The Travellers' Dilemma

An airline loses two suitcases, one belonging to Alice, the other to Bob. A manager explains that the airline is liable for a maximum of $100 per suitcase. She separates the travellers so they can't confer, and asks them to write down a value no less than $2 and no larger than $100. She tells them that if both write down the same number, she will treat that number as the value of both suitcases and reimburse both travellers that amount. However, if one writes down a smaller number than the other, she will take this smaller number as its value, and both travellers will receive that amount. Also, $2 extra will be paid to the traveller who wrote down the lower value and $2 will be deducted from the person who wrote down the higher amount. Alice and Bob each believe that the other will make a rational choice.

What value should each traveller write down?


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