Pigeons Outperform Humans at Monty Hall Dilemma

The Monty Hall Problem is a common mathematical fallacy based on Monty Hall's game show Let's Make a Deal. It works like this:

Imagine that you’re in a game show and your host shows you three doors. Behind one of them is a shiny car and behind the others are far less lustrous goats. You pick one of the doors and get whatever lies within. After making your choice, your host opens one of the other two doors, which inevitably reveals a goat. He then asks you if you want to stick with your original pick, or swap to the other remaining door. What do you do?

It's counterintuitive to many people, but switching doors will double your chances of winning:

The problem is that most people assume that with two doors left, the odds of a car lying behind each one are 50/50. But that’s not the case – the actions of the host beforehand have shifted the odds, and engineered it so that the chosen door is half as likely to hide the car.

At the very start, the contestant has a one in three chance of picking the right door. If that’s the case, they should stick. They also have a two in three chance of picking a goat door. In these situations, the host, not wanting to reveal the car, will always pick the other goat door. The final door hides the car, so the contestant should swap. This means that there are two trials when the contestant should swap for every one trial when they should stick. The best strategy is to always swap – that way they have a two in three chance of driving off, happy and goatless.

The bad news is that according to a scientific study, pigeons are better as this task than we are:

Each pigeon was faced with three lit keys, one of which could be pecked for food. At the first peck, all three keys switched off and after a second, two came back on including the bird’s first choice. The computer, playing the part of Monty Hall, had selected one of the unpecked keys to deactivate. If the pigeon pecked the right key of the remaining two, it earned some grain. On the first day of testing, the pigeons switched on just a third of the trials. But after a month, all six birds switched almost every time, earning virtually the maximum grainy reward.

Link via The Presurfer | Photo: Library of Congress

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I just had an ah-ha moment. I think I finally get this problem. It will definitely be the right choice to switch.

When you first choose, there is a 66% chance you picked the wrong door. This means there is a 66% chance that the correct door is one of the other two.

Then one of those two is eliminated. This does not change the fact that there is a 66% chance that one of those two is correct, except now you know one of them that isn't the correct door. So there is a 66% chance that the other door is the right one. You should switch.

I think the hang up for most people (certainly for me) is that it seems like the choice between two doors is independent of the original choice between three doors. What I just realized is that they aren't independent, because your first choice helps determine what doors will be available for the second choice.
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It seems to me that after the host has opened one of the doors, each remaining door now has equal odds of being the correct choice. Now, when the host asks whether you want to switch, he is effectively asking you to choose again, but now the choice is between two doors, rather than three. Choosing the same door you did in the first place after the "choose again" prompt is effectively the same as having not switched with the "Would you like to switch?" prompt. So the chance is 50/50. Am I wrong with this interpretation?

Of course, if the game is being manipulated in response to the player's choice, (ie, the car switches doors, the player is only given this choice if they chose correctly, etc) then questions of probability are irrelevant.
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It is this kind of scientific excess that is ruining both Science, and the economy. No matter how valuable the data we learn, we just can NOT afford to give pigeons new cars right now!
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I also disagree with any characterization of this study as "pigeons are smarter than humans", when in fact all it shows is that "some humans are dumber than pigeons". A subtle but important distinction.
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@houtx, @dj_nme,

I think the article was poorly worded. The first time I read it, I read this sentence: "If the pigeon pecked the right key of the remaining two, it earned some grain." to mean "If the pigeon pecked the key on the right hand side, it earned grain. Which isn't a correct model of the MH problem. I think it's safe to assume the researchers did correctly understand and model the MH problem, and it is the article that is ambiguous in the description, it should have stated "If the pigeon pecked the correct key of the remaining two, it earned some grain."
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