**The Traveller's Dilemma**Lucy and Pete separately went on the a vacation trip to a remote Pacific island. They both bought an identical exotic piece of handicraft as souvenir. When they returned, they found that their souvenirs were damaged in transit. They both filed a claim with the airline. The airline manager accepted the responsibility and was willing to compensate. However, there was no way to assess the value of the strange souvenirs as both Lucy and Pete did not have any receipt.

The Airline manager came up with a proposal. He was willing to pay an assessed value between $2 and $100. Lucy and Pete were asked to put down the value of the souvenir without conferring to each other. If they both claimed the same amount between $2 and $100, the airline would pay them both the amount. If their claimed amounts were different, the airline would only pay the lower amount. Furthermore, to reward honesty and to penalize cheating, the one who claimed the lower amount would get $1 more while the other would get $1 less. If you were either Lucy or Pete, what amount would you claim?

The scientific approach to this question is the use of the game theory. Based on the logical regression that the person who claimed a smaller amount would get a better reward than the other, game theory insists that rationality should lead the players to select $2. In the end, the airline manager would only pay $2 to Lucy and Pete, which would be the best choice for both of them.

When studying the payoff matrix, a table showing the payoff of each possible choice, game theorists rely on the Nash Equilibrium, named after John Nash (His story was made into the movie A Beautiful Mind). A Nash Equilibrium is an outcome from which no player can do better by deviating unilaterally. Other equilibrium concepts adopted by game theorists also come up with the same result with both players claiming $2.

However, in reality when people play this game, many of them choose an amount near $100. Many universities conducted experiments and surveys on this game. A common result is that a relatively large proportion of players chose a higher amount instead of the $2 equilibrium state. Some academics called this Sensible Irrationality. A new kind of reasoning is needed to gain a rigorous understanding of this rational choice not to be rational. The results of Traveler's Dilemma contradict economists' assumption that standard game theory can predict how supposedly selfish rational people will behave. They also show how selfishness is not always good economics.

This story illustrates an important distinction between ordinary decision theory and game theory. In the latter, what is rational for one player may depend on what is rational for the other player. For Lucy to get her decision right, she must put herself in Pete's shoes and think about what he must be thinking. But he will be thinking about what she is thinking, leading to an infinite regression. Game theorists describe this situation by saying that "rationality is common knowledge among the players." In other words, Lucy and Pete are rational, they each know that the other is rational, they each know that the other knows, and so on. The assumption that rationality is common knowledge is the source of the conflict between logic and intuition and that, in the case of Traveler's Dilemma, the intuition is right and awaiting validation by a better logic.

Researchers made some attempts to explain why a lot of people do not choose the Nash equilibrium. Some argued that many people are unable to do the necessary deductive reasoning and therefore made irrational choices unwittingly. This is not entirely satisfactory as the result is still the same with some games played by theorists. Some proposed that perhaps altruism is hardwired into our psyches alongside with selfishness. Many of us may not feel like letting down the other party just by trying to earn an additional dollar. In any case, it seems likely that altruism, socialization and faulty reasoning all play a part in guiding individuals' choices. Some people playing the game may just ignore the game-theoretic logic and select a large amount, assuming their opponents will play something similar. The interesting point is that this rejection of formal rationality and logic has a kind of meta-rationality attached to it. The idea of behaviour generated by rationally rejecting rational behaviour is difficult to formalize. It will be a subject for further researches.