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 obvious answer is $100, which if chosen by both people will net them each a crisp new Benjamin. But, if 1 of the persons choose $99 instead, and the other chooses $100, then the first person will instead get $101, and the other will get $97. And if the other person realizes this and then decides to choose $98, and the other person stays with $99, then the first person will get $100, and the other will get $96, and so on. In the article, which can be found here: The Traveler's Dilemma by Kaushik Basu, this is taken to its logical (illogical?) conclusion, all the way down to $2. To outthink oneself so horribly to come up with an answer that will net you at most $4 ($96 less than if you both chose $100), and then to justify it as the "logical" answer is ludicrous. It takes brains to be this stupid.
In the article Basu details how that when people choose a number other than $2 it goes against game theory. How in the world does trying to maximize the amount of money one gets, and assuming the other person will do the exact same, go against game theory? Choosing anything other than $100 will get you at most $101 (if you choose $99 and the other person chooses $100, which is a full $1 more, w00t!!!), and could get both of you much, much less. If anyone was dumb enough to follow the regress all the way down to $2, then hopefully they'd realize, "Hey, if I put $2 down, the other person is going to punch me in the face!" So they'd have to go up to $3, and realize, "Yep, I'd get punched for $3, too," and so on, all the way back up to $100. Unfortunately, the article doesn't cover this "Avoid Being Punched In The Face" (ABPITF) aspect of the Traveler's Dilemma. Hopefully someone will point this out in a nasty letter to the editor.
I could maybe see choosing $99, to try and get the extra dollar, or maybe even $98, which would at most get $100 as long as the opponent chose $99 or $100 (although I'd probably feel so bad about it afterwards I'd give them a few dollars so we'd both walk away with the same amount). But one would have to be a fool to go to $97, since at most choosing that would get you $99, which is less than if both people just chose $100. In a multiple trial scenario I think I'd go with $100 every time, and if my opponent wants to choose $99 to instead make $101, leaving me with $97, fine. I'd much rather get $97 each trial than get into some mutually detrimental bidding war where the only person that wins is the sucker who's paying out all the money. And if someone thought it through enough to choose the idiotic $2, then of course the game would have to pause for a few moments while they learned that if for no other reason, a person doesn't put down $2 or any other small amount to avoid being punched in the face.
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