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Think more, get less

May 30, 2007

There’s an article in the June 2007 episode of Scientific American, also published on their site.It is written by Kaushik Basu.
It’s about the curse of rational choices when playing the “Traveler’s Dillemma” game:

“Lucy and Pete, returning from a remote Pacific island, find that the airline has damaged the identical antiques that each had purchased. An airline manager says that he is happy to compensate them but is handicapped by being clueless about the value of these strange objects. Simply asking the travelers for the price is hopeless, he figures, for they will inflate it.Instead he devises a more complicated scheme. He asks each of them to write down the price of the antique as any dollar integer between 2 and 100 without conferring together. If both write the same number, he will take that to be the true price, and he will pay each of them that amount. But if they write different numbers, he will assume that the lower one is the actual price and that the person writing the higher number is cheating. In that case, he will pay both of them the lower number along with a bonus and a penalty–the person who wrote the lower number will get $2 more as a reward for honesty and the one who wrote the higher number will get $2 less as a punishment. For instance, if Lucy writes 46 and Pete writes 100, Lucy will get $48 and Pete will get $44.

What numbers will Lucy and Pete write? What number would you write?”The point is, when you don’t think much about it, you’d choose $100. However, when you start thinking more, you’d write down $99: because if you write down $99, and the other person will write down $100, you will get $101, while the other only gets $99. Then you start wondering if the other person thinks the same… and you obviously don’t want the other person to have more money. So you reduce your amount to $98 (thinking the other person will write down $99), so you’ll get $100, and the other person only $96. And so on. In the end, you’ll arrive at $2. In this way, you earn a lot less then the naively chosen $100.

This problem is the same, the article explains, as the Prisoner’s Dilemma: “… in which two suspects who have been arrested for a serious crime are interrogated separately and each has the choice of incriminating the other (in return for leniency by the authorities) or maintaining silence (which will leave the police with inadequate evidence for a case, if the other prisoner also stays silent).”

A Nash equilibrium, is when there’s no benefit (put simply) to change your strategy: in this case the equilibrium is at $2 (the absolute minimum of choices). This is what Game theory predicts; but it conflicts with our intuition.

In experiments, the choice people make depends on the reward: when the reward is low, the choice will on average be higher. On the other hand, when the reward is high, the choice will be lower. This makes sense, because when the reward is high relative to the choice, it’s more advantageous to lower your choice. For example, if the reward is not $2, but $50, you wouldn’t want your choice to be too high: the penalty you’d get would severely impact your profits.

Then why do we make these choices, based on our expectations that the other person will choose a high number? The article suggests:
“Perhaps altruism is hardwired into our psyches alongside selfishness, and our behavior results from a tussle between the two. We know that the airline manager will pay out the largest amount of money if we both choose 100. Many of us do not feel like “letting down our fellow traveler to try to earn only an additional dollar, and so we choose 100 even though we fully understand that, rationally, 99 is a better choice for us as individuals.”

The moral: sometimes it’s good not to think too extensively about seemingly simple questions.

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