Monty Hall strikes again!
Today’s New York Times has this article, by John Tierney, about the latest wrinkle in the Monty Hall problem. According to M. Keith Chen, an economist at Yale University, the results of certain psychological studies are called into question by a sytematic error in their methodology. And the error, it seems, is rather similar to the one found in the classic wrong solution to the MHP:
The Monty Hall Problem has struck again, and this time it’s not merely embarrassing mathematicians. If the calculations of a Yale economist are correct, there’s a sneaky logical fallacy in some of the most famous experiments in psychology.
Here are some of the details:
The Yale psychologists first measured monkeys’ preferences by observing how quickly each monkey sought out different colors of M&Ms. After identifying three colors preferred about equally by a monkey — say, red, blue and green — the researchers gave the monkey a choice between two of them.
If the monkey chose, say, red over blue, it was next given a choice between blue and green. Nearly two-thirds of the time it rejected blue in favor of green, which seemed to jibe with the theory of choice rationalization: Once we reject something, we tell ourselves we never liked it anyway (and thereby spare ourselves the painfully dissonant thought that we made the wrong choice).
But Dr. Chen says that the monkey’s distaste for blue can be completely explained with statistics alone. He says the psychologists wrongly assumed that the monkey began by valuing all three colors equally.
Its relative preferences might have been so slight that they were indiscernible during the preliminary phase of the experiment, Dr. Chen says, but there must have been some tiny differences among its tastes for red, blue and green — some hierarchy of preferences.
If so, then the monkey’s choice of red over blue wasn’t arbitrary. Like Monty Hall’s choice of which door to open to reveal a goat, the monkey’s choice of red over blue discloses information that changes the odds. If you work out the permutations (see illustration), you find that when a monkey favors red over blue, there’s a two-thirds chance that it also started off with a preference for green over blue — which would explain why the monkeys chose green two-thirds of the time in the Yale experiment, Dr. Chen says.
If I’m understanding this correctly, then the original scientists thought the monkeys valued all three colors equally. If they were right about this, then the monkey’s choice of red over blue did not indicate some inherent preference for red M and M’s, but merely some effectively random selection. But then in two thirds of the trials the monkey’s went on to select green over blue. This would be hard to explain by random choice alone. A possible explanation is that having already rejected blue once, the monkeys reject it a second time as a form of rationalization. I was not wrong to choose red over blue, thinks the monkey, I just don’t like blue!
But if we assume instead that the monkey does have a definite, but slight, ordering of preferences among the colors, then the monkey’s initial choice of red over blue is also telling us something about his relative preference for green over blue. If the monkey prefers red to blue, then it is twice as likely to prefer green to blue than the other way around. Just list all the possibilities (or consult the nifty diagram included in the article).
I confess I’m a little uncomfortable applying notions like cognitive dissonance to the workings of monkey brains. But it’s fascinating stuff nonetheless.