Monty Hall forever
A baffling decision-making problem that illustrates surprising irrationalities
Last Saturday, Monty Hall, a Canadian game show host died at the ripe old age of 96. In a career spanning over 70 years he presented more than 10 different programmes, both on radio and television. But his name will forever be associated with a probability problem bearing his name, which was central to the format of Let’s make a deal.
The problem is deceptively simple. Imagine three closed doors. Behind one of them, there is a big, desirable prize (like a car). Behind the other two, there are undesirable prizes (a goat). The host invites you to pick one of the doors, and you will get whatever is behind it. The host (who knows which of the three doors conceals the car) then opens one of the remaining two doors, revealing a goat, and asks you whether you want to switch from your original choice to the other closed door. Is it advantageous to do so?
Mathematically, the answer is yes. But this appears counterintuitive: there are two remaining closed doors, one with the car behind it, the other one hiding a goat. How could the probability of the car being behind either door be anything else than 50%?
And yet, it is: the chance of it being behind the door you originally picked is 1/3, the chance of it being behind the other remaining closed door is 2/3. (An explanation is at the end of this piece.)
The wrong kind of irrationality
This puzzle is often wheeled out to illustrate the ‘irrationality’ of not switching. But are people who just don’t know how to work out conditional probabilities truly irrational? When my nephew was four years old, he preferred two 5 euro banknotes to one 20 euro note. Quite reasonably, he assumed that two notes were worth more than one — because he did not understand that two very similar pieces of paper could have such a different value. That lack of understanding shouldn’t be really called irrational — and for the same reason, neither should the assumption be that the chance of a car being hidden behind one of two closed doors is fifty-fifty.
However, the Monty Hall problem is an excellent illustration of truly irrational behaviour, which has nothing to do with mathematical ignorance. Only a small number of contestants actually switched doors (and some of those probably had worked out it was better). If people truly believed there was an even chance that the car was behind either door, you would expect them to choose to stay, or to switch, in roughly equal proportion. How come they didn’t?
Maybe the most obvious reason for this aversion to switching is that people are led by the status quo bias. This describes our tendency to take the option with the least effort. We picked a door, the other remaining one (we think) doesn’t offer us an improved chance, so why bother? Arguably, even this is not irrational behaviour if we assume the chances are even: there is indeed a small cognitive effort needed to switch, for no discernible material benefit.
The right kind of irrationality
But some people might decide to stay put because of the endowment effect. This captures the observation that, on the whole, we perceive what we have to be more valuable than what we don’t have — even if both what we have and what we don’t have are identical. Once a contestant has chosen a door, that door becomes ‘theirs’, and even though there is no objective reason to prefer it over the alternative, they are reluctant to swap.
Another reason why people might stick with their original choice is regret aversion, which holds us back from making a decision that we could later regret. What is interesting is that we seem to regret an active decision that turns out wrong much more than a passive decision. Imagine the car was behind the door you first picked, but you switched and ended up with a goat. You actually, literally had the car in the bag with your first choice, and you blew it by switching! That is quite different from the alternative losing scenario, in which you first picked a door with a goat, and decided to stick with it. That is bad luck, whereas the alternative is a bad choice.
But the most striking irrationality related to the Monty Hall problem is not that of the contestants, but that of the people who believe that the probability of the car being behind either door is 1/2. What we see here is the backfire effect, a phenomenon that not only makes us blind to facts that contradict our beliefs, but that strengthens our mistaken belief.
When Marilyn vos Savant dealt with the Monty Hall problem in her Ask Marilyn column in Parade Magazine in 1990, she received many indignant replies from learned professors and doctors (some of which were shamefully sexist) claiming she was wrong to say the contestant would be better off to switch. The one positive takeaway from the unedifying comments to her piece at the time is that we shouldn’t be too downtrodden about our own propensity to fall foul of the backfire effect, if this is what it does to the great and the good.
So, why is it better for a contestant to switch, and indeed why is the chance that the car is behind the other door 2/3 and not 1/2? Imagine you are playing the game, and after you’ve picked one of the doors, instead of opening one of the remaining doors, Monty offers you what is behind both remaining doors. Should you switch? Of course: the chance the car is behind your door is 1/3, and so the chance it is behind one of the other two doors is 2/3. The fact that, in reality, he opens one of the other two doors before offering you the swap makes no difference: the chance the car is behind the doors you didn’t pick is still 2/3. And since it is not behind the door he just opened, the chance it is behind the other closed door is… 2/3.
If this explanation hasn’t helped (or if you are still battling the backfire effect!), there are plenty of internet sites which explain the conundrum in numerous ways. And if you do get why it is advantageous to switch, here is a twist to check if you really got it. Imagine your friend walks in just after Monty has opened one of the two remaining doors, and she hasn’t seen the discussion earlier: all she sees is two closed doors. Which one should she pick to maximize her chance of getting the car: the one you originally picked, or the other one?
Monty Hall is dead, but he will live forever. Let’s remember him with this joke that economist Alex Tabarrok tweeted:
Originally published at koenfucius.wordpress.com on October 6, 2017.
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