Let's say you're certain that the opponent has your hand beaten. There's still a chance that he or she will fold if you put in a bet, in which case you'll pick up the pot despite holding the worst hand.

Calculating your bluff equity

To figure out whether a bluff is good or bad, we must calculate the value of the bluff, that is, your bluff equity. In this calculation, we'll play around with the following parameters:

P = the pot size

B = the size of your bluff bet

f = the probability that the opponent folds if you bet

If you bluff and the opponent folds, you win the pot, P. If you bluff and get called, you lose your bet, B. If the opponent raises, you fold and lose B as well.

So the expected value of your bluff is given by this expression:

E = fP - (1-f)B

Folding a hand costs nothing, so the value of not bluffing is 0. Consequently, for the bluff to be better than folding, E must be > 0. This gives us the following requirement:

fP > (1-f)B

Which is the same as:

f > B / (P+B)

If the probability that the opponent folds is greater than the above ratio, bluffing is correct. Then, bluffing will earn you money (in the long run) and it would be a mistake not to bluff in this situation.

Example 1 - Pot Sized Bet

There's $100 in the pot. How likely must the opponent be to fold in order for a pot sized bluff to be correct?

P = 100, B = 100 -> f > 100/200 = 1/2

If a fold is more than 50% likely, a pot sized bluff bet earns you money in the long run.

Example 2 - Smaller Bet

There's $100 in the pot, you plan to bet a third of the pot, that is, B = $33. How likely must the opponent be to fold to this bet in order for the bluff to be profitable?

P = 100, B = 33 -> f > 33/133 = 1/4

If a fold is more than 25% likely, the small bluff will earn you money in the long run. (Always that long run...)

Example 3 - Bluff Sizing

There's $100 in the pot and you think there's a 20% chance that the opponent will fold to a bet. How much should you bet?

To solve this we have to isolate B in the expression fP > (1-f)B. We arrive at the following requirement for the maximum bet size:

B < P * f/(1-f)

That is, the bet must be smaller than the expression to the right, which we can read as the size of the pot times the fold odds (as opposed to fold probability).

In this example, the fold odds is 0.2/0.8 (which is the same as 2/8 or 1/4), and so the bet has to be smaller than P/4 for the bluff to be correct. You can bluff bet up to $25.

With fold odds of 1/2 (that is, 33% chance of folding), your bluff bet should be smaller than P/2.

With fold odds of 2/5 (40%), your bluff bet should not be bigger than 2P/5, that is, around 0.4*P.

Complications

In real life, things are a bit more complicated, as usual.

  • There's always the chance that your hand is actually a winner.
  • The probability that the opponent folds usually depends on the size of your bet. This dependence can be very complex.
  • Typically, bigger bets are harder to call than smaller bets.
  • Bets that are seen as either too small OR too big may often induce a call, while "normally" sized bets are less suspicious and easier to fold to.

We'll look at these complications in future articles on bluff equity.

/Charlie River

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