Let’s Make a Deal

by
Robert Waldmann
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This may be what Mueller is saying to Manafort, but I include those names just to try to trick google and get clicks.

I want to write about the very well known Monte Hall paradox. For the kids, there was this show “Let’s Make a Deal” featuring contestants and host Monte Hall who acted like a sleazy salesman trying to trick them. One constestant was the winner who got the final prize. They chose one of three doors. There as a big prize behind one of the doors. Monte would open another of the three doors showing a small prize, then he would ask the contenstant if he or she wanted to switch and take the prize behind the unchosen unopened door. I admit I watched the show. No one ever switched. Contenstants won the big prize about one third of the time (just as if they had guessed with no door opening or option to switch).

The funny thing is that if they had switched they would have won two thirds of the time. There were two doors left. The chance the big prize is behind the first door guessed by the contestant is 1/3 so the chance it is behind the unchosen unopened door is 2/3.

Bayes explains. Let’s say contestant chooses door 1 and Monte opens door 2. If the prize were behind door 3, he would open 2 with probability 1 (he always opened a door and the big prize was never behind it). so the probability that the prize was behind door 3 and Hall opens door 2 is (1/3)1 = =1/3 If it were behind door 1 he might open 1 or might open 3. *Assume* he chose at random in such cases. then the chance that the prize is behind door 1 and Hall opens door 2 is (1/3)/(1/2) = 1/6. So switching gives winning probability
(1/3)/((1/3)+(1/6))= 2/3.

This is extremely counterintuitive. Extremely. I remember advising my the contestants on my TV to not switch (I was very young). I recently learned Paul Erdos had to write a computer program to simulate the game to be convinced. That’s the Paul Erdos.

Why ?
1) loss aversion. People hate having something than losing it. Guessing right, then switching and losing is more painful than guessing wrong. I’m sure this is a factor, but it can’t beat two to one odds.
2) the cheater dectector (explanation due to John Tooby). During the whole show, if Monte Hall suggested doing something it was wise to say no. At the end, he operated according to simple rules. But it is natural to us to think “this man is trying to cheat me so I should say no). I think this might be it (being an actor Monte Hall managed to project more sleaze than Donald Trump himself). Very generally many paradoxes (or cases in which people don’t act as Bayes advised) make sense if one assumes that the experimenter might be lying. more on this below after the jump*
3) Our brains are hard wired to think about causation. The prize would be behind door 3 because 3 was chosen at random not because the contestent chose 1 and Hall opened 2. p is more likely if q is not the same as p is more likely because q and q caused p. This confuses us. We have trouble with correlation and causation. One fallacy is post hoc ergo propter hoc. Another is not propter hoc ergo no post hoc.
4. Did you see the stress on “assuming” ? Yes of course you did. The examples in which we should use Bayes formula and don’t always include strong assumptions about the data generating process (and Monte Hall US citizen, human being and free and equal agent was part of the data generating process). An older example is “I have 2 cards. one is black on both sides. one black on one side white on the other. I shuffle pick one and plop it down on the table. The upper side is black. The lower side is black with probability 2/3 not 1/2 as it is natural to guess. This works fine, because equal probabilities of one face up or the other make sense. Monte Hall is much more complicated. I can tell a story in which somone would have gained nothing by switching. Montey Hall opens the lowest number door which has not been chosen and doesn’t contain the large prize. If I choose 1 and he opens 2, then the probability that the big prize is behind door 3 is 1/2 (he opens 2 for sure if it is behind 1 or if it is behind 3). If he were to open door 3, then the probability that the prize is behind door 2 is 1. On average if one always swiches one wins 2/3 times but this is from 1/3 chance of winning for sure if one switches + 2/3 of winning half the time no matter what one does.

To calculate the probabilities, I need to know Monte Hall’s rule. I can tell (did tell) from watching let’s make a deal that he always opens a door and it never contains the big prize. But I don’t know he opened a door at random if the big priize is behind the first door the contestant chose. Now switching always gives at least as high probability of winning as staying, and the probabilities are only equal for a subset of measure zero of the manifold of may be Montes. But the sense that one needs to know something which one doesn’t know to do the calculations matters.

*more on cheater detectors

a) Allais paradox (cumulative prospect theory). People overweight low probabilities (Kahneman and Twersky) of extreme evends (Quiggin making sense of Kahneman and Twersky). 99% sure is,in our minds, much further from 100% sure than 89% sure is from 90% sure. Maurice Allais went to conferences on choice under uncertainty and managed to get people who had just given talks on expected utility maximization to make choices inconsistent with expected maximization of any function of winnings (choice which would be entirely rational if their sole aim in life were to amuse Allais). I have a rationalization. If someone claims something is 100% sure and it doesn’t happen, then there is proof beyond reasonable doubt that his claim was false. If he claims 99% there isn’t (notably lawyers, judges and jurors *really* can’t handle probability). This matters a lot in the real world where there is fraud and we think about whether it can be deterred.

b) Ellsberg paradox. there are 2 urns. One contains 50 purple balls and 50 orange balls. One contains 100 balls which are orange or purple. You can go to either urn and say “I win if I pull out a purple ball” or “I win if I pull out an orange ball” then pull out a ball and win if it is the color you said. The probability of winning if one choses the color at random is 50% for both urns. But we prefer the first urn. Why ? One explanation (Tooby type) is that the 50 and 50 statement is verifiable (by opening the urn). We prefer cases in which our counterparty says many things which can be proven to be lies. That makes trouble for crooks. The issue isn’t that we don’t know how many purple balls are in urn 2. The issues is that the experimenter knows and refused to tell us (ok refrained from telling us). In real life it is best to stear clear of such urns.

There are also fear of being lauged at (trying to pick a purple ball out of an urn with 100 orange balls is ridiculous). There is also the fundamental attribution errof of blaming people for bad luck. If I go to urn 2 and say purple and it has more orange balls, that is just bad luck. But I could have done better if I had guessed differently. That fact is painful.