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Thursday, April 17, 2008

Making monkeys out of experimental psychologists, plus a maddening Monty Hall paradox

Experimental psychologists are forever jumping to conclusions from experimental results whose implications they haven't thought through. John Tierney in the NY Times last week reported on a particularly satisfying smack-down:
For half a century, experimenters have been using what’s called the free-choice paradigm to test our tendency to rationalize decisions. This tendency has been reported hundreds of times and detected even in animals.
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.
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, 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.

While we're at it, here's a particularly confounding version of the Monty Hall problem (which Anna Mirer wrote about in a comment here a while ago), in my rephrasing:

I offer you two closed boxes, promising you that in one of them I have placed some amount of money, and in the other I have placed twice that much; let's call these values X and 2X.

You must choose one, and you do; you open it, and it contains some amount of money inside -- let's say $100. Now I offer you one last choice: you can keep that money, or take the other box and keep whatever is inside it instead. Which choice should you make: the current box, or the other box?

Keep in mind that the other box could contain half the amount you see now, or twice the amount: $50 or $200. Average these two equal possibilities together, and you get an "expected value" (probability of each outcome times the payoff if that outcome happens) of ($200 + $50) / 2 = $125, which is more than the $100 you could have now. Isn't it clear that you should switch?

But if you think you could switch, what would have happened if you'd picked the other box first? You'd have the same arguments, and you would have seen the logic of switching to the other box.

It's a paradox!

If you want to think about it further, suppose instead of X and 2X, the boxes contained X and 100X, or even X and 1,000,000,000X. Now what should you do if you see a certain amount of money in the first box?

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Anonymous Anonymous on Wed May 07, 12:58:00 PM:
Any time you speak of assigning a random value to a box, you have to explicitly state the distribution from which the random value is drawn.

In your statement of the problem, you are implicitly suggesting that the value is a random integer chosen uniformly between 1 and infinity. Unfortunately, this distribution does not exist. Uniform distributions require a finite interval.

So this isn't really a paradox at all. If you select the box values from a valid distribution, then a rational player can determine whether it is better to switch or to stay.
Blogger Ben on Thu May 08, 06:54:00 PM:
Precisely, thought the implicit suggestion is really 0 and infinity, not 1 and infinity.

Another way to look at the answer is to suppose that X could only be $100, $200, $400, or $800. If the first box you look at is any of the first three, you should switch -- though the advantage in switching is slight. If you find $800, though, you should definitely, definitely not switch; the disadvantage of switching away from the $800 there ($400 worse than how you'd do if you kept it) exactly offsets the sum of all the advantages you get in switching in the other five situations: when you find $100 ($100 more), $200 (either $200 more or $100 less), or $400 (either $400 more or $200 less).

Of course, you don't know the limits of the range I am picking X from, but you can use what you know about me to judge when a value is probably near the top of my range (say, if you saw a few hundred bucks in the first box, in which case you should keep it) or somewhere in the middle or low end ($10, say, in which case you should switch). If you don't even know the currency's exchange rate, though, the strategies of always switching envelopes or always keeping the first one have the same expected value.