Introduction

For a long time, most strategy for beginning and intermediate players for online poker has centered around taking advantage of loose and/or passive opponents. That’s a thought process that’s modeled after the older style of “fish,” but today’s players are better than that. More often than not, the players you’re going to be able to take advantage of the most have some basic understanding of aggression and essential poker concepts, so the way you beat them has to change as well.

Thankfully, the vast majority of players know just enough to get them in trouble if you know how to pull the strings.

We’re going to look at one way to use aggression against these players by inducing bluffs. This post is going to be a little math heavy, but if you’ve read my EV Calculations Tutorial series, then you should be alright. I haven’t had a post with a ton of math in it in a while, and I’m going to try to keep this from turning into one because I want this stuff to be accessible.

Basic Concepts

The most basic bluff catching scenario is when we check/call on the river against a single opponent to close the action. It’s important to notice that we have to be out of position to have this option in the first place for the river scenario. To decide if we should induce a bluff in a vacuum, we’ll want to evaluate checking compared to betting ourselves.

A Simple Example

I like to start with simple, restrained examples and then work backwards to more complicated ones. Understanding how things work on the micro level like this gives us a better idea of the inner workings of the larger picture.

Suppose we’re on the river out of position against a single opponent. The pot is $50, and we have $35 behind. If either player bets, then it’s all-in. Our two options are shoving or check/calling. Let’s look at the EV equations for each:

EV of Shoving = (Villain folds)($50) + (Villain calls)(Hero wins)($85) + (Villain calls)(Hero loses)(-$35)

If we call Villain’s fold percentage F and our equity when called C, we get the following somewhat simplified equation:

EV of Shoving = 50F + 85C(1-F) – 35(1-C)(1-F)

What’s important to note here is that both our opponent’s fold percentage and our equity when called affect the EV.

What About Inducing a Bluff?

Now let’s look at the EV of check/calling instead. You’ll note that there are multiple factors at work in this equation as well:

EV of Check/Calling = (Villain checks)(Hero’s equity)($50) + (Villain bets)(Hero wins)($85) + (Villain bets)(Hero loses)(-$35)

And again, we can call Villain’s check percentage H, Hero’s checking through equity A and Hero’s check/call equity E:

EV of Check/Calling = 50AH + 85E(1-H) – 35(1-E)(1-H)

What you’ll notice here is that these two equations are of the exact same form, but the variables are different. This is a really interesting equality of this situation, and it lends some insight into how we can compare these different scenarios.

What to Gain From This Analysis?

Look at the pairs of variable sections of the terms from one equation to the next, and you’ll see some interesting comparisons. We’re going to look at those comparisons here to show you why they happen and what they mean.

Comparison 1: F vs AH

Here we’re comparing our opponent’s fold percentage to the product of Villain’s check percentage and our equity. Our fold percentage will almost always be higher. For example, if we think Villain only folds 30 percent of the time in some spot, that might seem low. However, if Villain checks through 50 percent of the time we check, and we have 50 percent equity when he checks through, then the value of AH is 0.50 * 0.50 which is only 25 percent.

However, this isn’t a good thing if we have a hand that would be ahead of our opponent’s calling range because we don’t want our opponent to be folding so much when we are so strong.

Comparison 2: C(1-F) vs. E(1-H)

We’re comparing two things here, and to be clear, I’m going to list them like the following:

C(1-F): How often Villain calls multiplied by our equity when called. E(1-H): How often Villain bets multiplied by our equity when he bets.

What you’ll notice here is that we’re essentially comparing the same exact thing here. We want to know what percentage of the time we win if either player bets. This can be a trade-off. For example, if betting gives us 40 percent equity with Villain calling 60 percent of the time, that will contribute to the EV in the same way as if our opponent would have bet 40 percent of the time we check giving us 60 percent equity.

The lesson here is that our equity matters just as much as how often the money goes in for the total EV of our plays.

Comparison 3: (1-C)(1-F) vs. (1-E)(1-H)

Again, I’ll spell out what these variables mean for the sake of clarity:

(1-C)(1-F): How often Villain calls multiplied by Villain’s equity when called. (1-E)(1-H): How often Villain bets multiplied by Villain’s equity when he bets.

This is more or less the opposite situation of the second comparison above. We’re looking at Villain’s equity in each of the two situations when the money actually goes in. Along similar lines, how often the money goes in each way is just as important as the equity we have when it does.

Overview

Inducing bluffs is generally best under a number of specific conditions, but it always has to be taken in context to value betting (and vice versa). Considering the following points and their converses:

The more your hand beats your opponent’s calling range, the more likely you should be to value bet.

The more your hand beats your opponent’s betting range, the more likely you should be to check.

The more your opponent calls when you value bet, the more likely you should be to value bet (if your hand beats his range).

The more your opponent bets when checked to, the more likely you should be to induce bluffs (if your hand beats his range).

Where to Go From Here

Start off by posting your thoughts in this forum thread where I’ll add additional notes soon.