# Traveler's dilemma

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Traveler's dilemma

In game theory, the traveler's dilemma (sometimes abbreviated TD) is a type of non-zero-sum game in which two players attempt to maximise their own payoff, without any concern for the other player's payoff.

The game was formulated in 1994 by Kaushik Basu and goes as follows: [Kaushik Basu, "The Traveler's Dilemma: Paradoxes of Rationality in Game Theory"; "American Economic Review", Vol. 84, No. 2, pages 391-395; May 1994.] [Kaushik Basu, [http://www.sciam.com/article.cfm?chanID=sa006&colID=1&articleID=7750A576-E7F2-99DF-3824E0B1C2540D47 "The Traveler's Dilemma"] ; "Scientific American Magazine", June 2007] :

"An airline loses two suitcases belonging to two different travelers. Both suitcases happen to be identical and contain identical antiques. An airline manager tasked to settle the claims of both travelers explains that the airline is liable for a maximum of \$100 per suitcase, and in order to determine an honest appraised value of the antiques the manager separates both travelers so they can't confer, and asks them to write down the amount of their value at no less than \$2 and no larger than \$100. He also tells them that if both write down the same number, he will treat that number as the true dollar value of both suitcases and reimburse both travelers that amount. However, if one writes down a smaller number than the other, this smaller number will be taken as the true dollar value, and both travelers will receive that amount along with a bonus/malus: \$2 extra will be paid to the traveler who wrote down the lower value and a \$2 deduction will be taken from the person who wrote down the higher amount. The challenge is: what strategy should both travelers follow to decide the value they should write down?"

One variation of the original traveler's dilemma in which both travelers are offered only two integer choices, \$2 or \$3, is identical mathematically to the Prisoner's dilemma (often abbreviated "PD") and thus TD can be viewed as an extension of PD. The traveler's dilemma is also related to the game Guess 2/3 of the average in that both involve deep iterative deletion of dominated strategies in order to demonstrate the Nash equilibrium, and that both lead to experimental results that deviate markedly from the game-theoretical predictions.

For the traveler's dilemma, game theory predicts that both will write down the value '\$2' if their strategies were purely rational. The \$2 in this instance is the Nash equilibrium point for the game. However, when the game is played experimentally, most participants select the value '\$100' or a value close to '\$100', including those who have not thought through the logic of the decision as well as those who understand themselves to be making a non-rational choice. Furthermore, the travelers are rewarded by deviating strongly from the Nash equilibrium in the game and obtain much higher rewards than would be realized with the purely rational strategy. These experiments fail to show either that most people use purely rational strategies nor that they would be better off financially if they were to do so. The paradox has led some to question the value of game theory in general, whilst others have suggested that a new kind of reasoning is required to understand how it can be quite rational ultimately to make non-rational choices.

Payoff matrix

The canonical payoff matrix is shown below (if only integer inputs are taken into account):

References

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