bluffing 5CD

[nolimit]

hi all--

on 2/26/07 bigpooch posted a very informative reply to a post. There was a snippet on opt. bluffing frequency which I did not understand and I wonder if bigpooch (or anyone) would provide clarification. I am a newbie to 5CD and poker theory but I figure this might be informative to others as well. Here is the snippet:

[ QUOTE ]
your optimal
bluffing frequency is your legitimate betting frequency
divided by (P+1) where P is the size of the pot before you
bet in terms of the number of big bets

[/ QUOTE ]

What kind of a number do you come out with at the end of this calc and how do you use it? Could someone provide an example? Thanks in advance.....

[MarkGritter]

Let's say you are drawing to a flush. There are then 9 cards you could draw that complete your hand (and some additional ones that make pairs you won't bet.)

So you are legitimately betting about 9/47 of the time.

If the pot is 2 big bets (P=2) then you want to bluff 1/3 as often as you are betting, or (9/47)/(2+1) = 3/47. You can interpret this as betting with 3 specific cards in addition to the 9 that make your hand.

If the pot is 4 big bets then you should bluff 1/5 as often as you are betting, or (9/47)/(4+1) = 1.8/47 = about 4%. (Pick one specific card to bluff with always and other to bluff with 80% of the time, for example.)

What happens when you bluff this amount is that you become indifferent to your opponent calling or folding.

At 2 big bets: When you bet, you have a flush 9/12 of the time and garbage 3/12 of the time. So if your opponent calls you win the pot + 1 bet 9/12 of the time, and lose one bet 3/12 of the time, for a total of 9/12 * 3 + 3/12 * -1 = 2 bets. If he folds you earn the pot, which is 2 bets.

Similarly, at 4 bets, when you bet with the above frequency you have the flush 5/6 of the time and are bluffing 1/6 of the time. If your opponent calls you win, on average, (5/6)*(4+1)+(1/6)*(-1) = 4 bets, the size of the pot, the same as if he always folded.

(Chen and Ankenman like to use ex-showdown equity instead to demonstrate how much you come out ahead by playing the balanced strategy compared with just showing down after the draw.)

[Big Limpin]

I understand how this makes villain's action indifferent to hero, but not why that's important, nor why this is necessarily "optimal frequency".

I should be able to outperform this by bluffing more into tightwads, and less into calling stations, correct?

[MarkGritter]

Yes, if you know your opponent's strategy sufficiently well, you can outperform the game-theoretic optimal strategy by exploiting his mistakes.

The bluffing frequency is "optimal" in the sense that it is unexploitable. If you choose some other frequency, your opponent can cost you money. So it is the correct frequency to use against an unknown or tricky opponent.

Once you have decided to deviate from this frequency based on your knowledge of what the opponent will do, there is no immediate reason to bluff anything other than 100% or 0% of the time. (You might want to try disguising the fact that you are exploiting your opponent, I suppose, but if you're that much inside his head you should also be able to predict when he catches on...) It doesn't make sense to say, "well, he's more likely to call so I'll bluff 10% less." There are really only three values that make any sense.

The optimal bluffing frequency is a rather strong mathematical statement about what the endpoint of thinking and meta-level thinking ends up looking like, rather than a prescription for maximally exploiting weak opponents. It's not just "here is a simple strategy that will work against anybody" but "expert-level play must necessarily end up looking like this."

If your exploitative strategy is providing you as much profit or more than the game-theoretic optimal, great. If it's less, then there's an easy fix. [img]/images/graemlins/wink.gif[/img]

[bigpooch]

Here's an example from 6-handed $2-$4 fixed limit 6-handed
draw poker.

(In the calculations, we ignore the "bunching effect" as a
result of the action/inaction of the opposition.)

You are the big blind and after the utg limps, you draw
three cards as does the limper (you are just heads up). You
can't hold AA in this spot as you always raise a limper with
AA and you never raise with a weaker one pair hand when
heads up with the utg player.

You decided in advance that in this spot you will bet aces
up or better and that you will only draw three to pairs (so
you draw four to an ace with AK, etc.).

There are 84480 combinations of each pair and you could have
one of twelve ranks (you couldn't have AA as you would have
raised the limper) for a total of 1 013 760 combinations of
hands. It turns out exactly 1/4 of these pairs have an ace
kicker which makes your chances of improving to aces up or
better only (2064+126)/16215. For the other 3/4 of these
pairs, you make aces up or better with a probability of
(2064+246)/16215 so the "weighted average" of improvement is
2280/16125.

You make aces up on "effectively" (2 280/16 215)(1 013 760)
or 142 545.35 combinations.

You then must choose "effectively" 4/9 as many combinations
of bluffing hands or about 63 353.49 combinations of hands
to bluff with.

You get unimproved "deuces" on "effectively" 84 480 x
(11 559/16 215) = 60 222.28 combinations. If you were to
bluff with these, you would be bluffing "almost optimally"
(it's actually only about 95.06% of the GTBF = game
theoretical bluffing frequency).

In practice, you probably don't want to bluff like this
for several reasons.

First of all, if you bluff with only deuces, if your
opponent is astute enough, he can decide to call if he
hasn't seen a deuce but tend to fold if he has seen a
deuce because of the way you have decided to bluff. Thus,
in practice, you might want to bluff with about 1/4 of the
"deuces", 1/4 of "threes", 1/4 of "fours" and 1/4 of the
"fives". To make your bluffs more likely to work with
these pairs, you may also want to have seen a "key card"
that is likely to improve your opponent's hand: e.g., if
you have seen this player limp in with KK, you might want
to choose your bluffing hands where you have started with
a king, which conveniently makes it 1/4 of the time you
would have a pair with a king as a side kicker, so that
if you don't improve with 22K??, 33K??, 44K?? or 55K??,
you simply bluff.

Secondly, you probably want to tailor your bluffing
frequency in accordance with your opponent. If, from your
observations, your opponent doesn't call with unimproved
pairs, you would want to bluff very frequently. On the
other hand, if your opponent is a calling station and
never seems to fold, you simply never bluff.

You may "autobluff" against certain opponents on the end,
but by doing so, you might get caught and some players will
react by raising your blinds indiscriminately, putting the
"squeeze" on you and making your decisions more difficult.
Maybe the upper bound on your bluffing frequency should be
about four times the GTBF, but of course, that's a matter of
personal choice. If you want to exploit to the maximum, you
can bluff even more, but remember this: if at any time, your
opponent thinks you are bluffing too often, he may look you
up more often. Now you wish you didn't bluff quite as often
as you did! If your bluffing in certain spots remains
unnoticed, you can continue to play exploitively with the
side benefit of lower fluctuations as in other spots you
seldom bluff to give you more credibility on your attempted
bluffs.

[nolimit]

wow--- thanks to you 5cd/poker theory monsters: very informative, very impressive! Again thanks for taking the time to post, much appreciated.