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The Math of Bluffing

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By Tony Guerrera

When most players think of bluffing, they think of one player (the bluffer) exerting psychological dominance of the other (the bluffee). Even though bluffing will always have some mystique in the poker community, bluffing is really just part of a tactically sound game plan. Provided that you know your opponents’ hand distributions well enough, you can use math to determine the profitability of a bluff.

Whenever you play a hand, you have two ways to win. First, play can go all the way to the river, and your hand can hold up in a showdown. Second, you can bet or raise such that your opponent will fold before showdown. Your expectation value, the amount of money you expect to win on average when playing a hand, is the sum of the EVs when you go to showdown and when you attempt to bluff. In general, when you have good hand, most of your +EV will be from when you get to showdown, and when you have bad hands, the only +EV that you’ll derive will be the result of you executing successful bluffs.

The question, therefore, is what exactly is a successful bluff? Without putting much thought into it, most would say that a successful bluff is a bet that forces an opponent with a better hand to fold. Defining a successful bluff like this may seem good, but we actually need to refine the definition a bit by more directly applying the concept of EV. A successful bluff is a bluff that is +EV, and the best bluff is the bluff that yields the highest EV.

When you bluff, there’s a probability that your opponent will call or raise (i.e. your bluff will fail) and there’s a probability that your opponent will fold (i.e. your bluff will succeed). There’s a payout corresponding to a failed bluff attempt and a payout corresponding to a successful bluff attempt. The EV for a bluff is therefore expressed by the following equation:

Consider the following example. You are heads-up on the river. You have position on your opponent, and he checks to you. The pot is currently $100. Your opponent will fold to a $25 bet 25% of the time, and he’ll fold to a $75 bet 40% of the time. Which bet is better?

The EV for the $25 bet is the following:

The EV for the $75 bet is the following:

These EV calculations show that the $25 bluff is better. This situation shows that the best bluff isn’t necessarily the one that makes your opponent fold more often.

How do we come up with P(successful) and P(failed)? To deduce these important probabilities, you need to assign hand distributions and calling/raising distributions to your opponent. Your opponent’s hand distribution is the set of all possible hole cards he may hold based on all information available to you. Your opponent’s calling/raising distribution is a subset of your opponent’s hand distribution, and it’s the set of all possible hole cards that your opponent will call or raise with based on all information available to you.

To find P(failed), take the number of combinations in your opponent’s calling/raising distribution and divide by the number of combinations in your opponent’s hand distribution. Since your bluff either fails or succeeds, P(successful) is the complement of P(failed). In other words, to find P(successful), simply take P(failed) and subtract it from 1.

Suppose that your opponent’s hand distribution contains 80 combinations of hole cards, and that for a specific bet size, your opponent’s calling/raising distribution contains 30 hole cards. This means that  and .

Let’s take an example hand and go through the entire process of calculating the EV of a bluff, including the process of enumerating the combinations in your opponent’s distributions.

Suppose you’re playing $500NL hold’em ($2-$5 blinds). Action folds to you preflop, and you raise to $20 from the button with QJs. The small blind folds, and the big blind calls. The flop comes K♦8♥5♥. Your opponent checks, and you check. The turn makes the board K♦8♥5♥5♦. Your opponent checks, and you check. The river makes the board K♦8♥5♥5♦T♥. Your opponent checks, and you are contemplating a $20 bet.

A reasonable hand distribution to assign a generic opponent given the action in the hand is [AQ,A8]||[QJ,QT]||[JT]||[TT,22]. We can debate for hours about whether hands should be added to this distribution, but at least it gives us something to work with. Given this hand distribution, a reasonable calling/raising distribution is the following: [AT]||[A8]||[QT,JT]||[TT,88]||[55].

With our opponent’s hand distribution and calling/raising distribution clearly stated, our next step is to figure out P(failed) and P(successful). To do this, we must tabulate the number of combinations that exist for each hand in your opponent’s hand distribution. When doing this, make sure you account for cards that are already known (your own hole cards and the board cards).

For AQ, there are 4 aces left in the deck and 3 queens left in the deck, meaning that there are  AQ combinations. For AJ, there are 4 aces left in the deck and 3 jacks left in the deck, meaning that there are  AJ combinations. You can deal with the other unpaired cards in your opponent’s distribution in a similar manner.

Let’s now talk about dealing with the pocket pairs in your opponent’s hole card distribution. For TT, there are three tens left in the deck. Using the method we used to find the combinations available for unpaired hole cards, we’d do . However, the problem with this is that we are actually double counting because we are counting hands like T♣T♦ and T♦T♣ separately. To take care of this problem, we need to divide by two. Thus, there are 3 TT combinations. Since there are four nines left in the deck, the number of combinations available for 99 is .

The table below breaks down your opponent’s distribution in terms of combinations available to each set of hole cards:

Hole Cards

Combinations

AQ

12

AJ

12

AT

12

A9

16

A8

12

QJ

9

QT

9

JT

9

TT

3

99

6

88

3

77

6

66

6

55

1

44

6

33

6

22

6

In total, your opponent’s hand distribution contains 134 combinations. Meanwhile, your opponent’s calling/raising distribution has 55 combinations. Your payout for a successful bluff is +$42 since the pot contains $42, and your payout for a failed bluff is -$20 since that’s the bet amount you’re contemplating. Therefore, the EV of a $20 bluff is the following:

When using this model to calculate the effectiveness of your bluffs, you need to be aware of two concepts. First, your opponent’s calling/raising distribution will be a function of your bet size. Different bet sizes will cause your opponent to have different calling/raising distributions. Second, your opponent’s calling/raising distribution needs to include a correction that accounts for the probability that your opponent will rebluff you.

The first correction is very easy to account for theoretically. In practice, it’s tough to do precise analysis on multiple calling/raising distributions when you’re in the heat of battle. When you’re at the table, you’ll probably just use ballpark figures. However, you should do rigorous analysis away from the table, so that you can return the to the tables better informed.

The second correction is tougher to account for…you have to somehow figure out the probability that your opponent will rebluff given that he doesn’t actually have a good hand. The good thing is that most of your opponents are not willing to rebluff. Against opponents willing to rebluff, you simply need to make an on-the-fly adjustment to P(successful) and P(failed).

With all this knowledge, you’re now in a great position to make tactically sound bluffs at the table. May your EV always be positive!

Tony Guerrera is the author of Killer Poker By The Numbers and co-author of Killer Poker Shorthanded (with John Vorhaus)

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