Assume a river bet where villain will either call or fold. It is fairly well known that if if you make a pot size bet you should bluff 1/3 of the time (e.g., see Theory of Poker). This is a GTO play for villain will be indifferent between calling and folding for his EV is zero for both actions (ignoring rake). This is also true for any probability mixture of the two actions.

What often is not stated is that it is assumed if you make the value bet you will always win if called and if you bluff and are called you will always lose. These are somewhat extreme since bluffs sometimes win and value bets sometimes lose.

Using the same break-even EV analysis as was done for the above case (set the EV equation to 0 and solve for the bluff frequency), I calculated the required bluff frequency for various combinations of value bet-win and bluff-win probabilities. The results are shown below: [90/5 = value bet wins 90% of hands, bluff bet wins 5% of the time]

Required Bluff Frequency For GTO Play
Probabilities - Value Win / Bluff Win

Bet Size	100/0	90/5	80/7.5	70/10
0.5 Pot	25.0%	23.5%	24.1%	25.0%
Pot	33.3%	33.3%	35.6%	38.9%
1.5 Pot	37.5%	38.2%	41.4%	45.8%

Example: If you make a pot size bet with good cards that have an 80% chance of winning if called, then you should bluff 35.6% of the time with cards that win 7.5% of the time.

Note that departures in probability from the 100/0 case do not lead to very significant changes in bluff frequency as compared to the change in frequency for different bet sizes. Also note the curious results for the ½ pot case where the farther the departure from 100/0 the closer the bluff frequency is to the 100/0 result. I don’t have a good explanation for this. Finally, being a GTO strategy, it doesn’t take advantage of poor villain play. If villain is a calling station, bluff less; if he usually folds, bluff more.

I don't think it is assumed nor required that a value bet always wins. It needs to win over 50% of the time when called.

It is assumed that a bluff always loses when called. I think that's a good assumption for correct play.

In the half street (0,1) game where one player can bet pot or check and the other player can only call or fold the GTO strategies are:

Bet top 2/9 bluff bottom 1/9.
Call top 4/9.

Value bets do not always win. Nevertheless, value bet to bluff ratio is still 2 to 1.

If a bluff sometimes wins when called, wouldn't it follow that checking is more profitable than bluffing?

Not if the bluff makes him fold often enough with hands that would have beaten you.

So now he's calling worse and folding better? Makes no sense.

That is assumed when people talk about a GTO 2:1 river ratio of value to bluffs.

From the villain's standpoint, EV of calling = (1/3)*2 - 2/3 = 0

But if hero's value hands only win 60% of the time then Villain's EV becomes (Edit#3: fixed):
2[1/3+(2/3).4] - (2/3)(.6) = +4/5 pot

To make it break-even, hero would need to increase her value:bluff ratio.

Edit: Maybe he's calling worse some % of the time to remain unexploitable (edit#2: because presumably hero's win% with bluffs is an average since hero will also be bluffing with 0%-value hands), and likewise folding some of his weaker better hands some % of the time. I haven't thought about how realistic that is but to me it doesn't sound absurd on its face.

Rather than get into a semantic discussion of value and bluff win probability definitions, I just wanted to show that the GTO ratio of betting ‘good’ hands to ‘marginal hands’ with a polarized range depends on the equity values you use for each part of the range. The 2 to 1 good to marginal ratio often mentioned does follow if you always win with a good hand and always lose with a bluff although other values may also work. With this assumption. a 2 to 1 ratio and with a pot bet, EV from villain’s perspective is

EV_villain call = 2/3 *(-Pot ) + 1/3*(2*Pot) = 0

Therefore if villain is limited to calling or folding, it makes no difference to hero what villain does – an optimal strategy in the GTO sense.

Edit - heehaw beat me to it - not unusual

Hm idk, instead of that, villain could eliminate his lightest calls and increase the number of slightly heavier calls so that Hero's bluffs never win when called. I think Bob was right, having >0% equity with a called bluff only applies to earlier streets.

Edit: It's possible there aren't enough heavier calls available in V's range to make Hero's bluffs unprofitable, forcing V to make some calls with hands Hero sometimes beats. However, that would also mean V is never folding any better hands, therefore once again Hero would be better off checking.

