Isn't it better to think in terms of ranges?
If 25% of the hands we play are "bluffs" then do we arbitrarily pick some time to just raise w/ absolute air?
If however we pick a range for a position, so that 3/4 of that range are > 50% equity, then the remainder (the 25% that are bluffs) have less than 50% equity, but will still win some of the time.
This doesn't consider that from time to time you will raise with a flush draw, for example.
Bluffing also works in reverse. For example, if you NEVER limp, you can't limp to induce a raise when you have a monster -- a pat flush eg.
So every so often I'll limp QQ from EP "just because". Even if I fold it w/o showing it, it tells other players I'll limp/fold.
I've been trying to construct such a range for the Button in 5-Card Draw.
If you raise a hand from the button, you have two players behind you.
The SB is getting 5:1, the BB is getting 2.5: 1 to call.
I read in an article that you can raise 99 from the BTN if you also hold AK
The lowest hand that fits this description is 99AK2.
There are 624036 hands that already have this beat.
-- approx 24%
Therefore, the chances that one or the other of them has us beat ALREADY is 48%, and about 6% of the time, they both do.
Not only that, but there is the draw to consider, and our chances of improving on the draw.
If we stick with the pair of 9s example,
We will improve 29% of the time, since we hold a pair.
We can guess what hands they hold by what they draw:
Assuming they both call
I break it down like so:
Draw 3: Pair, thus same chance to improve
Draw 2: Pair with kicker -- better chance to hit two pair (and likely a better one) but slightly worse chance to improve over all
Draw 2: Already has trips -- much less chance to improve, but we're already beat
Draw 2: Cathop flush draw -- don't worry, be happy
Draw 1: Two pair, we're already beat, but only 9% they'll improve
Draw 1: Flush or straight draw -- roughly 18% chance they'll improve.
Now let's say we DO improve...
Lots more decisions.
That's a pile of stuff to consider without even taking into account the tendencies of the people in the blinds -- do the fold a lot?
Call? 3Bet?
BTW, from the above, I concluded that I don't like opening 99AKx from the BTN unless the blinds fold a awful lot <== I consider it a bluff.
I know I didn't answer the question "how do I calculate a mathematically formula for how often to bluff?", but it's the best I can do.
Also, I'm pretty sure it isn't "very simple math" -- It took me over a week of putzing about with Excel and reference tables even to get the stuff above.
Finally, in limit games, the big blind is getting 2.5 : 1 to call ALWAYS, and the odds of improving a pair drawing 3 are 2.48:1
Fold equity is consequently quite low. You might think that they will fold any hand w/ no pair, but in that category are a whole mess of hands that will rise up and eat you for lunch -- flush draw, straight draw and so on.
If there is a raise and two callers ahead of the BB, he now has correct pot odds to call with those.
All the best,
--VS