A lot of the time in poker, we get caught up on the idea of wanting to know if something is profitable or not. We often make mistakes in our game because we’re taking an aggressive action that we have deemed to be profitable since we have made a critical mistake in our thinking: Instead of considering which option is the most profitable, we have jumped at the aggressive choice simply because we think its profitable on its own. Our goal in this week’s edition of [spoonitnow strategy] is to consider the alternative to the aggressive action.

River Bluffing HU IP

A lot of people know that an all-in river bluff heads-up is profitable when our opponent folds more than the value of our bet divided by the sum of our bet and the initial pot size on the river. For example, if the pot is $12 and we shove $8, we need our opponent to fold more than 8/(12+8) = 0.40 or 40 percent of the time. However, meeting this criterion doesn’t mean that bluffing is the best option. Consider the following.

An Example Scenario

Suppose that we have a pot of $10 with stacks of $7.50 left behind, and our opponent has checked to us on the river in a heads-up pot. What happens if our opponent folds 46 percent of the time on the river (assuming we never win when he calls)? Our EV of bluffing is:

EV of bluffing = (0.46)($10) + (0.54)(-$7.50)
EV of bluffing = $4.60 – $4.05
EV of bluffing = $0.55

So that by itself shows that bluffing is profitable in the vacuum. However, instead of having a knee-jerk reaction to go for the aggressive option, we should consider the alternative.

Alternative: River Checking

Suppose that we have E equity if we check through on the river instead of bluffing. Then our EV becomes the following:

EV of checking = (E)($10) + (1-E)($0)
EV of checking = 10E

So how much equity do we need to have in this situation for checking to be a better play than bluffing?

EV of checking > EV of bluffing
10E > $0.55
E > 0.055
E > 5.5 percent

That might seem like a very small amount of equity to you, so let’s check it really quickly to see what our EV of checking is when we have 6 percent equity at showdown after checking:

EV of checking = (E)($10) + (1-E)($0)
EV of checking = (0.06)($10) + (0.94)($0)
EV of checking = $0.60

Ta da.

River Bluffing HU OOP

River bluffing heads-up from out of position is a little different because of one key change. That change is that the value of our checks is going to be lower since we have to let our opponent act after we check. Because the value of checking is lower, with all else being equal, we’re going to bluff more often out of position in heads-up pots on the river.

Why Checking OOP is Different

Consider the EV equation for checking in position that we had earlier:

EV of checking IP = EV from winning at SD + EV from losing at SD

Since we don’t have a bet size, our EV from losing is always zero. Therefore, our EV of checking is just our EV from winning which is our equity multiplied by the size of the pot:

EV of checking IP = (equity at showdown)(size of pot)

Now consider the EV equation for checking out of position assuming that we are intending to check/fold:

EV of checking OOP = EV from facing a bet + EV from winning at SD + EV from losing at SD

Like above, our EV from losing at showdown is zero. Our EV from facing a bet is also zero since we haven’t made a bet size. That means our EV of checking OOP is the same as our EV from winning at showdown which is the following:

EV of checking OOP = EV from winning at SD
EV of checking OOP = (% our opponent checks)(equity at showdown)(size of pot)

Because our opponent is going to check less than 100 percent of the time, we have to scale down our EV from checking OOP accordingly. Let’s look at this in an example.

Example of Checking OOP

Let’s consider the same situation as before where we have a $10 pot with $7.50 stacks left behind heads-up on the river, but this time we’re out of position. If we again assume that our opponent will fold to our bluff 46 percent of the time, then our EV of bluffing is still $0.55. Now let’s look at the EV of checking and when it’s going to be better than the EV of bluffing.

EV of checking > EV of bluffing
(check %)(equity at SD)($10) > $0.55
(check %)(equity at SD) > 0.055

So what we’ll notice here is that we have basically the same equation and same concept as above except we have to compensate for how often Villain checks. The more he checks behind, the less often we’re going to bluff, and the less he checks, the more often we’re going to bluff. It’s kind of anti-intuitive in a way, but it makes sense when you look at it through the thought processes we have shown here, then it makes a lot more sense.

Checking OOP Against Multiple Players

If you’re checking against multiple players on the river, then your EV of checking will depend on how often the players left to act will check behind you. It looks like this:

EV of checking = (Villain 1 check %)(Villain 2 check %)(…)(equity at SD)(pot size)

You’ll need to compare it to the EV of bluffing against those players like this:

EV of bluffing = (% all Villains fold)(…)(pot size) + (% all Villains don’t fold)(-bet size)

It’s a bit more complicated because of the way folding percentages work for multiple players, but you’re typically going to be checking a lot more in these situations as a general rule.