The Three Pillars of Poker Theory

 

Expected Value, Minimum Defense Frequency, and Optimal Bluffing Frequency. These are, in my opinion, the three most important concepts in poker theory, which is why I begin all of my poker courses with them; I consider them prerequisites for teaching my students virtually anything about poker strategy.

In the era of solvers, it is possible to study poker and get better without truly understanding these concepts, but you will mostly be learning through memorization. If you only study solver outputs without understanding the fundamental reasoning behind them, you’ll have a very hard time thinking for yourself in real situations.

1. Expected Value (EV)

Every poker player needs to know how to do a basic expected value calculation. Any time you do an EV calculation, it follows the same format:

EV = probability of winning * amount expected to win - probability of losing * amount expected to lose

Where probability of winning + probability of losing = 100%

Example

You get to the river with a pure bluff and you make a 3/4 pot bet to try to steal to pot. If your opponent folds 60% of the time, what is your expected value?

Let probability of winning = 60% (opponent's fold frequency)

Let amount expected to win = 1 PSB [pot-sized bet] (i.e. the dead pot)

Let probability of losing = 100% - 60% = 40% (opponent's defense frequency)

Let amount expected to lose = 0.75 PSB (your bet)

EV = 0.6 * 1 PSB - 0.4 * 0.75 PSB

EV = 0.6 PSB - 0.3 PSB = 0.3 PSB

Answer: Your EV is 0.3 PSB or 30% of the pot. This bluff is significantly +EV (because your opponent’s fold frequency was higher than the minimum defense frequency).

Note: EV calculations can have more than two terms. When this is the case, the sum of the probabilities of each term must equal 100%, just like in the example above.

2. Minimum Defense Frequency (MDF)

Minimum defense frequency is the frequency that the defender must call and/or raise in order to keep the aggressor’s bluffs from auto-profiting in a vacuum. If a player defends exactly at MDF, the opponent’s bluffs will be breakeven (0 EV).

Use the following equation to calculate the minimum defense frequency facing a river bet:

MDF = 1 / (1 + b)

Where b is the bet size in pot-sized bets (PSB).

Example

You get to the river with a pure bluff and you make a pot-sized bet to try to steal to pot. What is your opponent’s minimum defense frequency?

MDF = 1 / (1 + b)

MDF = 1 / (1 + 1) = 1/2 or 50%

Answer: Your opponent’s MDF is 50%. If your opponent folds exactly 50% of the time, you will break even on this bluff; if he folds more, your bluff will be +EV; if he folds less, your bluff will be -EV.

Notes:

It is common to see MDF referred to as “1 - α” where α is the fold frequency. The α symbol is pronounced “alpha.”

MDF is a much simpler model than reality. In fact, a true game theory optimal defense does not make the aggressor indifferent to bluffing in a vacuum. Instead, it makes the aggressor indifferent to bluffing relative to checking.

As a result, PioSOLVER will end up defending less than MDF in all river lines. In most lines, this ends up being about 5% less than MDF. (This varies slightly depending on the bet size.) In some lines, however, PioSOLVER will defend even less than that.

Additionally, this formula only works to approximate the river. It cannot be used on the flop or turn. In the pre-solver era, no one really knew what MDF was in pre-river scenarios. However, some were able to approximate it with pretty impressive accuracy.

Reference Table

The table below gives the minimum defense frequency for various river bet sizes:

Many poker players study theory very rigidly, and they don’t spend a lot of time studying what happens when an opponent deviates from the GTO strategy. As a result, they miss mathematical gems like the table below.

This table shows the EV in pot-sized bets of a bluff if the opponent defends 5% less than MDF for various river bet sizes.

Note: α is the minimum fold frequency and α’ is the minimum fold frequency plus 5%.

As you can see, the EV of a bluff when facing a 5% overfold increases as the bet size increases.

One way you can think about this is: as the bet size increases, MDF decreases. Therefore, overfolding by 5% vs. an overbet is missing a larger share of MDF proportionally than overfolding by 5% to a small bet.

This is actually quite a dramatic effect. In fact, if your opponent overfolds by 5% vs. a 2x pot river bet, and by 10% to a 1/3 pot river bet, the EV of the 2x pot bluff is actually slightly higher than the EV of the 1/3 pot bluff! Most poker players assume just the opposite is true.

3. Optimal Bluffing Frequency (OBF)

Optimal bluffing frequency (also known as the bluff-to-value ratio) is the frequency that a player must bluff in order to make the opponent’s bluff catchers breakeven (0 EV) in a vacuum.

Optimal bluffing frequency is essentially the pot odds the defender is being offered, which is a function of the bet size. Optimal bluffing frequency is also the same as the defender’s minimum equity required to call a bet.

If you do not understand the concept of pot odds, please look up an article about it before reading on. There are articles all over the internet which can help you with this.

Use the following equation to calculate optimal bluffing frequency vs. a river bet:

OBF = b / (2b + 1)

Where b is the bet size in pot-sized bets (PSB).

Example

Your opponent bets 1/2 pot on the river. What is his optimal bluffing frequency?

OBF = b / (2b + 1)

OBF = 0.5 / (2*0.5 + 1) = 0.5/2 = 25%

Answer: Your opponent should be bluffing 25% of the time. If your opponent bluffs exactly 25% of the time, you will break even when you call with a bluff catcher; if he bluffs more, your call will be +EV; if he bluffs less, your call will be -EV.

Notes:

Like the formula for MDF, this formula only works on the river. OBF increases with each additional street left to play in the hand.

Additionally, this formula assumes the bettor’s range contains only nut hands and air hands, and your range only contains pure bluff catchers. In reality, there will be more medium-strength hands in the betting and calling range, but the simplified model is still a strong approximation.

Reference Table

The table below gives the optimal bluffing frequency for various river bet sizes:

Notice that the optimal bluffing frequency is only 4% higher for a 4x pot overbet than it is for a 2x pot overbet. This is because pot odds is an asymptotic relationship. As the bet size approaches infinity, the optimal bluffing frequency approaches, but never exceeds, 50%. See the graph below for a visual represenation.

Optimal Bluffing Frequency vs. Bet Size (PSB)

As in the previous example with MDF, we will now look at the EV in pot-sized bets of calling with a bluff catcher when facing a 5% overbluff for various bet sizes.

Note: β is the optimal bluffing frequency and β’ is the optimal bluffing frequency plus 5%.

As you can see, just like with MDF, when the margin of overbluffing remains constant, the EV of bluff catching increases as the bet size increases. Large bet sizes are not only harder to defend against, but they are also harder to balance.

Conclusion

If you understand the three pillars of poker theory, you’ll be able to understand, at least on a basic level, the reasoning behind the game theory optimal strategies in any hand of poker that you study.

If you’re a professional poker player, then you should be able to do these calculations in your sleep. You should have the minimum defense frequency and optimal bluffing frequency memorized for all common bet sizes. When I first started playing poker, I wrote the reference tables above on flash cards and taped them above my monitor.

Remember to not just study these concepts at equilibrium, but also to test what happens when your opponent deviates from the game theory optimal strategy. This is the key to being able to think for yourself at the poker table rather than robotically trying (and failing) to play like a solver. It will also help you become more creative in finding ways to exploit your opponents, which is where the real money in poker is made.

Resources

Click here to view the EV calculator spreadsheet that was used in the examples in this article.

Note: Please go to File > Download in Google Sheets if you want to make edits to the spreadsheet.

 
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