EV = (1-F)(.4*3.5 + .6*-2.5)
0 = (1-F)(.4*3.5 + .6*-2.5)
0 = (.4*3.5 + .6*-2.5) - F*(.4*3.5 + .6*-2.5)
0 = -0.1 + 0.1F
0.1 = 0.1F
F = 1
i.e. folding 100% gives you 0EV.

(If there is any step you don't understand, I can elaborate)

thanks a lot! So no way, we can call his bet and ever end up profitably.

Well maybe there are still benefits to calling some % of the time, although -EV. To keep him from bluffing too much, maybe thats what heehaw refered to

No, I think heehaw was asking you to think about the size of the pot where your call would be break even - that's the bet size that makes you indifferent.

hm yea maybe. I just dont really understand his post: why should we fold more often than we could call break even, in order to save money long-term... dont get it ...

Villain's play is imbalanced because you aren't indifferent to calling or folding. If you always fold and if he doesn't suck and if you'll be playing enough hands with him, then he'll adjust that 60:40 ratio. If he adjusts closer to the equilibrium point (which would make you indifferent to calling or folding), that's bad for you in the long run. If you can prevent that adjustment just by calling once in a while, you might want to consider it. It may or may not be a good idea in this particular example, but you asked if there could be a reason not to fold, so I provided one.

From Villain's standpoint, if he bets 2.5x pot, then when he's bluffing he's risking 2.5 to win 1, so he needs you to fold 5/7 of the time in order to break even on his bluffs. If you fold exactly 5/7 of the time, then when he has a losing hand, he is indifferent to bluffing or checking, because it's 0ev for him either way.

Since he's not bluffing often enough, you exploit him by folding more often than 5/7. By "exploit" I mean you lose less money than you would if he were bluffing at exactly the equilibrium frequency (the frequency which would make you indifferent to calling or folding).

So maybe you can keep him from adjusting by folding 90% (a good amount more than 5/7) of the time instead of 100%.

Actually though, he's bluffing 40% of the time which is already very close to the frequency which would make you indifferent (5/12 or 41.67%). If you fold 100%, he's probably more likely to over-adjust farther from 5/12 than to move closer to it. Farther is better for you (as long as you notice it before he steamrolls you). Then you'll be playing a game of adjust / counter-adjust, and it will be a pendulum approaching 5/12 bluffs by him and 5/7 folds by you.

thanks for the detailed response!

by adjusting closer to the equilibrium point, you mean him betting his bluffs slightly bigger than 2 times the pot, but smaller than 2.5 times the pot, so hes losing less on his bluffs, when we are calling? And him never betting smaller than 2 times the pot because then he will lose on his bluffs?

if thats not what you ment, it sounds a bit contradicting, because if hes adjusting closer to the equilibrium point, where we are closer to getting the right odds to call, his bluffs will go through less often, right? So how would that be bad for us and good for him?



So practically, 1 out of 10 times, we are willing to make a -EV call, so villain sees that although hes laying us incorrect odds, we are still calling and that way we are trying to stop him from bluffing us too often, right? Would you advise that ratio (1/10 times) generally when we are not getting the right odds or is that strongly related to that 1:2.5 scenario?

He could reduce his bet size (with his entire range in that spot) and/or increase his bluffing frequency. I was talking about the possibility of him only doing the latter.
Suppose his 60/40 is 30% and 20% of total rivers respectively, and that he check/folds the other 50%.

If he sees us always folding and then adjusts by bluffing 5/12, he'll do so by adding more bluffs to his range. Whenever he bets, his EV will still be Pot, but now he'll be betting more often than 50%, so he'll win the pot more often than before he adjusted. This is why, with polarized betting range on the river, it's best to make the largest bet that you can balance. A larger bet allows you to bluff more often and take more pots.

If, instead of bluffing more often, he reduces his bet size, that's different. If he reduces it to the size that makes you indifferent, then whenever he bets, his EV is Pot, but he's still only betting 50% of the time and so this adjustment did not benefit either player.

