July 28, 2010
The goal of determining an optimal (in the game theory sense) bluffing frequency is to make our opponent indifferent to calling or folding.Â This makes the frequency dependent on two factors, our hand and the pot odds (and in no limit games we can play with the latter).Â In practice, our opponent won't usually call with the exact frequency he needs to in certain situations.Â We can adjust our bluffing/valuebetting frequencies to account for this (an exploitative adjustment) and readjust if he seems to be calling/folding with different frequencies as result.
Let's take a look at a simple spot where we always either bet (for value and as a bluff) or check/fold (we never check/raise or check/call) against a hand we suspect is somewhere near the median of predraw pat hands, say something like a T8753 (but we don't know exactly what it is).Â How often should we bluff?
Say we have the monster of all draws, 7432.Â Â There are 5 ranks of cards that are good for us (so 20 cards max) and we likely on average have somewhere around 17 outs to improve.Â If we bet half pot (giving Villain 3 to 1) we need to bet for value 3 times as often as we bluff to make Villain indifferent.Â So with a 17 out draw, we'd want to bluff 5.66 cards.Â It usually makes sense to do this with the worst of our catches (even though in this situation we're assuming we have no showdown value with anything worse than a T and could choose randomly as long as we were consistent).Â So maybe we'd bluff on 7,7,7, the 4,4 not of the same color of the 4 in our hand all the time, then 2/3 of the time we could bluff the other 4 (and we could do this 2/3 randomization in various manners).Â Say we notice after a few times in this spot (or similar spots) that this player never/rarely folds when we bet.Â We can exploit this by bluffing less (if we don't want want it to be too noticeable) or not at all (and if we stop bluffing against this player we need to be vigilant about how this looks to other players in the game).
Now, if instead of betting half pot, we decide to overbet 2x pot.Â Villain only gets 3:2 and we should be bluffing twice as many (11.333) cards (so maybe 77744433322 and 1/3 [via some randomization] of the 2 of the same color in our hand).Â In this case we may find our opponent is never calling and we should increase the number of times we bluff until Villain catches on.
We can use these ideas in other circumstances as well and to think about how we should establish our various ranges in the different drawing spots, though they require more rigor.
Here's a table you can use to interpolate: