Chinese Peasants, Logic, and Poker

Mathematics, Strategy

September 28, 2007

There is a classic logical fallacy, the Emperor’s nose fallacy, that all poker players need to be aware of because it appears so often in poker reasoning (especially in poker books and forums) and has become increasingly common of late.

According to legend, there was a Chinese peasant who wanted to know how long the Emperor’s nose was. However, law forbid him from going to court and gazing on the royal beak directly. So instead he asked everyone he knew how long the emperor’s nose was, and averaged the answers. He was very proud of the result – being the average of so many answers, it must be highly accurate. It even had nice properties of numerical stability – it wasn’t heavily influenced by any one person’s response. As such, the peasant was very surprised and felt more than a little silly when the emperor traveled through the village and his nose bore no resemblance to the averaged result.

The point of this, of course, is that averaging numerous inaccurate values gives you an accurate value if and only if the inaccurate values are distributed such that their mean is in fact the accurate value. In the peasant’s case, since everyone he asked was as ignorant of the Emperor’s nose as he was, there was no such guarantee, and his result was worthless no matter how many people he asked.

This fallacy usually shows up in poker when reasoning using hand ranges. Here, you calculate the odds you would have of winning against each candidate hand, and then average. However, what you should know is that this average is inaccurate unless the candidate hand range more or less brackets (In terms of odds) the actual hand he has. You would hope this is the case, but often it is not. In particular, when someone tries to use a “sufficiently large” hand range for villain, they are usually able to add only more weak hands, because all hands up to the nuts have already been included in the range. And as such, as the range is expanded, your “average” odds inevitably improve. You might think that, by averaging more hands, the result becomes more accurate, but instead it just becomes more optimistic. By following this line of reasoning you can almost always find a call in any scenario where the fundamental theorem says you should fold. In other words, it’s nothing more than a very mathematically complicated justification for calling station behavior.

Moral of the story: beware hand range math, and be doubly concerned about a wide range, and be triply concerned when someone widens the range to find a call. Like that Chinese peasant, you’ll feel pretty silly when villain flips over the solid hand you should have known he had. We’ll talk about how to do hand range math correctly in a future article.

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