Nope, this is wrong.

You have to think about what you are trying to make indifferent here.

I'm trying to make indifferent villain's actions in response to a river bet - either calling or falling assuming they are his only two choices.

Please show me where the values I got are wrong with this objective in mind.

This is a more basic question but it seems like a good thread for it.

If you're supposed to bluff 1/3 of the time with a pot bet, 40% of the time with a 2x pot bet etc, how does this make sense in terms of villain's minimum defense frequency?

So when I bet 1 into 1, if villain folds more than half the time then I can auto profit with any hand right? But I'm only supposed to have 1/3 bluffs? So how does it make sense for him to call 50%? (assume nuts/air)

I know I'm missing something obvious here lol. Is there some equation that relates value/bluff ratio to MDF?

No but what hand's in villains range you are trying to make indifferent.

You can't just say V has only bluffcatchers and our valuerange is beat sometimes. Either he has only bluffcatchers and our value has 100% equity or he has also better hands and now our value has less equity. But the point is, you can't make his hands indifferent that beat some of your valuerange, they always have a +EV call.


When V isn't defending 50%, we don't give a **** about value:bluff ratio as the game isn't stable anymore as we can improve EV by bluffing 100%.

I think you are trying to change the problem. Clearly if villain is acting in an exploitable way you should exploit him. I mentioned this in the last two sentences of the Op.

"Finally, being a GTO strategy, it doesn’t take advantage of poor villain play. If villain is a calling station, bluff less; if he usually folds, bluff more. "

I'm not trying to make his hands indifferent; i'm trying to make his decision strategy indifferent. He can always call, he can always fold or he can choose any probabilistic weighting of calling and folding. Whatever strategy he elects, his EV (long term profit/loss) is zero with the calculated value/bluff ratio for hero's bet.

Game theory doesn't "care" what hand villain has. Exploiting your opponent is a different objective.

Making hand indifferent to call/fold = making his decision strategy indifferent. And obviously you can't make his strategy totally indifferent when the ranges aren't totally polarized ie. his "bluffcatchers" beat some of our valuerange (meaning he has some slowplays). Slowplays are always profitable, because they aren't bluffcatchers.

Ok so the equilibrium is us bluffing 1/3 with pot sized bet and him calling 50% right?

Can you explain how this makes sense? Villain facing a pot sized bet needs 33% equity to call, yet needs to call 50% to prevent us profiting on bluffs.

Is it simply that the 33% number kind of just irrelevant game-theory wise? Where does villain's extra 17% calls come from?

Thanks for your help.

If you are using the term slowplay as I understand it (sandbagging, trapping) that is not an option under the conditions I imposed. Hero bets on the river and villain can either call or fold. See first sentence of OP.

50% of the hands in your range must have >33% equity.

No by slowplaying i mean that v has some% nuts/strong hands OTR in addition to the bluffcatchers. As our valuerange didn't have 100% equity, that has to be the case.

They don't have to be slowplays. Instead they might just be hands that could be bet for value but win more as a check call, such as all the hands with 33%-50% equity.

If they win some % vs our valuerange, they aren't bluffcatchers, it's in the name itself.

Then call them valuecatchers.

Point is those are not going to be indifferent to calling so we don't need to focus on them when figuring out value :bluff ratios.

Wouldn't this mean your total EV is much less than pot? I think I prefer playing GTO and winning the whole pot. :/

On the river, villains calling range can be split into two parts.
(1) The part that is weaker than hero's worst value hand. This part has 0% equity vs hero's value range.
(2) The part that is stronger than some of hero's value bets. This part has some equity vs hero's value range.

Your math above applies only to (1). Since these hand's have 0% equity vs hero's value range your math leads to 2:1 value:bluff ratio for pot-sized bets.

Since (2) has equity vs hero's value range, his value range is < 100%.

So for pot-sized bets the 2:1 ratio hold's even when hero's value range is less than 100%.

I think the problem is you are trying to make caller's entire range 0 ev rather than making his worst calling hand 0 ev.

But I'm not sure. What formula are you using exactly?