Winner Poker Player - News Detail

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 he never calls your bets, you can safely bet every time and win every pot. 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 a 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 for 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

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) - p*B

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

That is:

b = [p / (1-p)] * [B / (T+B)]

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.

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.

So let's assume that we always make pot sized bets 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 o

For ther values of p we have the following bluffing frequencies:

Some pointers: 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.

Limitations

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.

We assume that your bluffing isn't 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