Hi,

1. villain bets half pot on the river. Calling frequencies are at around 66%. But what does that mean? Calling with 66% of those hands that beat a bluff AND ARE TOO WEAK to raise, or defending with 66% of ALL hands that beat a bluff (either raising OR calling)?

2. If pot is $10 on the river, hero is IP. Villain checks, we bet $5 and villain checkraises to $15. I'm struggling with working out the frequencies in checkraise situations, so
- What is villain's bluff frequency?
-What should be hero's call/defend frequency?

Thanks so much

1. Yes you're only defending hands that have enough equity to at least call (don't forget about including rake in your calculation). If a hand breaks is exactly 0 ev you can fold or call with it depending on your needs as either option has the same EV.

Whether or not to raise depends not only on equity but how your continuing ranges are otherwise constructed and if other hands can belong in your raising range. It also depends on your opponent's ranges.

Keep in mind the idea of minimum defense frequencies (i.e. I defend Y amount of the time vs an X sized bet) is really an idea based on toy games with specific rules enforced and there may be better ways to conceptualize playing a hand or range of bluff catchers.

2. Hero's defense frequency is 1-B/(P+B) where B is the bet size made by villain and P is the pot size before villain bets. In this instance that would be 1-($15/($15+15) = .5 × 100 = 50%. The other way to look at it is that villain is risking $15 to win $15 so his bluff is getting 1:1 on it's money then you can convert the odds into a percent (50%).

Villain's bluffing frequency "should" be (amount you have to call)/(Total pot after betting). The reason for this is because that is the minimum equity your hand must have to call villain's raise. So to make your bluff catchers in different between calling and folding in this instance it would be $10/($40) = .25 × 100 = 25%.

I use should above because with more cards and betting to come this estimate breaks down because it assumes a showdown.

Thanks!
Regarding the first part of my question. My question was more about if we first define our defending range and than do whatever we want to do (e.g. raise with value hands and call with bluff catchers), OR if we exclude those our hands that we'd raise from the calculations. Basically then we'd define our value raising range, then define our bluff catchers and call with the respective (calling frequency). Difference to the first case is, that we calculate the defending frequency on ONLY the hands we call (in addition we defend also with hands that we raise). I think I'm just confused by the wording. What's called calling frequency is actually defending frequency.

Why bluff 25%? Shouldn't it be 33%? We basically get 3:1 10$/30$ = 0,33?

Well without going into too much detail the whole toy game is set up so that your range is ONLY bluff catchers. When your range has stronger hands in it that not only impacts your defending but villain's bluffing.


$10/($15 pot +$15 villain's raise+10 your call).

So this is the GTO approach to bluff-catching using the (1-A) equation and why is villain's bluffing frequency dictated by the pot odds OTR? That seems so confusing to me, so you are saying that villain's bet sizing determines his bluffing frequency?

Am i missing something or just understanding what you are saying incorrectly.

If he wants to make you indifferent to calling or folding then the sizing of his bet would indeed correlate with his bluffing frequency.

Because if we look at bluff catching on the river from our perspective vs a bet size B we need B/(P+2B) equity for a call to break even (ignoring rake).

Since hand values on the river are known (i.e. there are no cards to come to change hand values) that can be translated directly into the relative frequency of bluffs in villain's range (i.e. the percentage of bluffs relative to other hands in villain's range).

If he bluffs at the appropriate frequency then we are indifferent to calling or folding, because we win against his bluffs, but pay off an equivalent amount to his value range.