Using the [0, 1] Model to Look at Polarized Ranges

Last week, we took an initial look at the [0, 1] model for ranges and looked at a couple of examples of how it can be put to work for us. We tied this into our previous discussion of how to calculate the EV of different types of situations, and we looked at how to use this model to break down a situation into its different possible outcomes as we do following the method laid out in the EV Calculations Tutorial series. This week, we’re going to continue our discussion of the scenario we laid out.

Using a Polarized Range

In last week’s example, we had a player who was checked to on the river in a heads-up situation. After the check, both players have the same range. The IP player was going all-in for $20 into a pot of $25 with his top 40 percent of hands, and the OOP player was calling with his top 60 percent of hands. We found that there were five possible outcomes and that the IP player had an advantage of $3.20 when the above strategies were followed.

One change that the IP player could make to his strategy is to employ a polarized betting range. Instead of betting just the top 40 percent of hands and checking the rest, what if he bet with the top 30 percent and bottom 10 percent of hands instead? With a clear value betting range and a clear bluffing range, it will change the IP player’s EV. We can easily calculate the EV of him using this new strategy with the following outcomes:

IP checks, IP wins
IP checks, OOP wins
IP bets, OOP folds
IP value bets, OOP calls, IP wins
IP value bets, OOP calls, OOP wins
IP bluffs, OOP calls, OOP wins

Note that we can break up the value betting and bluffing cases for OOP calling to make the situations a little easier to work with since IP always loses when he bluffs and is called.

The Resulting Calculations

There’s one particularly tricky part when it comes to these calculations, and that’s thinking about what happens when IP checks. Note that IP’s checking range is [0.3, 0.9], and since OOP is checking his entire range, OOP’s range in those situations is [0, 1]. You can look at each possible case for OOP’s range to figure out how often IP and OOP are winning, respectively, when it checks through.

If OOP has a hand on [0, 0.3], then he always wins. If he has a hand on [0.9, 1], then he always loses. For the remaining part of the range [0.3, 0.9], they will split it. This means that OOP will win 60 percent of the time when it checks through, so IP has to win 40 percent of the time.

I’m not really a big fan of just filling these articles with endless calculations, so I’ll just give you a quick run-through of the EV from both the IP player’s standpoint and the OOP player’s standpoint. For the player in position, it’s:

(.24)(25) + (.36)(0) + (.16)(25) + (.09)(45) + (.09)(-20) + (.06)(-20)
6 + 0 + 4 + 4.05 – 1.8 – 1.2 = $11.05

And for the player out of position it’s:

(.24)(0) + (.36)(25) + (.16)(0) + (.09)(-20) + (.09)(45) + (.09)(45)
0 + 9 + 0 – 1.8 + 4.05 + 4.05 = $15.30

The net advantage is for the OOP player to the tune of $4.25.

Analyzing Why This Happened

In the original situation, we had IP winning by a good sum, and now it’s the OOP player who has the advantage after IP switched over to a polarized betting range. So why is that?

Let’s start off by looking at the profitability of IP’s bluffs. If we do our bet/(bet+pot) calculation here, we’ll see that we would need the OOP player to be folding at least 20/45 = 44.4 percent of the time for a bluff to be profitable. Since the OOP player isn’t folding that often, and only folds 40 percent in fact, then it’s a mistake for the IP player to bluff that much. As it turns out, IP’s entire bluffing range is a massive mistake.

Now let’s think about value betting. If the OOP player is calling with [0, 0.6], or the top 60 percent of his range, then it’s going to be profitable for IP to value bet with any hand on the range [0, 0.3]. This is because our value betting hands have to beat at least half of our opponent’s range to be profitable, and 0.3 is the halfway point of OOP’s calling range. When IP is value betting [0, 0.4] in the non-polar scenario from last week, those value bets from 0.3 to 0.4 are mistakes. However, they’re not as big of a mistake as the bluffs from this week’s polarized strategy example.

Lessons to Draw From This

While we’re using these examples to help you learn how to use the [0, 1] model for poker hand ranges, there are some other lessons to draw from this as well. As we can see from comparing the two scenarios that we have looked at with polarized and non-polarized ranges, we notice that betting a polarized range is worse against a player who isn’t folding enough compared to betting a non-polarized range.

What this means is that if you’re up against an opponent who isn’t folding very often, then you should be value betting them like crazy. It also means that you shouldn’t worry quite as much about value betting too often because even if you are making a mistake, it’s going to be a relatively small one.

Along these lines, we can also quickly figure out a “perfect” or completely exploitative strategy for the IP player. Since we can’t ever bluff profitably, we never will if we want to exploit our opponent to the max in this scenario. Since we know that our only profitable value bets are on [0, 0.3], we will only value bet that range and check the rest. This is illustrative of the way to play against calling stations that will maximize your profitability on the hand.