Close Decisions

It's only natural to put more thought into close decisions - decisions for which the correct alternative isn't immediately obvious. At the same time, close decisions are almost by definition individually less important than obvious decisions. If your two alternatives differ in expected value (EV) by only a tenth of a big bet, you may linger over the decision for a long time even though your expectation for the hand hardly depends at all on your decision.

Of course, that doesn't mean that close decisions as a whole are unimportant. If the majority of the decisions that separate you from your opponents are the close ones, that's where you'll have to realize your advantage. So, getting them right at a higher rate is critical to winning, even though it seems like each individual decision is relatively unimportant. Just as important as getting them right is minimizing the size of the errors you make when you do err.

Often, close calls depend to some extent on your read of your opponent. In a simple example, you're deciding whether or not to call a final bet in limit poker, and you remember that your opponent raised on the turn. If the only hands that can beat you are hands with which your opponent would never have raised, you can call with impunity (sorry, no raise, because you're out of chips). If your opponent is unknown to you, but probably would have raised with a variety of hands that can beat you, a call is more questionable. Knowing how your opponent would have played which hands at just that moment is crucial to making correct decisions. You often don't have enough information to make a particularly accurate guess, especially when playing against unfamiliar opponents.

Suppose we restrict the issue to calling a bet to reach showdown. All you need to know in order to make the decision perfectly is your odds against winning and the pot odds (which you know exactly). If your odds against winning are greater than the pot odds, you fold; otherwise, you call. How well you do in the long run depends only on the accuracy of your estimates.

It may be useful to have a sense for just how accurate you need to be. For example, you might think you have about a 5 percent to 10 percent chance of winning the hand. That seems like a pretty specific estimate. But that's a big range in odds - from 19-1 to 9-1. Being off by just a few percentage points can make the difference between a clear fold and a marginal call. Of course, just because the number falls on just the wrong side of the pot odds doesn't mean you're giving away the store by calling. Even when a misestimation leads you to make the wrong decision, small errors won't cost you much in EV.

It's still worth getting a sense for how much you give up when you misestimate your chances of winning, whether it's expressed in percentages or odds. It's sometimes easier to appreciate these things graphically than to sort out a list of facts about numbers. So, let's start with odds. We can chart the EV of a call as a function of both pot size and the odds against your winning:

Some obvious things are easy to see: Your best EV comes when the pot is large and your odds against winning are short. A smaller pot or longer odds makes a call less attractive. When the lines dip below zero, you're better off folding (in which case your EV will be exactly zero).

One thing that's important to notice is where the lines cross the zero EV point, which represents the point at which it doesn't matter if you call or fold (that is, your EV is exactly zero either way). The larger the pot size, the farther right the crossover is, which just reflects the fact that you can call with longer odds when the pot gets large.

You can also think of the absolute distance between the line and the zero point as the size of the mistake you'd be making if you made the wrong decision. To the left of the zero crossing, mistakes mean folding when you should call (a missed opportunity). To the right, mistakes mean calling when you should fold (a bad investment). The biggest possible mistakes (where the lines stray the farthest from zero) are on the left: folding when the pot is large. Conversely, you can't make really large mistakes when the pot is small - the line doesn't get too far from zero, indicating that the worst investment you can make still doesn't even cost you a whole bet. That doesn't mean the mistakes are unimportant. Especially for small pots and long odds, making bad calls can cost you close to a full bet in expectation, which is a large hit to take in one hand.

You're most likely to make mistakes when it's a close call, not when the correct decision is obvious. Close calls are near those zero crossings. Notice that for large pot sizes, the graph is relatively flat around the crossing point, whereas for smaller pot sizes, it's a bit steeper. This suggests that if you mistakenly call when you should fold, or fold when you should call, you're going to tend to make a larger EV mistake when the pot is small than when it's large.

Well, sort of. The chart is in odds, and assumes that the difference between 10-1 and 11-1 is the same as the difference between 1-1 and 2-1. If those mistakes are equally likely for you, the chart probably has some relevance. But most people are probably a lot better at telling the difference between 2-1 and even odds than they are at 10-1 vs. 11-1. What happens if we graph it by percentages instead?

Now we see the opposite. The slope is steeper for large pot sizes, regardless of where you look. That suggests, more intuitively, that errors are liable to be more consequential when the pot gets big. Which chart is right? That depends on whether percentages or odds are a better basis for comparing your errors in judgment. On the whole, I think percentages are the better of the two, because my intuition is that whether the true percentage is 50 percent or 10 percent, I tend to be accurate to within almost the same tolerance (say, 5 percent), or close. At least it seems like a much better approximation than using an odds scale, although I'm willing to entertain other suggestions.

However you slice it, these graphs lend some concrete imagery to warnings about folding on the end in limit poker. As the pot starts to get even moderately large, not only is calling much more often correct, but the balance between the possible errors you might make weighs strongly in favor of calling. Unless you're relatively certain about your probability of winning, there's a lot more upside to calling than there is downside. A few too many calls will cost you a lot less than a few too many folds. Although calling too often at other stages of the hand can get you into a lot of trouble, calling too often on the end, especially in looser games, may approximate correct strategy more closely than folding too often, especially when you have very little idea of what your true probability of winning is. Even in tight, passive games, the same applies, only not as strongly. The pot has to be pathologically small before the largest risk associated with a bad call exceeds the largest risk associated with a bad fold.

Although there's not much conveyed by these graphs that isn't captured by oft-given advice about the dangers of making too many "good" folds when the pot gets big, I find that having an image in mind helps me keep in mind the real costs of making errors with difficult decisions.