## Think you’re rational? Better think twice!

28 Dec, 2015 at 16:23 | Posted in Economics | 10 Comments

My friend Ben says that on the first day he got the following sequence of Heads and Tails when tossing a coin:
H H H H H H H H H H

And on the second day he says that he got the following sequence:
H T T H H T T H T H

Which day-report makes you suspicious?

Most people I ask this question says the first day-report looks suspicious.

But actually both days are equally probable! Every time you toss a (fair) coin there is the same probability (50 %) of getting H or T. Both days Ben makes equally many tosses and every sequence is equally probable!

And in mainstream economics one of the basic assumptions is typically — still — that people make rational choices …

1. Well if q is the probability of heads, then Pr(q=0.5) for the first sequence is much smaller than Pr(q=0.5) for the second. In fact any reasonable estimator would estimate q to one in the first and q to 0.5 in the second.

So if you instead of asking “what is the probability of the data occurring given that the coin is fair” ask “what is the probability that the coin is fair given the data” you get a very different answer. As the latter is really the hypothesis testing procedure, and that people have a gut feeling towards asking it, doesn’t really show that people are irrational. Quite the opposite.

• My point, but succinctly 🙂

2. I don’t know that I fully understand the point of the example. In particular, I did answer that the first sequence was more suspicious, even though I was already quite aware that if this was a 50-50 Bernoulli trial, as I assumed was the intention, that both sequences were equally likely. The use of the word suspicious led me to the following train of thought (it was in fact subconscious, but I lay it out in more detail now):

Either Ben is a thorough worker and actually does the trials, or he is lazy and just makes some result up. In the latter case he might be more inclined to make up something “human”, like all heads. In that case, Bayesian updating leads me to considering his result more likely to be the outcome of cheating in the first case and less likely in the second case. Thus one might refer to the first sequence as more suspicious.

For example, let’s suppose that if Ben cheats, he reports either all heads or all tails (50-50). Suppose that beforehand I don’t know if he cheats and think it’s 50-50. Then the probability of him cheating after having observed sequence 1 is .5/(.5+.5^10)=.999. And the probability of him cheating after the second sequence is actually zero, as that would never happen under the cheating hypothesis (as I postulated it here for illustrative purposes).

While writing this it occurred to me that there is an even more natural sense in which the first sequence is more suspicious: when viewed as evidence for deciding whether the coin is fair. The first sequence would lead me to suspect that the coin is not fair, the second would not. That is because I would naturally test the hypothesis of the fair coin against some hypothesis like: it always comes up heads, whereas the alternative hypothesis that it always goes
H T T H H T T H T H
is not as natural. I have never seen a coin which does that (although in fairness, I would not recognize it unless I was really looking for it 🙂 ).

In short, I think it’s quite reasonable to say that the first sequence is suspicious. The notion of suspicion is sufficiently vague that it allows some reasonable interpretations. I do not consider myself less rational for that opinion (although I do not consider myself especially rational). But if your point was that a) people cannot handle probabilities and b) therefore they are not rational, then to a) I would say that examples abound (I like the Monty Hall paradox, but there’s so many examples of easy situations where the correct answer simply doesn’t feel right, even to educated people), while to b) I would say, no not really, but how much does it matter? An open question, I think, in most circumstances. Lots of room for research.

P.s. Bruce Wilder, I didn’t really understand your point. Could you give me some example of the sort of information that we are omitting here? Or is your point that these examples are stylized.

3. To my mind, the comments all raise similar objections relating to the abstractions entailed at the outset by statistical reasoning. The first step in all statistical reasoning is the construction of the statistic, and the second step a wave of the hand to justify the representativeness of the statistic. These two steps sweep away a vast array and depth of information as irrelevant, and frequently leave the “thinker” asked to puzzle out an answer to a question posed in statistical terms with no ground on which to fix a perspective. The result is dizzying.
.
There is a lesson here for economics, which treats experience as unreal until it has been distilled into some essence of statistics.

• Great comment 🙂

4. I really don’t like this behavioural stuff. The questions are always unspecific. They’re clearly loaded.

Take this one. What does “suspicious” mean? It’s not specific. Hell, even if we were more specific and ask which coin toss is “less probable” the question is not clearly formulated.

Take “suspicious” first. It depends on your perspective. The second toss may look “suspicious” as the first toss led to a hypothesis – quite reasonable – of a biased coin. Maybe you left the room after the first toss and when you came back in you “suspected” that the coin was replaced by another coin. “Suspicious” is always context dependent and the “subject” that you ask the question to will likely answer in line with the context that they imagine. Thus different people will formulate different answers in line with their different imagined contexts. The space for imagination is opened up by the vagueness of the question.

And even if you say “less probable”, which is more technically accurate it doesn’t work. Why? Because you have seperated the tosses. You have two distinct sets of tosses. This begs the question: why? Was one undertaken before the subject left the coin in a room? Was it undertaken two days later? And so on. The fact that the tosses have been separated begs the question: why? The human mind responds to this “why?” by formulating imaginary contexts and scenarios.

It’s a perfectly rational response. But the behaviourists are con men. They play the same games as the dude with the three cups at the fairground. And they get their intellectual “payoffs” through vagueness and trickery.

5. There is something akin to opportunity cost here. I. e., the first sequence has a plausible alternative to a fair coin, namely, a two headed coin. The “cost” of believing that it was produced by a fair coin is the disbelief that it was produced by a two headed coin. There is also the question of Ben’s veracity. He may be your friend, but I don’t know him.

Now, the first sequence is all heads, while the second sequence is half heads and half tails. My guess is that both are fudged. 😉

I once attended a talk given by a mathematician who had analyzed some psychokinesis experiments from, IIRC, Princeton University. One of his findings, to which he attached great significance (in the non-statistical sense), was that the distribution of results was closer to a (statistically) normal distribution than expected. To him this was further evidence of paranormal activity than the experimenters had claimed. To me it was evidence of fudging the data, making sure that it looked like a normal distribution. 😉

6. Lars, I appreciate that your line of reasoning is quite common, but I don’t get it. You seem to conflate probability with ‘reason to be suspicious’. But why?

Suppose that Fred is suspicious about sequences that satisfy predicate S( ) and that the probability of satisfying S( ) is suitably low for a sequence chosen at random then isn’t ‘satisfying S’ evidence that a sequence has not been chosen ‘at random’ (according to Keynes)? To put it another way, if the null hypothesis is’ chosen at random’ then wouldn’t it be a statistical commonplace to reject the null after a sufficiently long sequence of Hs? How is this irrational?

This seems to me a lot like behavioural economics, where someone claims that some strange behaviour is rational and then observes that people generally aren’t so strange.

Happy New Year.

• Dave, according to the ‘standard’ probability theory, Pr(sequence1) is (0.5)^10, and Pr(sequence2) is (0.5)^10 …
Happy New Year!
Hope to hear from you also next year 🙂

• Lars, the missing bit is that ‘if Pr(X) = Pr(Y) then it is irrational to be suspicious about X but not Y’. I get that many people think that this is true, or to put it another way ‘grounds for suspicion = improbability’. But I am not aware of any logical argument to this effect, and I do not believe it.

If a roulette wheel constantly came up with a ‘1’ I would suspect that it was rigged. If it came up with a ‘seemingly random sequence’ I would not be suspicious. You seem to deny that the concept of ‘seemingly random sequence’ is meaningful. I’m with Ben.

Regards.

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