Hi everyone— before we start, just wanted to remind you, the Math Mutation book is out! To order, you can follow the link at mathmutation.com or just search for it on Amazon. And if you ever pass through the Portland, Oregon metro area, I’ll be happy to autograph your copy. If you like it, posting a positive review on Amazon would be really helpful. Now on to today’s topic.
As you may recall, one of the topics we covered in some previous podcasts, and in the Math Mutation book, is the idea of “Cognitive Biases”. These are well-known ways in which the human brain has a natural instinct to think in ways that violate basic laws of logic and mathematics. One classic example is the Anchoring bias: if asked a question that has a quantitative answer, you will tend to give an estimate close to numbers you recently heard. For example, suppose I arrange separate discussions with two people to estimate how many listeners Math Mutation has. With the first one, I start by asking “Does Math Mutation have more than 100 listeners, or fewer than 100?”. But with the second one, I open with “Does Math Mutation have more than 1 million listeners, or fewer than 1 million?” If I then ask both of them to estimate the total number of listeners, the first will probably come up with a much smaller estimate than the second, even though neither has any objective information to justify a particular number.
After reading the chapter in my book, my old Princeton classmate Tim Chow pointed out that calling this a “Cognitive Bias” might not be justified. Sure, the listener technically has no information to support the larger number in the second case— but in cases where we are talking to another human being, we trust them to provide relevant information. This includes both direct statements of facts, and implications that might not be directly stated. If I ask you whether Math Mutation has more or fewer than 100 listeners, I am implicitly communicating the information that the 100 number is pretty close, even though I have not rigorously declared this to be a relevant fact. So if this number isn’t close, I have essentially misled you with false information— the fact that you trusted me and used the wrong number is my fault, not some flaw in your mental logic. Thus, this “Cognitive Bias” is really a social manipulation.
Now, if you’re familiar with the literature on this topic, you might point out an interesting experiment that seems to refute this. In this experiment, subjects saw a roulette wheel spin, then were asked the percentage of the United Nations countries that were in Africa. Even though there is no logical reason for them to suspect the roulette wheel had advanced knowledge of geopolitics, their answers were still biased towards the results they saw on the wheel. Many similar experiments have been carried out. Doesn’t this provide irrefutable proof that this really is a cognitive bias?
Not so fast. This is a very artificial situation. Maybe when asked to guess a number about which they have absolutely no idea, they just grab any arbitrary number they can think of, which will tend to be one they saw recently. They aren’t following some flawed cognitive process, they just don’t have any reason to pick any particular number. Again, this doesn’t really indicate a mathematical flaw in their reasoning— they don’t think the number they picked has a particular logical justification. Not knowing an answer, they just defaulted to what was at the top of their head.
Most of the other well-established Cognitive Biases are open to similar criticisms. Another example is the Conjunction Fallacy: suppose I tell you that Joe is a Princeton mathematics graduate and chess champion, and then ask you to choose the more likely of two statements. 1. “Joe is now a physics professor.” 2. “Joe is now a physics professor and head of the local Math Mutation fan club.” You will likely choose option #2, since it seems like this kind of guy should be a Math Mutation fan. But on reflection, option 2 must be strictly less likely than option 1, as it takes the same basic fact and adds an additional, more restrictive, condition. But again, there is information being communicated between the lines: if I give you those two choices, you probably interpret #1 as implicitly stating that Joe is NOT president of the Math Mutation fan club. I didn’t say that, but the additional choice in the second option made this a very reasonable inference. Once again, it can be seen as more of a social manipulation, where I leveraged typical communication conventions to imply something without actually stating it, and the implication is not strictly justified by mathematical logic.
We should point out, though, that even if the so-called “Cognitive Biases” are not truly flaws in the logic of the human brain, they are still important psychological effects to be aware of for many reasons. For example, let’s take a look at some practical applications of the Anchoring bias. It’s well known that when negotiating prices in business, your opening offer can set an anchor that affects the entire discussion. Negotiating business contacts usually have some level of trust in each other, and taking advantage of this to establish a good anchor is a smart, though slightly manipulative, technique. On another note, suppose you’re a surveyor trying to get accurate estimates in a survey or questioning experts on a difficult topic. You need to be careful not to include some kind of number in the question that might unintentionally influence the result. So knowing about Anchoring is still very useful, whether you call it a true cognitive bias or a simple persuasion technique. In general, I still believe the Cognitive Biases are worth studying and raising awareness of, though maybe more as social or linguistic phenomena than as true flaws in the human mind.
And this has been your math mutation for today.
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