What is the 'Traveler's Dilemma'
The traveler's dilemma, in game theory, is a nonzerosum game in which two players attempt to maximize their own payoff, without regard for the other. The game demonstrates the “paradox of rationality” — the irony that making decisions illogically or naively often produces a better payoff in game theory.
BREAKING DOWN 'Traveler's Dilemma'
The traveler’s dilemma game, formulated in 1994 by economist Kaushik Basu, presents a scenario in which an airline severely damages identical antiques purchased by two different travelers. The airline manager is willing to compensate them for the loss of the antiques, but as he has no idea about their value, he tells the two travelers to separately write down their estimate of the value as any number between $2 and $100 without conferring with one another.
However, there are a couple of caveats:
 If both travelers write down the same number, he will reimburse each of them that amount.
 If they write different numbers, the manager will assume that the lower price is the actual value and that the person with the higher number is cheating. While he will pay both of them the lower figure, the person with the lower number will get a $2 bonus for honesty, while the one who wrote the higher number will get a $2 penalty.
The rational choice, in terms of the Nash equilibrium, is $2. However, most people pick $100 or a number close to it, either without thinking the problem through or while fully aware they are deviating from the rational choice. So, while most people intuitively feel that they would select a much higher number than $2, this intuition seems to contradict the logical outcome predicted by game theory — that each traveler would select $2. By rejecting the logical choice and acting illogically by writing a higher number, people end up getting a substantially bigger payoff.
How can $2 Be the Rational Choice or the Nash equilibrium?
The reasoning goes as follows. Traveler A’s first impulse may be to write down $100; if Traveler B also writes down $100, that is the amount she will receive from the airline manager. But upon second thought, Traveler A reasons that if he writes $99, and B puts down $100, then A would receive $101 ($99 + $2 bonus). But A believes that this line of thinking will also occur to B, and if she also puts down $99, both would receive $99. So A would really be better off putting down $98, because then she would get $100 ($98 +$2 bonus) if B writes $99. But since this same thought of writing $98 could occur to B, A considers putting down $97 (because then she would get $99 if B writes $98), and so on. This line of backward induction will take the travelers all the way down to the smallest permissible number, which is $2.

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