Traveler’s Dilemma
The cover story is that: Two travelers travel by airplane. They have identical luggage, a valuable antique vessel. Unfortunately, the airline somehow lost both pieces of luggage.
Now, the travelers claim compensation. The problem is that the value of the antique vessel is impossible to determine for the airline. They have to rely on the statements of the passengers. But the maximal possible value is 100 EUR.
So, they ask both travelers to write down the value. It should be somewhere between 2 EUR and 100 EUR. The passenger that noted the lower value receives this value plus 2 EUR. The traveler who wrote down the higher value has to pay a penalty of 2 EUR for attempting to cheat. If both travelers note the same value, they both receive the value they wrote down.
What value would be rational to write down?
We can try to solve this with backward induction. For that reason, we make a table with all possible values and pay-offs.
| Stated values of A / B | 100 | 99 | 98 | 97 | … | 3 | 2 |
| 100 | 100;100 | 97;101 | 96;100 | 95;99 | … | 1;5 | 0;4 |
| 99 | 101;97 | 99;99 | 96;100 | 95;99 | … | 1;5 | 0;4 |
| 98 | 100;96 | 100;96 | 98;98 | 95;99 | … | 1;5 | 0;4 |
| 97 | 99;95 | 99;95 | 99;95 | 97;97 | … | 1;5 | 0;4 |
| … | … | … | … | … | … | … | … |
| 3 | 5;1 | 5;1 | 5;1 | 5;1 | … | 3;3 | 0;4 |
| 2 | 4;0 | 4;0 | 4;0 | 4;0 | … | 4;0 | 2;2 |
If we start in the upper left corner {100; 100}, both travelers will receive a compensation of 100. But for traveler A (and also for traveler B), it would be beneficial to state a value of 99. He would receive 101 EUR, and B would receive only 97 EUR (the lower offer mins the penalty).
In such a situation, B could decide to write down 99 EUR. This would lead to a pay-off of 99 EUR each. But again, each traveler has an incentive to state a lower price than 99 to reap the 2 EUR reward.
The whole process stops at {2;2}.
But in experiments, people usually don’t state just 2 EUR. They provide higher values. The reason could be that they didn’t understand the game, or they understood it and thought that losing 2 EUR was not that much. Therefore, they try higher values, hoping that the other one thinks similarly. This is an example where backward induction reaches its limit.
