Poker Math: Are Bad Players Tougher To Beat Than Good Players?

Poker Math: Are Bad Players
Tougher To Beat Than Good Players?
by Martell

If you've played low-limit poker at all, you've probably heard someone complain, “I can't beat this game—the players are all too bad!” Maybe you've even said something like this yourself. The idea is that, since a) the bad player will play almost any two cards, and b) the bad player will stay in the hand longer than is reasonable, it is hard to gauge when your hand is beat, as you will often be beaten in the most unimaginable ways.

Typically, this complaint is written off as being a myth, a short-term statistical illusion. I mean, how could a bad player really be tougher to beat than a good player? It just doesn't make any sense. Either the complainer isn't as good as he thinks he is, or he's just seen some short run bad luck, right? This is how most people think, and the typical response one hears is something along the lines of “Hey, you want that guy in your game!” or “You gotta love it!”

Something about this easy dismissal didn't sit right with me. Of course, it made sense—after all, conventional wisdom always does. But one thing I've learned over the years is that, if a bunch of gamblers who have put in thousands and thousands of hours all believe in the same illogical superstition, it probably warrants further investigation. There may very well be some mathematical principle at work that causes this pattern to recur so frequently. And indeed, there is.

Let's begin by defining what “bad players” and “good players” are. We will define “bad players” as being those players who will play any hand they want to play, even if the pot has been raised. We will define “good players” as those who will only play a raising hand when the pot has already been raised—but will never fold that hand before the flop. (This is in contrast to “great players,” who will frequently fold a raising hand if the raiser is someone who deserves respect, but may reraise with the exact same hand if the raiser has low raising standards. I think there are very few people who believe that bad players are harder to beat than great players; as a result, they are not a part of this study.)

Okay, so how can it be possible that someone who plays all kinds of bad hands can be harder to beat than someone who only plays good hands? Let's do a little math. Let's say there is a raise from middle position—the type of hand a good player might call with would be something like AQ or AJ offsuit, or maybe even KQ suited. A bad player, on the other hand, would probably call with hands like JT offsuit, pocket 5s, or even 87 suited (and a ton of other hands too—these are just a few examples). Let's compare how these hands do versus a common raising hand—AK offsuit.

Of the three choices given, which of the good player's hands would you rather have? Conventional wisdom would say AQ offsuit. But before the flop, AQ will beat AK just 26.0% of the time, while AJ will beat AK 26.8% of the time. That's right—if you are up against AK, AJ is slightly better than AQ. And KQ suited—a hand with which some good players wouldn't even call a raise—is the best of them all, beating AK 30.3% of the time.

Okay, what about the bad hands, you ask? Most people know a small pair is a slight favorite to two overcards, and indeed pocket 5s beats AK 55.0% of the time. But JT offsuit beats AK 37.4% of the time—much better than any of the good player's calling hands—and 87 suited beats AK a whopping 41.8% of the time. It turns out it's actually much more likely that one of these “bad” hands will take down AK than one of the “good” hands.

Of course, there is more to the hand than calling before the flop, but that is often where the “good players” place the blame. “How could you call a raise with a hand like that?” they ask. Hmm, probably because they do better with hands like that. But let's look at what can happen after the flop.

Let's look at a good flop for AJ: A 7 4 rainbow. Chances are, if a good player hits a flop like this with AJ, he's stuck in the hand until the end. But which hand do you think has a better chance of catching up with AK—AJ or 87 suited? It should be easy to see that 87 suited is in a much better position here: this player has three 8s and two 7s to improve his hand, as well as the backdoor straight and flush draws. By contrast, the player with AJ only has three cards to improve his hand. Overall, the 87 suited will catch AK 24.3% of the time, while AJ will only win 12.7% of the time.

What about a hand like pocket 5s? Let's compare it to AJ on the same flop—A 7 4 rainbow. Only a lunatic would call a bet on the flop with a hand like pocket 5s, right? Actually, pocket 5s will beat AK on this flop 13.6% of the time, compared to 12.7% for AJ. The backdoor straight draw, combined with the two other 5s, turns out to be a better draw than AJ's three other jacks. Yet, if pocket 5s misses on the turn, he likely dumps the hand, saving the bets on the turn and river, but if AJ misses on the turn, it doesn't matter—he's calling bets on the turn and river no matter what. The worse draw loses the maximum, while the better draw saves money when they miss.

So indeed, there are times when a bad player is tougher to beat than a good player. But doesn't the fact that he is called a “bad player” imply that he's going to lose more money than a “good player,” since that's what “bad” and “good” typically mean in a poker context? And if so, how does that reconcile with what we've just seen? Yes, it does mean that the bad player will lose the most money. Figuring out how to beat the bad player, though, requires one to approach the problem from a different angle. Next time, we'll do just that.

Martell can be reached at martell@babblog.com.