Sep 202013
 

I was at dinner the other night with a few friends, when one of them, Nick, proposed the following problem:

Imagine you meet Dr Wise, who says he is an expert mind reader. To prove it, he has an experiment. He takes two boxes, a Green one and a Red one, and asks you to choose: you can either open both boxes or open just the Red one. Everything you find inside will be yours. Then, in front of you, he puts 1,000 dollars in the Green box. He won’t show you what he puts in the Red box, but he says: I will put nothing if I think you will open both boxes; and I will put a million dollars if I think you will open just the Red box.

Dr Wise is very accurate. Nobody knows how he does it, but he has run the experiment many times and here are the certified results: he correctly identified 99% of those who opened both boxes, who therefore got only 1,000 dollars; and he correctly identified 99% of those who opened just the Red box, who therefore got a million.

What would you choose? – asked Nick.

The first answer came from cash-strapped Pierre. “Do I get this right? I know there is 1,000 dollars in the Green box. So if I open both boxes I get 1,000 dollars for sure. On top of that, with a bit of luck, I could get a million dollars in the Red box. I’d go for both.”

“Wait a minute” – said Chuck – “assuming Dr Wise is really that accurate, if I open both boxes I have a 99% probability to get 1,000 dollars and a 1% probability to get 1,001,000 dollars. But if I open only the Red box, I have a 99% probability to get a million and a 1% probability to get nothing. I’d go for the Red box all day!”

“Hang on a second” – said Dominic – “Think about it: there is already 1,000 dollars in the Green box. Whatever Dr Wise decides to put in the Red box, he can only do one of two things: either put nothing or put a million, right? So, I figure: if he puts nothing, I get nothing if I open only the Red box and 1,000 if I open both boxes; if he puts a million, I get a million if I open only the Red box and 1,001,000 if I open both boxes. So in both cases I get 1,000 dollars more if I open both boxes. I should go for both”.

This is the much-debated Newcomb’s paradox. What would you do? Let’s see.

You know Dr Wise is very good at reading your mind. But you are also trying to read his. How does he do it? You must assume that the experimental results are real and honest. You must also assume that Dr Wise has no ‘supernatural’ power to influence your choice: whatever you decide, it is your decision – and therefore it doesn’t matter whether it comes before or after Dr Wise’s prediction: what matters is that the prediction is accurate. What he does is basically this: he looks into your eyes and, somehow, is able to figure out whether you are like Pierre, like Chuck or like Dominic. If he thinks you are like Pierre, he puts nothing in the Red box and leaves you with a compassionate grand. If he thinks you are like Chuck, he rewards you with a million in the Red box, for acknowledging and surrendering to his powers. But if he thinks you are like Dominic, the smart ass who dares to challenge his ability, he punishes you and puts nothing in the Red box, leaving you with a pitiful morsel, rather than the full prize you were aiming at.

Let’s put the problem in our framework. Dr Wise is evaluating hypothesis H: This person in front of me (you) is a Dominic (let’s subsume poor Pierre under wily Dominic, as they both end up making the same choice). Otherwise, he is a Chuck. Dr Wise makes his prediction on the base of evidence E: You look like a Dominic. Otherwise, you look like a Chuck. He is an expert with a very high symmetric accuracy: A=TPR=99%. This is the probability that he says you are a Dominic, given that you are. Under symmetry, it is also the probability that he says you are a Chuck, given that you are. But Dr Wise is not infallible: 1% of the times he meets a Chuck, he mistakes him for a Dominic and puts nothing in the Red box; and 1% of the times he meets a Dominic, he mistakes him for a Chuck and puts a million dollars. He dislikes both errors: leaving a faithful Chuck with nothing, but especially granting insolent Dominic the full prize.

We know Dr Wise is very accurate. But what we need to know is the probability that he decides to put a million dollars in the Red box. This depends on whether he reckons you are more likely to be a Dominic or a Chuck, in the light of the evidence resulting from his insight. As we know from Bayes’ Theorem, these posterior probabilities depend on accuracy, but also on the Base Rate: the prior probability that you are a Dominic or a Chuck. We also know that, if BR=50%, then PP=TPR: if, before looking at you, Dr Wise assumes that you are equally likely to be a Dominic or a Chuck, he will conclude that the posterior probability coincides with his accuracy. In that case, he will confidently call you a Dominic or a Chuck, depending on whom he thinks you are, and will act accordingly.

