Deriving Sklansky's Formula for the Optimal Calling Frequency

Question

The optimal calling frequency is the calling frequency which makes your opponent's expected value of always bluffing equal to his expected value of never bluffing. Theoretically, if you call bets with the optimal calling frequency, whether or not your opponent employs a strategy of bluffing or not shouldn't matter to you (since his expected value will be the same either way).

After reading this article on optimal calling frequencies, I am having trouble understanding why if your opponent bets B into a pot of P, your optimal calling frequency is

1 - B/(B+P)

if you think he is bluffing at a rate of B/(P+B) or higher and

0

otherwise. Here is the passage it comes from in the article:

Question 1: What justifies this? The author states this rule comes from Sklansky's Theory of Poker, but I can't find it anywhere in the book (all I can find is stuff about optimal bluffing frequencies, not optimal calling frequencies). Also, I understand the author's lengthy argument at the beginning of the article for a particular case, but then generalizing to this rule isn't justified and it just comes out of nowhere.

Question 2: Let's say you apply this formula in a particular situation and it comes out saying you should optimally call 10% of the time. Does it matter whether or not you only call with bluff catchers or not? Or could you literally call 10% of the time with any two cards (including occasionally with 23o that didn't hit), and fold 90% of the time (including occasionally AA), and still force your opponent's EV to be invariant to whether or not he bluffs?

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EDIT: I understand that if our opponent is assumed to be bluffing on one particular hand, and he bets B into a pot of P, then us calling him 1 - B/(B+P) of the time with a bluff catcher makes his long-run expected value equal to 0 (in this spot only, when he is bluffing). But I don't see how this fact takes us to a generalized optimal calling frequency, where the opponent's expected value is the same whether he always bets (regardless of whether he is bluffing or not) or always only bets with made hands.

Answer 1

Question 1: Let:

p = probability villian has better hand than you

c = probability you will call

bet = the bet amount

pot = the amount in pot

If villain doesn't bluff:

--when you call (probability = c), he wins: bet+pot

--when you fold (probability = 1-c), he win: pot

So his EV for don't bluff is simply the the chance he wins (p) multiplied by the two situations above with you calling:

p*[c*(bet+pot) + (1-c)*pot]

If villian does bluff:

--when you call (at rate c), then he wins bet+pot at p chance, and loses bet at (1-p) chance.

--when you fold (at 1-c), he wins pot

So his EV for bluff is:

c*[p*(pet+pot) - (1-p)(bet)] + (1-c)(pot)

The un-exploitable position will be when the above EV's are equal:

p*(c*(bet+pot) + (1-c)*pot) = c*(p*(bet+pot) - (1-p)(bet)) + (1-c)(pot)

You can then plug this into a solver, like wolfram-alpha (b=bet, t=pot):

p*((c*(b+t)) + ((1-c)t)) = c((p*(b+t)) - ((1-p)*b)) + ((1-c)*t), solve for c

The p's actually cancel out, and the solution is:

c = pot / (bet + pot)

In poker this is known as the minimum defence frequency. You can also write it as:

c = 1 - bet/(bet + pot)

Now you have to work out the EV of you as the caller, against this system.

His over all win chance is: Vw = p/ (p+(1-p)*(c))

Your hero EV is: Hev = (1-Vw)*(bet+pot) - (Vw)*(pot)

Now look at the above carefully. When is your EV exactly zero? It is when his bluff rate is equal to bet/bet+pot. At lower bluff rates your EV is negative, and you should not be calling any bluffs.

Of course that applies to one hand. He may then exploit that, but obviously his bluff rate would go up, and when it does, you would call him.

Now on the other hand... if you have NO IDEA what his bluff rate is... then it would make sense to call at the c rate. Because we give him the credit to be playing at optimum until be learn what his bluffing frequency is.

Question 2: This comes back to the axioms. Villian must be in situation where they are making a decision between: checking (definitely losing), or betting (and winning at the calculated rate).

Therefore it does not apply to your holding 23o if you only have high cards, because he would not lose at showdown, regardless of if he thought he would. So yes, it only applies in situations where you have a literal bluff catcher - that is - anything that beats his bluff.

It works with any hand that beats him if he doesn't bluff.... which is.... well if you are calling 10%, might as well pick the strongest 10% you could get from your range in any situation.

Answer 2

Not a full answer.

But the bluff rate is not 100%. There is an equation for optimal bluff rate from GTO (Game Theory Optimization).

Just as your opponent puts you in tough spot by bluffing. You need to turn it around and call back at an optimal call rate from GTO.

If you never call back then your opponent will stop playing GTO and play exploitative. Always bluff.

If you always call back with a bluff catcher then your opponent will play exploitative and never bluff.

I suggest you read up on GTO.

I don't agree with that bluff rate of B/(P+B) or higher. Where did you get that?

Answer 3

If you try to stop someone profiting from a profitable situation, then you are burning money. Hey - this guy is 3b me with a linear range of 5%. I should stop him making a profit when oop with half the deck lol. It's situational. You have your own 3b range.

Auto profit is right in some spots. Six people started the hand. Not just you and the villain! Position and Hand Strength matter.