Why We Use Ranges

I was playing in my regular game last night and it was a good one. Five people took the flop for $40 in a $5-$5 blind game. The flop was 10 8 7. One of the blinds bet $55 out of his $400 into the $200 pot. One player folded, then the second EP limper jammed for $300. The original pre-flop raiser thought for a while and folded, and so did everyone in between. With the action back on the guy who bet $55, he turned up 9-6. I presumed he had made a verbal declaration to call and I missed hearing it somehow. It happens. I can’t hear everything everyone says at the table. The dealer mucked his cards. I asked the lead bettor if he folded and he said “Yes, of course, I haven’t been hitting anything all day.” I spoke up only to make sure that his hand didn’t get mucked on accident and perhaps I had an incredulous look and tone of voice because he explained some more about his play. All the while, the guy in between us was muttering assent! I couldn’t believe what I was hearing and seeing. Getting more than 2-1, someone folded the second nuts on a draw heavy flop to one raise and no more action. The table all started offering their opinions.

We’ll get to those opinions next issue so let’s just look at the math to see how bad this fold was. Our snakebit hero has bet $55 into $200, putting $255 in the pot. His opponent raised all in to $300. There is now $555 in the pot and our hero has $245 to call. 555-245 is 2.26-1. In order to find out what percentage equity we need to call, we divide the 1 by 3.26 since it happens 1/(2.26+1) times, which results in 30.6 percent. So, if our hand has > 30.6 percent equity against our opponent, our call is profitable. It’s easy to construct a range that we don’t have 30.6 percent equity against: J-9. Even against that hand, we tie 6 percent of the time! J-9 is not a range though, that’s a hand. It’s very rare, especially on the flop, that we can narrow an opponent’s range down to one specific hand. There are some instances, but this is not one of them.

I’d say a typical opponent in this situation will jam with very good draws, two pair, sets, straights, and perhaps some top pair hands. So, 10-8, 8-7, 10-10, 8-8, 7-7, J-9, 9-6, all nut flush draws, 9 7, Q J, K 9, Q 9, A-10, and 10-9. We have 64.37 percent equity against this range. (.6437)(555)-(.3563)(245)=357.2535-87.2935= 269.96. So, we make a profit of $269.96 when we call against this range. There are few spots in no limit hold em when we can get in 60 big blinds on the flop with more than 53 big blinds of expected profit.

Let’s say though, that our opponent is pretty tight and will jam a much smaller range. Versus a range of 10-8, 10-10, 8-8, 7-7, A 9, A 7, 9 7, J-9 and 9-6, we have 46.78 percent equity. When we plug that into our previous equation, we get a profit of $129.24 or about 26 big blinds.

Finally, let’s say we’re playing against someone who only jams with sets, straights, and top two, someone incredibly tight and rare to encounter. This doesn’t really change much as our equity against this range is 46.057 percent, essentially the same as our last calculation.

There are many dangers we run into when we put our opponent on a hand instead of a range. Since it’s a game of incomplete information, we can almost never have a read that is nuanced enough to narrow an opponent down to one specific holding. In addition to that, humans have many different informational biases that are difficult to get over. The most relevant one when we’re talking about putting opponents on hands is confirmation bias. This means we’re seeking out things that confirm our perceptions or preconceptions.

Remember what our hero said after he folded? “Yes, of course, I haven’t been hitting anything all day.” He finally hit something. In fact, he flopped one of the best possible hands he could with his two starting cards. Instead of thinking “Oh man, this is amazing, I’m going to get the money in as a huge favorite, and I’m getting more than 2-1 on my money.” He thought “I’ve been losing all day, he HAS to have J-9.” This is also an example of recency bias where we tend to think that what’s been happening most recently will continue to happen. In fact, that’s not true. Each individual situation is its own and unrelated to previous ones from that day. If, instead of trusting our flawed brains we trust math, we’ll realize that we have a wonderful spot here. The way we trust math in this instance is by putting our opponent on a range instead of a hand and making the calculations to decide whether we have a profitable situation. Our brains are wonderful things but they can influence us in ways that we don’t even realize. If we can stay calm and logical when putting our opponents on a range, we can trust in the math and realize our opportunities. ♠

Gavin Griffin was the first poker player to capture a World Series of Poker, European Poker Tour and World Poker Tour title and has amassed nearly $5 million in lifetime tournament winnings. Griffin is sponsored by HeroPoker.com. You can follow him on Twitter @NHGG