Advanced Value Betting in Simple Scenarios With Game Theory

Last week, we looked at one of the scenarios that helps to form a foundation of poker understanding. This scenario was when we are on the river, heads-up, in position with a single bet left behind. More specifically, we looked at the bluffing frequency of the player IP (in position) and the calling frequency of the player OOP (out of position). We can summarize our findings as follows:

Assume we have bet size B with pot size P. OOP has checked to IP on the river with one bet left behind.
When IP bets, B/(B+B+P) tells us the percentage of the time that he should have a bluff to be unexploitable.
OOP should fold to IP’s bets B/(B+P) percent of the time to be unexploitable.

One aspect of the situation that we did not cover is how often IP should be betting overall. We know how often he should be bluffing as a percentage of his total bets, but we don’t know how often he should be betting overall. This is the same thing as asking how often IP should be value betting, and that’s what we’re going to look at this week.

The Nature of a Value Bet

There are a couple of characteristics of value bets that should be understood. First, a value bet is not intended to always win if the opponent calls in this scenario. If IP value bets in this scenario and OOP calls, IP is making money on the bet (relative to checking the hand) as long as the hand wins at showdown more than 50 percent of the time.

There’s a relatively new term called value cutting that gets thrown around from time to time. The term basically means when you value bet with a hand that’s low enough in value that you actually lose more than 50 percent of the time against your opponent’s range.

The second characteristic of value betting that you need to understand is that it requires a little more forethought than bluffing. With bluffing, you only have to be bothered to consider your opponent’s fold percentage against your bet. However, when you’re value betting, you have to do an evaluation of your opponent’s calling range and how it stacks up against your hand.

With that in mind, let’s get into how this relates to game theory and our specific river situation that we have been analyzing.

Unexploitable Value Betting

If we want to exploit our opponent as much as possible, then here’s what we would do. By definition, the most exploitative strategy would value bet with every profitable hand without value betting any hands that are better off checking. To determine if a hand is profitable, then we would stack it up against our opponent’s calling range. So how do we determine what the unexploitable value betting range is?

We look at the most exploitative value betting range that we can have against our opponent when our opponent is folding at an unexploitable rate.

If that seems confusing, think about it for a moment. We know that OOP is folding B/(B+B+P) percent of the time when he’s calling in an unexploitable way. To match that with unexploitable value betting, we simply value bet with every hand that is profitable against his calling range.

Let’s use the same example we had in last week’s column. OOP checks to us on the river with a $10 pot and $7.50 behind. Our opponent’s unexploitable folding percentage is 7.50/17.50 = 42.9 percent, so our unexploitable value betting range is all of the hands that would be profitable against that 57.1 percent calling range.

Exploiting Opponents With Value Betting

Remember that unexploitable strategies are really just a guideline for us that tells us how we should exploit our opponents. In the above example, if our opponent is calling more than 57.1 percent, then we should adjust by value betting more often than whatever our unexploitable range would be. Along similar lines, if our opponent is calling less than 57.1 percent, then we should tighten up a bit. This is how exploitation works with value betting.

Figuring Out the Size of Your Unexploitable Bluffing Range

Consider the following. In the above, we figured out the size of our unexploitable value betting range. In last week’s column, we figured out that B/(B+B+P) told us the percentage of our total bets that had to be bluffs for us to be unexploitable. Using those two pieces of information, it follows that we should be able to figure out the size of our unexploitable bluffing range. Unfortunately, the actual process involves a bit of math. Since we’re not really trying to be all about a lot of math in this series, you’ll be happy to know that there is a very simple shortcut.

Step 1: Find the value bet/(bet+pot).
Step 2: Take that percentage of the size of your unexploitable value betting range.

So for example, if you had a $10 pot with a $7.50 bet size, then bet/(bet+pot) is 42.9 percent. The size of your unexploitable bluffing range will be 42.9 percent of the size of your unexploitable value betting range. If you decide that your unexploitable value betting range is about 28 combinations, then your unexploitable bluffing range will consist of about 42.9 percent of that which comes to 12 combinations.

A Summary of the Process So Far

We’ve done a lot of work over the past two weeks, so let’s summarize what we would do in a river scenario like this one to quickly work out our unexploitable strategies. First we would imagine our opponent’s unexploitable folding range, which is the bottom B/(B+P) percentage of his hands, and we would imagine all of the hands that we hold that beat that corresponding calling range. This group of hands is our unexploitable value betting range. Then we use the size of that range to determine the size of our unexploitable bluffing range based on the percentage found with bet/(bet+pot).

Next week we’re going to talk about where these ranges come from inside of our total, overall river range and how to do these calculations quickly while at the table.