Intro to Bluff Theory

Bluffing is an essential skill that any competitive poker player will need to master. Many players learn bluff-value ratios on the river (polarized river bets) but fail to bluff on other streets. Let’s start with the very basics.

To begin, we need to understand indifference. Indifference is when two actions are equal in value. For example, we might have a hand with a 50-50 split between calling a river jam and folding. This hand is typically a bluff catcher. Or another hand that is split between checking and betting because it is thin value. Identifying these boards are crucial to understanding indifference.

Identifying indifference will enable us to learn what balance is. However, we are not trying to be balanced; instead, we understand the line to know how to deviate relative to our opponent’s suboptimal plays.

Let’s visualize a mini river game.

Say the board is 7♠7♥7♦7♣2♥. The pot is $200 and both players have $100 behind.

Suppose hero can only have AA or QQ in this spot and hero shove. Also suppose villain has KK every time here. This means that hero’s AA is shoved for value whereas QQ is shoved as a bluff. Villain’s KK is a pure bluff-catcher every time. Let’s try to solve this mini game.

In the case we have AA, we are guaranteed to win in our mini game. We should be value jamming this hand every single time. However, we cannot just fold QQ even though we lose to villain’s KK. If we have value hands in 100% of our jams, we will never get paid. On the other end, if we jam QQ every time, villain will snap call every time because we give them the odds to call. If we give villain a 50-50 chance to win, they will be profitable since the pot odds are bet $100 to win 400.

This means that hero should bluff equal to the pot odds laid which in this case is 25%. Villain should call equal to the minimum defense frequency. The MDF in this case is:

MDF = 1 - (AggressorBluffPercentage / (1-AggressorBluffPercentage)).

Since hero jams for half pot, we are giving 4:1 odds. We need to bluff 25%. That means villain’s optimal MDF is 1-(0.25/(1-0.25)) = 66.67%.

Let’s do another example to make sure we get it. Say there is $100 in the pot and hero jams river for $100 effective. Since we are jamming 100% pot, we are giving 3:1 odds. This means we need to bluff 33% of the time. Villain’s MDF is 1-(0.33/(1-0.33)) = 50%.

Let’s transition to talking about turn bets. Let’s use the same example we just solved but instead of river jam, we pot turn and have a pot-sized bet left on the river. So let’s say we have 66% value and 33% air, we pot the turn and have exactly pot behind. What is villain’s calling percentage if they have a pure bluff catcher? If you said 50%, you would be totally incorrect. In fact, villain should fold 100%.

What the heck! How can the difference between one street completely change how we bet? In the river example, villain’s MDF was 50%. So why does a pot sized bet on the turn change things? This is because villain is effectively facing a 400% pot bet. Hero will always bet the river if they polarize and bet the turn. This means you are doubling the pot twice by the river.

Figure 1

I created a little diagram to highlight how villain will be unprofitable calling hero’s turn bet with a pure bluff catcher. Alright, so we now know that villain can comfortably fold because hero is under bluffing. We don’t want to be exploited so how do we calculate hero’s turn bluffing percentage? Since SPR is 4 and we have 66% value, we need to have around 55% bluffs for villain to break even on calls.

Real poker is a lot more complex than this mini game. In this hypothetical, we represent pure polarized equity and assume that geometric bet sizings are always available to us. In the real world, it is a lot more complicated. This does not mean our Nash Equilibrium strategy is useless. We can still harness the compounding effect of polarized equity and ranges to get good odds for ourselves and exploit our opponents. By understanding balance, we can use these polarized sizings to exploit our opponents. We can adjust our betting strategy based on if they are overbluffing or underbluffing. If our opponent is unaware of the compounding effect of pot odds on MDF, we can gain a massive edge and take massively profitable lines.