The 1:2.5 scenario, but really neither. My point was academic, partly because 40% is already close to 5/12, partly because he might adjust his bet size instead, and partly because you don't know his exact ratio of winning/losing hands. What I'm talking about in these posts is a toy game of nuts/air range vs bluffcatcher with only one bet size allowed.

But the general concept of exploiting less than maximally, for the purpose of delaying the counter-adjustment, can apply. How much less depends on your read of how badly you can get away with exploiting him without him adjusting.

thanks again for the detailed answere, trying my best to understand what youre saying.

How exactly do I define the largest bet that I can make? If I know how many value-hands I'll have in that spot compared to how many possible bluffs, then figure out the ratio and somehow determine the betsize?



This might be another stupid question as I'm not sure if what you're talking about already is GTO. Unfortunately I dont know much about GTO, but by my understanding, GTO presents a solution without knowing villains ranges or tendencies (bluff frequencies, betsizes). So since all of what you're saying sounds interesting but hardly applicable to in-game poker, I wonder if theres an easy solution that allows us not to become exploitable, without having any informations about villains tendencies (GTO)?


Another solution might be to study common pool tendecies to see what ratio a pot or overbet turned out to be a bluff/value-hand. How do you think about that?

Suppose the effective stacks on the river are 10x pot.

If you've shown up to the river with a range of 4 winning combos for every 1 losing combo, then betting 1/3 pot allows you to bet your entire range while making him indifferent. If you bet any less, he'll always call and you won't profit as much overall. If you bet less and bluff less often to compensate, he'll be indifferent, but you won't take as many pots, so again you'll profit less overall. (Btw, when I say "taking pots" in this context, I mean "Galfond pots" or G-pots if you will. When he calls your bluff, you didn't take the pot, but you did take the G-pot because he's also sometimes paying you off with your winning hands, which comprise the majority of your range.)

In that situation I'd advise betting a little more than 1/3. If she uses pot odds and knows your range's ratio of winners to losers, she'll always fold, but it won't cost you anything and it will give her a chance to make a mistake. It won't cost you anything compared to betting exactly 1/3, because when you bet exactly 1/3, it makes her indifferent, which means your EV is Pot regardless of what she does. When you bet more than 1/3 and she always folds, your EV is still always Pot (except without variance), and you're still collecting that EV with your entire range. She's not exploiting you by always folding; she would be if you weren't already betting your entire range with this bet size.

If you have 6 value hands for every 5 losing hands, you can bet your entire range by betting 5x pot.

To bet all-in you'd need a ratio of 21:10 value to bluffs.

You always need more value than air in your betting range, so if you showed up to the river with more air than value hands, you'll have to surrender with some of your air rather than always bluff.

How to figure out these sizes, well you need to make Villain indifferent to calling, so make her EV of calling equal 0.

Her chance of winning is equal to your bluffing frequency, f.
Her payout is 1+x, where x is the size of your bet in pot units.

(1+x)f - (1-f)x = 0
...
x = f / (1-2f)

If f is a fraction a/b, I prefer: x = a / (b - 2a)

E.g. in the 6:5 example, f=5/11 so x = 5

Make the opponent indifferent. Not easy in general, but easy on the river. In your OP example, you're unexploitable if you call 2/7 of the time, so that's the GTO calling frequency. You can derive it by setting the EV of Villain's bluff to 0.

Regardless of whether you're the bettor or caller, you don't need to know Villain's range to determine your GTO frequency. However, to know whether your actual hand falls in that frequency requires that you know own range, which most people don't well enough. For instance, if you're trying to call with 2/7 of your range, you want it to be the top 2/7. If you don't know your own range, and your hand is top 3/7, you might make the wrong call.

Yeah that can help against unknown villains if you're trying to exploit them from the start.

Edit: I made up a phrase "Galfond pots" without referencing Galfond Bucks or "G-bucks", which were coined here (and an alternate use was written about here). And then there is this good article about g-bucks and beyond.

thats very elaborated, thanks a lot that you took the time to explain me. awesome!