say I choose specifically 6 key cards to bluff with.
That means I will bet 24 times. 18 of those times
I have the best hand, and 6 of those times I am bluffing.
Therefore, the odds against my bluffing are exactly
3-to-1. The pot is $200, and when I bet, there is
$300 in the pot. Thus, your pot odds are also 3-to-1.
You are calling $100 to win $300. Now when the odds
against my bluffing are identical to the odds you
are getting from the pot, it makes absolutely no difference
whether you call or fold. Furthermore, whatever you
do, you will still lose exactly $600 after 42 hands.
If you were to fold every time I bet, I would beat
you out of $100 24 times when I bet and lose $100
to you 18 times, when I don't bet, for a profit of
$600. If you were to call me every time, you would
beat me out of $200 six times when I'm bluffing and
$100 18 times, when I don't bet, for a total of $3,000;
but I would beat you out of $200 18 times when I bet
with my good hands for a total of $3,600. Once again
my profit is $600. So other than being a psychic,
there is no way in the world you can prevent me from
winning that $600 per 42 hands, giving me a positive
expectation of $14.29 per hand. Bluffing exactly 6
times out of 24 has turned a hand that was a 4-to-3
underdog when I didn't bluff at all into a 4-to-3
favorite - no matter what strategy you use against
We can now move to the heart of game theory and bluffing.
Notice first that the percentage of bluffing I did
was predetermined- one time every 19 bets or 5 times
every 23 bets or 7 times every 25 bets. Notice secondly
that my bluffing was completely random; it was based
on certain key cards I caught, which my opponent could
never see. He could never know whether the card I
drew was one of my 18 good cards or a bluff card.
Finally, notice what happened when I bluffed with
precisely six cards - which made the odds against
my bluffing in this particular instance identical
to the pot odds my opponent was getting. In this unique
case my opponent stood to lose exactly the same amount
by calling or folding.
This is optimum bluffing strategy - it makes no difference
how your opponent plays. We can say, then, that if
you come up with a bluffing strategy that makes your
opponent do equally badly no matter how he plays,
then you have an optimum strategy. And this optimum
strategy is to bluff in such a way that the odds against
your bluffing are identical to the odds your opponent
is getting from the pot. In the situation we have
been discussing, I had 18 good cards, and when I bet
my $100, creating a $300 pot, my opponent was getting
3-to-1 odds from the pot. Therefore, my optimum strategy
was to bluff with six additional cards, making the
odds against my bluffing 3-to-1, identical to the
pot odds my opponent was getting.
don't really want to just call here because you don't
have many chips left, and you don't want to have to
be faced with a decision if you miss the flop. You
don't want any callers.
What to do depends a lot on the stack sizes of the
players who haven't folded yet. You don't really want
a caller. The players who are most likely to call
an allin bet are those with very large stacks or those
with very small stacks.
So, although folding might not be a mistake, a raise
is probably the best thing. But the worst thing to
do here is make a small or medium-sized raise. A raise
is essentially committing yourself. You don't have
enough chips left to do anything but call if someone
reraises, and a small raise doesn't have the power
an allin raise does. An allin raise both makes it
less likely you'll get another player to play with
you and protects you from having to make a very tough
decision on the flop.