# Traveller’s Dilemma

Some game theory paradoxes can only be resolved by assuming that people are not rational, and the Traveller’s Dilemma is one of them. This game shows clearly that sometimes it is smart to be dumb, or in other words, it is rational not to be rational…

Suppose an airline loses two bags belonging to two different travellers, Lucy and Pete. Both bags are identical and contain the same items. An airline manager, responsible for solving this problem, explains that the airline is liable for a maximum of 100€ per baggage. Simply asking the travellers for the real price is worthless because they will inflate it in order to get a higher reward.

The manager is clueless about the price of the items, and in order to determine a fair estimate the manager asks each of them to write down the price of the items between €2 and €100 without conferring together. If both write down 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 smaller number is the true value of the items and that the person writing the higher number is lying. In this case, both travellers will receive the smaller amount along with a bonus and a penalty. €2 extra will be paid to the traveller who wrote down the lower amount and a €2 deduction will be taken from the person who wrote down the higher amount as a punishment.

The challenge is: what strategy would you follow if you were one of the travellers? What number between 2 and 100 would you write down?

The Nash Equilibrium of this game is at (2, 2). This can be obtained through the following logic. If one player, for example Lucy, chooses €100 (the maximum) to be the value of the bag, Pete may also choose €100, and then they would get €100 each. But if Lucy were to choose €99 (while Pete still chooses €100) she would now have the lowest value and therefore would receive €101, a higher payoff, and Pete would receive €97. Pete could improve this payoff by saying €98, hence receiving €100. Continuing with this reasoning, the value of the travellers’ bags will spiral down until it gets to the smallest possible outcome (2, 2).

Through the use of backward induction, the equilibrium (2, 2) is also the result. Assuming Pete chooses 2 and Lucy chooses 3, Pete receives €4 and Lucy receives nothing at all. So she is better off by choosing 2 as well, which yields her a payoff of €2. If Pete chooses 2 or 3, Lucy gets the best result from choosing 2, and vice-versa. Note that when restricted to choices of only 2 and 3, the game becomes equivalent to Prisoner’s Dilemma. On the other hand, if Pete chooses any number between 4 and 100, Lucy gets a better result from choosing a number higher than 2. In this game, (2, 2) is also a strict equilibrium and a perfect equilibrium.

The game’s logic dictates that 2 is the best option, yet most people pick 100 or a number close to 100 – both those who have not thought through the logic and those who fully understand that they are deviating from the “rational” choice. Most of us feel that we would play a much larger number than 2 and would, on average, make much more than €2. In fact, by acting illogically, people end up receiving a larger reward. Thus, there is something rational about choosing not to be rational when playing Traveller’s Dilemma. Our intuition seems to contradict all of game theory.

The game and our intuitive prediction of its outcome also contradict economists’ assumption that standard game theory can predict how selfish rational people will behave. They also show how selfishness is not always good economics.

Afonso Queiroz Aguiar 722

### Author: studentnovasbe

Master student in Nova Sbe