Jason_t's flight home...

[CallMeIshmael]

As some are aware, today was Jason's flight home from Denver. He recounted a rather remarkable story to me over aim, and I thought I'd relay it to OOT.

While in Denver, Jason found himself in an antique shop. Having never done much antiquing before, Jason was somewhat wary of being sold an overpriced good. His girlfriend assured him that this would not happen. As it turns out, Jason spent $90 on a vase, which he was told was the 'only one of its kind.' Jason proceeded to tuck the vase away for safe keeping (the original purchase was over a week ago) until his trip back to LA.

Despite the most careful of planning, when Jason arrived back in LA, he found the vase was smashed on the airplane. But, this is merely where the story gets interesting.

Jason then opted to take the now smashed vase to an airport authority. He was instructed to enter a room at the other end of the terminal. When he entered, he found an irate older gentleman holding a broken vase. The EXACT SAME vase that Jason now had in his left hand. The man in charge addressed Jason, then looked at the vase and noticed that the older gentleman's claim that his was unique was patently false. He assured both that they would be reimbursed for the loss, and asked if either could produce a proof of purchase for the vase, and that if either could, they would BOTH be paid that much, as it seemed reasonable to assume they each paid the same amount for the vase (it was revealed that both were on a plane from Denver, and presumably purchased the vase at the same store). Neither could produce a receipt.

The manager then proceeded to explain his rather unique solution to this problem.

He would ask both Jason and the older gentleman to walk to separate sides of the room. Each would write a number on a piece of paper representing how much they claim to have paid for it. He also said he does not feel the vase is worth more than $100 but feels it is certainly worth at least $2. As such, they can only write numbers from 2-100.


Here is how they would be reimbursed:

- If they wrote the same number, the manager would assume that that was the honest price, and pay each of them that much money.

- If they wrote different numbers, the manager would assume that the person who wrote the lower number was being honest, and pay them both that much. BUT,
to reward him for his honesty and punish the other for trying to cheat him, he would take $2 from the cheater and give it to the honest player.


So, for example, if one person wrote 80, and the other 70. The player who wrote 70 would get $72 (70 + 2) and the other would get $68 (70 - 2).



But, right before the game could be played, the older gentleman turned around, and Jason recognized him immediately. It was none other than John Nash, perhaps the most influential game theorist of all time. Interestingly enough, Jason was sporting his "I [img]/images/graemlins/heart.gif[/img] Game Theory" t shirt at the time. Nash's eye caught the shirt, then he and Jason made eye contact. Much happened in this instant, however. Nash must assume that anyone wearing that shirt will play rationally. Nash must also assume that anyone wearing that shirt knows who John Nash is, and will thus assume that Jason knows he is playing a rational player. Jason will of course know who Nash is, and assume he is playing rationally. On top of that, Jason can assume Nash knows that Jason knows who Nash is, and that he is playing a rational opponent. This line continues, and complete rationality is assumed.



What number did Jason write? What about John?


EDIT: just to be absolutely clear, it is possible to achieve a payment of $101.