Imagine, just for the discussion, that you never bluffed. You only bet if you thought you had the best hand. Of course, opponents would quickly catch up with this and never call your bets.
But if your opponent never calls your bets, you can safely bet every time and win every pot. But if you bet every time, he will start calling you every time he has you beaten. Somewhere in between these extremes is a frequency of bluffing that gives you the best chances.
A simple example
We'll use game theory to explain the value of bluffing through a simplified example.
You're on the last street in an online poker game. The opponent checks and now you can either check it down or make a bet. If you bet, the opponent can either fold or call.
In this simplified example we neglect the risk of a check raise.
We use the following notation:
- T - pot size
- B - bet size
- p - the probability that you have the best hand
- b - your bluff frequency
Note that b is the share of your non-winning hands that you bet as a bluff. We're trying to figure out how often you should bet when you don't have it.
If the opponent calls your bet, he either wins T+B when you're on a bluff, or loses B when you have the best hand. His EV in this case is:
EV = b(1-p)(T+B) - pB
Finding the optimal bluffing frequency
If he folds, his EV is zero. According to game theory, you play an optimal strategy when the opponent's EV is the same no matter what he chooses to do.
In this case, optimal strategy means choosing a bluffing frequency such that:
b(1-p)(T+B) - pB = 0
Which gives the optimal bluffing frequency:
In other words, the odds that you have the best hand times the pot odds created by your bet.
If you bluff with a frequency that matches this expression, it doesn't matter for the opponent if he calls or folds. He has the same EV.
Again remember that with bluffing frequency we mean how often you bet when you don't have the best hand, compared to how often you check the hand down. (In the simplified example, the opponent always checks to you.)
The value of bluffing
To see what optimal bluffing is worth to you, let's compare your EV when bluffing optimally with your EV if you never bluff.
If you only bet winners, we can assume that the opponent won't pay you off. You'll win the pot when you have the best hand and lose it when you don't.
Your expectation with no bluffs is:
EV( no bluffs ) = pT
On the other hand, if you bluff optimally, your EV is:
EV( optimal bluffing ) = p(T+B) - b(1-p)B
By substituting b from above we get:
EV( optimal ) = pT * [ 1 + B/(T+B) ]
It's easy to see that this is always more than pT, our EV when we never bluff.
Bluffing frequencies for pot sized bets
Since B/(T+B) increases when B increases, we see that betting bigger increases our expected value of an optimal bluffing strategy.
So let's assume that we always make pot sized bets (whether we're bluffing or value betting) and see what optimal bluffing frequencies we arrive at.
To find this out, we substitute B = T in the expression for b:
b = [p / (1-p)] * [B / (T+B)] = 1/2 * p / (1-p)
Your optimal bluffing frequency should be half of the odds that you have the best hand in the situation at hand.
For all p > 2/3, b = 1. This means you should always bluff if the probability for your having the best hand is better than 2/3. Which isn't really surprising.
For other values of p we have the following bluffing frequencies:
Some pointers - From the table we can learn a couple of things: If you have no chance of holding the best hand, don't bluff. If it's a coin flip, bluff half the time when you don't have the best hand.
Comparing with Sklansky's Theory of Poker
In his "seminal" work Theory of Poker, David Sklansky says this about optimal bluffing frequency:
"Optimal bluffing strategy is to bluff in such a way that the chances against your bluffing equals his pot odds."
Sklansky deals with the ratio of bluffs to value bets while we try to figure out how often you should bet when you don't have it - a figure that' seems more useful. But if you inspect the result we found in this article you'll see that it fits perfectly with Sklansky's description.
If you make a pot-sized bet, the opponent's pot odds is 2-to-1. According to Sklansky, your bluffs should be a third of your total bets. That is, you should bluff half as much as you bet for value.
Look at the line for 20% in our table: 12% of 80% is around 10 which is half as much as 20 - it fits. Or the 40 line: 33% of 60 is 20 which is half of 40.
Another example. Say you want to bet half the pot in a coin flip situation, what is your optimal bluffing frequency?
The opponent's pot odds are 3-to-1 so your bluffs should be 25% of your total bets according to Sklansky.
According to our formula, b = (50/50)*(1/3) = 1/3. A third of 50% (our losing hands) equals a third of our winning hands and therefore 25% or our total bets - it fits.
Or we can see this by doing the maths properly. If we call Sklansky's bluff frequency bs (no offense), it relates to our bluff frequency b as follows:
bs = b(1-p) / p
With the expression for b above, we get:
bs = p/(1-p) * B/(T+B) * (1-p)/p = B/(T+B)
Our result matches Sklansky's, but is more complete. Sklansky doesn't go into the math, and his description hides some of the complexity.
These calculations are valid for a simplified situation and build on a number of assumptions that are more or less realistic.
- We assume that you always bet when you have the winning hand. You never check and win.
- We assume that you know how often you have the best hand in the current situation. This is not obvious, and it's not "how poker works". In a later article we'll attempt to find the optimal bluffing frequency in a more realistic example.
- We assume that your bluffing is randomized and not related to the kind of hand you have or the kind of board you're looking at. This assumption is unnatural.
- We assume that the opponent never check raises. We'll involve check raising in a later article.
On top of this, we need to remember that optimal strategy isn't the most profitable, in general. It's a strategy that prevents opponents from exploiting your game, but on the other hand it doesn't let you exploit their game either. We'll see in a later article how our EV changes if we adjust the optimal strategy to an opponent who calls too often or too rarely.