A Note from Two Plus Two: the following is an excerpt from our upcoming
book, Analytical
No-Limit Hold ‘em by Thomas Bakker.

In the previous chapter, we studied optimal betting under the assumption that our opponent
would exploit our weaknesses. Here, we’ll study optimal calling frequencies. When your opponent
bets and you will not raise, the decision that needs to be made is whether to call or fold.
If this is not balanced, for example, calling thin river bets too frequently, your opponent
can exploit this by always (or never) betting.

Consider the following situation: We are in position on the river holding the

A  J

and having played the hand straight-forwardly, our opponent knows that our most likely hand
is a medium strength such as top pair. In fact, since the board is the

A  T  3  5  8

top pair is our precise hand. Furthermore, we expect our opponent, who is first to act,
has a weak hand about 80 percent of the time, and the flush the remaining 20 percent. The
pot is $300 and we are faced with a pot-sized bet. Do we call?

Normally, when in a situation like this where we have to make a decision, we do the following:

Estimate our opponent’s range prior to betting.
Decide which hands he would bet.
Calculate how often our opponent now has us beat.
Compute the expected value of calling.

In this example, we have already done the first step which told us that we are beat 20 percent
of the time. We could now continue with the next steps, find some decision based on the estimate
we make in Step No. 2, and then come to a decision that would have the highest expected value
– we would either call or fold.

But there’s a problem here. This approach would make us predictable in this situation. Specifically,
with a bluff-catching hand on the river in such a spot, our strategy would be to either always
call or always fold. If our opponent noticed this, he could use it to exploit us by bluffing
often or rarely in such situations.

So the solution to this, and the topic of this chapter, is to find the optimal calling frequency.
That is, we should find the proper frequency to call with to make us unexploitable. And to
do this, it will be necessary to change some of the steps of our plan outlined above:

Estimate our opponent’s range prior to betting.
Calculate how many of these hands beat us.
Calculate the optimal calling frequency.

Since the first two steps are already done, let’s continue with the third. First note that
“optimal” here means “unexploitable.” That is in this situation, our expected value, when
we call, will be a function of two variables:

1. Pflush: The probability that our opponent has us beat, which in this
example is 20 percent.
2. b: Our opponent’s bluffing frequency. (Note that he will clearly always
bet if he has the flush, so this is the probability that he bets when he does not have a
flush.)

Since our opponent will always bet when he has the flush, we first calculate, using Bayes’
theorem, the probability that our opponent has the best hand given that he bets:

Filling in the known value for Pflush we get:

The expected value of calling is now simply

which simplifies to

We can plot this equation:

We see that if our opponent bluffs less than 12.5 percent of the time (which is of misses,
or in other words 10 percent of the total hands), calling is unprofitable, and that if he
bluffs more often, it’s profitable to call. Unfortunately, we do not know this bluffing frequency,
and since our assumption is that he will adjust to our strategy, we’ll have to try a different
approach.

If you look at the above graph, you see that depending on our opponents bluffing frequency,
we can play exploitatively by always calling or always folding. Similarly, if we play some
strategy where we call a certain percentage of the time, our opponent can play an exploitative
strategy that consists of either never or always bluffing.

So for instance, if bluffing 10 percent of the time is profitable for our opponent, bluffing
100 percent of the time would be even more profitable if we do not change our strategy in
response. Since we want to find such an optimal constant strategy, we will use this knowledge
and first find the expected values for our opponent of both always and never bluffing.

If our opponent never bluffs and we call c percent of the time, our opponent’s
expected value is:

Note that we multiply by 0.2 to account for the fact that our opponent will only bet when
he has the flush, which in this example is 20 percent of the time. If he does not bet, he
does not win the pot.

If our opponent always bluffs, he will have the best hand 20 percent of the time. This gives
us the following formula for his expected value:

This simplifies to:

Now since our strategy needs to be unexploitable, that is our opponent does not gain by
choosing one strategy over the other, we need to solve the following equality for c:

and solving this simple linear equation gives us c = 0.5.

So the optimal calling frequency at this river is 50 percent. This frequency is correct
if our opponent has the flush 20 percent of the time. In fact, it’s correct no matter
how often our opponent actually makes the flush as long as it’s below 67 percent.1 If
our opponent makes the flush more often, the optimal strategy for us is to always fold since
it becomes impossible for him to bluff enough that we need to use this approach.

It’s important to realize that this 50 percent is only correct if our opponent makes a pot-sized
bet. The optimal calling frequency changes for different sized bets relative to the pot.
But the answer is that the smaller the bet, the more often you should call, and the larger
the bet, the less often you should call.

(As in the previous chapter, we again note that David Sklansky in The Theory of Poker explains
a second way to get the same result. First, notice that since the opponent is getting 1-to-1
odds on his bluff [since he is betting the size of the pot] that must be your calling ratio.
That is in this example, the optimal calling strategy would be to call 50 percent of the
time.

For another example, suppose your opponent makes a half pot bet and you are deciding whether
to call or fold. Since the bettor is getting odds of 2-to-1 from the pot, that must also
be your call to fold ratio. So now you would call two-thirds of the time.)