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.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:
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.
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?