But we should not commit the Inverse Fallacy. Dr Wise’s decision on whether or not to put a million dollars in the Red box is not based on accuracy but on posterior probabilities. If he thinks you are more likely to be a Dominic than a Chuck, he will put nothing; otherwise, he will put a million. Only under prior indifference will posterior probabilities coincide with accuracy. In that case, the probability that Dr Wise will put nothing in the Red box equals the probability that he thinks you are a Dominic; and the probability that he will put a million equals the probability that he thinks you are a Chuck. But if BR is not 50%, posterior probabilities are not the same as accuracy – and can in fact be significantly different.

While we know Dr Wise’s accuracy, we don’t know his BR. Presumably, it should be the percentage of Dominics he has met in the course of his experiments. Maybe it is 50%. Indeed, in the original paper that kick started the whole debate, Robert Nozick referred that, having proposed the problem to a large number of people, he found that they “seem to divide almost evenly, with large numbers thinking that the opposing half is just being silly”. However, under prior indifference Chuck’s argument seems hard to refute: the expected value of opening both boxes is .99×1,000+.01×1,001,000=11,000 dollars, much lower than the .99×1,000,000+.01×0=990,000 expected value of opening just the Red box. Dominic is also right: choosing both boxes leaves him always 1,000 dollars ahead, irrespective of Dr Wise’s prediction. However, by assumption, that prediction is 99% correct. Hence, by choosing both boxes Dominic will indeed be always 1,000 dollars ahead but, alas, very likely ahead of 0, not of a million.

But here is where it gets interesting. If Dr Wise is prior indifferent, Chuck’s argument seems unassailable. However, for this very reason, it could well be that, impressed by Dr. Wise’s accuracy and allured by the million, most participants choose to be a Chuck, and only a small minority, say 1%, dare to be a Dominic. This would mean, for example, that in the last 10,000 experiments Dr Wise met 100 Dominics. Thanks to his accuracy, he called 99 of them correctly and only one managed to fool him, getting away with the full prize. Of the other 9,900 Chucks, he called 9,801 correctly, and mistook 99 of them for Dominics, leaving them with nothing:

We have seen a similar table before. Compare Dr Wise to the coach in our child footballer story. The coach is also very accurate at sorting talented from ordinary children: TPR=100% and FPR=5%. In his mind, the probability that he will be proven right is as high as his accuracy. But that’s because he has fallen prey to the Prior Indifference Fallacy: BR=50%. If he hadn’t, he would have realised that, despite his confidence, since champions are rare (BR=0.1%) the probability that a child will turn out to be a champion is a mere 2% – twenty times higher than the initial 0.1%, but still low in absolute terms.

Things are not as extreme in Dr Wise’s case. Here we have TPR=99%, FPR=1% and BR=1%. If you are a Dominic, Dr. Wise will spot you with 99% accuracy. As a result, the probability that, in his eyes, you are a Dominic will increase substantially, from the 1% Base Rate up to 50%, and the probability that you are a Chuck will decrease by the same amount, from the 99% Base Rate down to 50% – reflecting the fact that, in the last 10,000 experiments, half of his Dominic calls were right and half were wrong. Therefore, Dr Wise will be undecided as to whether you are one or the other, and indifferent on whether to leave the Red box empty or put a million in it. But, either way, he needs to decide. Therefore, the smallest move from the 1% equilibrium BR (which, under symmetry, equal the error rate FPR) will be sufficient to tip the balance: if BR is more than 1%, Dr Wise will conclude that you are more likely to be a Dominic and will leave the Red box empty; if BR is less than 1%, he will conclude that you are more likely to be a Chuck and will put a million.

On the other hand, if you are a Chuck, Dr Wise will also spot you with 99% accuracy. Hence, the probability that you are Chuck will go from the 99% Base Rate to virtually 100%, and the probability that you are a Dominic will go from the 1% Base Rate to virtually 0%. No doubt in this case: if you look like a Chuck, you certainly are.

Conclusion: If BR>1%, Dr Wise will put nothing in the Red box if he thinks you are a Dominic, and one million if he thinks you are a Chuck – just as he would do under prior indifference. But if BR<1%, he will put a million regardless of whether he thinks you are a Dominic or a Chuck. Therefore, being a Dominic will grant you the full prize.

You should be a Dominic only if you think you belong to a rare elite of people who have figured out how to outsmart Dr Wise. If not, you are better off joining the Chuck crowd.

As a corollary, if Dr Wise is perfectly infallible (TPR=100%) there is no way he can beaten.

This – after some thought – is my answer to Nick, and my take on the Newcomb’s paradox.

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