Saturday, April 17, 2010

Short to Long: Birds Know How to Make a Deal

The Short

Apparently pigeons are better than undergraduate students at the Monty Hall problem. I'm not sure whether this reflects well on pigeons or badly on undergraduate students. I don't recall pigeons doing well at checkers.

The Monty Hall problem is based on part of the game show Let's Make a Deal that Monty Hall co-created and hosted. The problem goes like this:
  • There are three closed doors on stage.
  • Randomly placed behind one of the doors is a prize. The prize is hidden and will not move to another door.
  • The contestant picks a door.
  • The host, Monty Hall, opens one of the two remaining doors showing it to be empty.
  • The contestant can then either still hope the prize is behind door they first picked and stick with their initial choice or switch to the remaining closed door.
  • The prize is revealed. If it's behind the door the contestant picked the contestant wins.
That's it. There isn't much to it. It all boils down to: do you stay or switch after one of the closed doors you didn't choose is opened? And why do you stay or switch?

Image yourself in the situation. You've picked one of the doors. One of the remaining two doors has been opened. You now have a choice to make. There are three possibilities:
  1. You have a better chance staying with the door you first picked.
  2. You have a better chance switching to the unopened door.
  3. It doesn't matter if you switch or not.
So? What's your answer? Not sure? Well try it yourself. Here's two online versions for you to try. One's from the New York Times the other from the University of California, San Diego.

The Long
The Monty Hall problem seems simple. Most people get to an answer pretty quickly. Many people quickly get to the wrong answer though. I once spent part of an afternoon working through the problem with my boss. He 'knew' that the right answer was to switch but he couldn't see why. We went through it several times in several ways and finally he had his aha moment. All of a sudden it made sense.

It's telling that while the Wikipedia the page for Monty Hall is 1,100 words long and the page for Let's Make a Deal is roughly 5,200 words long, the page for the Monty Hall Problem is roughly 7,000 words long.

Books have been written about the The Monty Hall Problem. Entire books. (Remind me to pick up a copy of the first one and disappear for a few days). Articles have been written that patiently try to explain why you should switch doors. Every time it's talked about in a magazine or online there seems to be a flood of letters or email saying the author is wrong or that they don't know what they're talking about.

So why do pigeons do better than us poor advanced apes?

My guess is that they don't try and analyze what to do. Instead they just learn that they're more likely to get a reward when they switch and they keep switching. They don't try and understand it. We try and understand the puzzle. We think it through. We act on what our understanding is. If we end up understanding the puzzle wrong... well... we keep acting on our understanding. We aren't good at keeping track of how well we're doing. Sometimes it may be better to be a birdbrain.

What everyone forgets is that the Monty Hall problem is not the problem Monty Hall put to contestants on Let's Make a Deal. The reality was much more subtle than that. Monty Hall was acutely aware of human psychology and he knew how to exploit it mercilessly.

Don't believe me? Read all about Monty Hall's encounter with the Monty Hall problem. Picture Monty Hall in his living room. He's just seen the problem performed - staying ten times and then switching ten times. He's just read one of the first articles that explained the problem to the public and...
After the 20 trials at the dining room table, the problem also captured Mr. Hall's imagination. He picked up a copy of Ms. vos Savant's original column, read it carefully, saw a loophole and then suggested more trials.
Now watch a master at work. If there's a loophole a smart man will exploit it. He then started to perform his own version of the Monty Hall problem. One that exploited the loophole to make the contestant lose most of the time. He created a variant of the 'true' Monty Hall problem.

To the credit of the people who wrote the Monty Hall Problem page on Wikipedia there is a section on variants. In some variants you should switch. In others you should stay. In other variants you might as well flip a coin because your odds are even. Monty would be smart enough to keep you guessing so you'd never know if there was a strategy that was better than 50/50.

I think he'd even outsmart the pigeons.

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