Poker/Bluffing

Bluffing is a seriously overrated concept in poker. Yes, it is important, and yes every good player is a good bluffer. But it is not the be-all and end-all of the game. Certain types of players will make frequent bluffs - perhaps every 4 or 5 hands they play. This is a high variance way of playing, and only the truly great players are significant winners in this manner.

The mathematics of bluffing dictate this. For example, let's say that the pot is currently $10. I have nothing at all, and you have a moderate hand, a top pair ace kicker. I bet $5 as a total bluff. Whether this is profitable or not depends on the chance that you will fold.

I am betting $5 to win $10. I must therefore succeed one time from three to make this a positive expectation move (see expected value). If you fold 30% of the time, it is a losing move from me. If I bet $10 into the $10 pot, you must fold 50% of the time - folding 45% of the time loses me money in the long run.

Optimal bluffing frequency

The reason for bluffing can be explained using a branch of mathematics called game theory, which also gives an insight into the optimal frequency for bluffing (though it's usually only possible to work out the math in simplified situations).

Suppose you are playing 5-card draw poker against a single opponent. There is $10 in the pot from antes, and after the draw you have the opportunity to bet another $10 (no raises or other bet sizes are possible, to keep this simple). Your opponent gets:

for a made straight, while your hand is:

Your opponent stands pat, so you know he has at least a flush or straight to beat your two pair. You have four cards in the deck that can help you, so you are roughly a 11 to 1 shot to improve to a full house and beat your opponent's straight. Ignoring the possibility of your opponent betting or either of you raising, you have two options: bet only if you make your hand, or bet whether you make your hand or not. Your opponent also has two choices: call or check and fold. The expected payoffs are as follows:

	You bluff	You don't bluff
Opponent calls	$20\times {1 \over 12}-10\times {11 \over 12}\approx -\$7.50$	$20\times {1 \over 12}\approx \$1.67$
Opponent checks or folds	$\$10$	$10\times {1 \over 12}\approx \$0.83$

If you bluff and your opponent calls, you will win $20 some of the time but lose $10 the majority of the time, so you end up with a negative expectation (the wost outcome here). If you bluff and your opponent folds you win $10 every time, giving you the best outcome here. Not bluffing gives a much lower expectation, but on the other hand it can't be negative, since you'll never put money into the pot with a hand that risks losing.

The trouble is, if we assume that your opponent is smart and knows the way you play, he can always choose the option that makes your expectation the worst: if you bluff he'll call you, but if you don't bluff he'll check or fold. If you assume that your opponent reads you correctly, then the best you can expect to make from this is 83 cents. Other than playing stupider opponents, there's no way around this.

But hold on: you can actually do better than this, if you notice that you have more than just the two options mentioned above (always bluff or never bluff). Suppose you bluff with probability $1 \over 20$. If your opponent always folds to a bet, then you make more than if you hadn't bluffed:

${\text{Expected gain}}=10\times {1 \over 12}+10\times {1 \over 20}\times {11 \over 12}\approx \$1.29$

If your opponent calls all your bets, then you still do better:

${\text{Expected gain}}=20\times {1 \over 12}-10\times {1 \over 20}\times {11 \over 12}\approx \$1.21$

Your opponent will choose to call rather than fold to your bets, since this minimizes your gain (and his loss), but notice that this is a much better outcome than the 83 cents you would average if you never bluffed. To optimize your expectation, you want to choose your bluffing probability so that your opponent's loss is the same whichever of the two options he chooses. It doesn't matter how well your opponent knows you or how well he has read your hand, he can't worsen your expected gain by his choice of action.

The semi-bluff

Semi-bluffing is a less well understood but vital concept. A semi-bluff is when you bet a draw to the hand that will win the pot, but you do not have anything yet. For example, suppose you are playing hold 'em with:

The turn comes:

You are very unlikely to have the best hand as it stands. You have four cards to a flush, with around a 20% chance to hit your flush on the next card. If the pot is very small (say against a single opponent) you may not be getting sufficient odds to make a call profitable. However, if you judge that the other player has a 20% chance of believing you have a pair of aces to his pair of kings and folding, the combined expected gain of either making your opponent fold, or making your hand if he doesn't, may make the bet profitable.

To simplify the math, we will assume that if you hit your flush you will win the pot without any further betting, and if you miss your flush but your opponent stays in you will check and fold on the river. We will also simplify the probability of hitting your flush to $1 \over 5$, and assume that your opponent will not raise your bet.

${\begin{aligned}{\text{expected gain}}&=({\text{size of pot}})\times P({\text{opponent folds}})\\&\qquad +({\text{size of pot}})\times P({\text{opponent calls and you make your flush}})\\&\qquad -({\text{size of bet}})\times P({\text{opponent calls and you miss your flush}})\\&=({\text{size of pot}})\times \left[P({\text{opponent folds}})+P({\text{opponent calls}})\times {1 \over 5}\right]\\&\qquad -({\text{size of bet}})\times P({\text{opponent calls}})\times {4 \over 5}\\&=({\text{size of pot}})\times \left[p+(1-p)\times {1 \over 5}\right]-({\text{size of bet}})\times (1-p)\times {4 \over 5}\end{aligned}}$

If $p$ is the probability that your opponent folds given the hand and community cards he's faced with, then $(1-p)$ is the probability that he will call, since we've assumed he will not raise. Observe that the value of this is dependent on the value of $p$, which you obviously don't know since you can't see your opponent's hand. However, if you estimate that $p$ is high enough, then the value above will become positive and the semi-bluff will become a profitable play.

For another example, I am again betting $5 into a $10 pot. You again have top pair with an ace kicker, and I have the nuts (best) flush draw. Here however, I do not need you to fold 33% of the time. Even assuming you fold should I hit my flush (rather than pay me more money - implied odds), I now only need you to fold around 25% of the time for my move to be +ve, and thus profitable in the long run.

Bluffing most often now happens in two distinct ways — the BIG bluff, and the small bluff. A continuation bet where you have missed the flop completely is a small bluff.

Aggressive players will semi bluff a lot more than they make straight out bluffs. The fact that once you hit you may still get more money from your opponent is a large part of this, and makes the semi bluff a powerful move in any player's arsenal.

Poker Bluffing Resources

Wikipedia.org Definition For Bluffing

Short Poker Bluffing Tutorial