Traveler's Problem

Problem and Action Space

The traveler’s dilemma is a game where an airline loses two identical suitcases from two travelers. The airline asks the travelers to write down the value of their suitcases, which can be between $\$ 2$ and $\$ 100$, inclusive.

Reward Function

If both put down the same value, then they both get that value. The traveler with the lower value gets their value plus $\$ 2$. The traveler with the higher value gets the lower value minus $\$ 2$. In other words, the reward function is as follows:

\[\begin{aligned} R_i(a_i, a_{-i}) = \begin{cases} a_i & \text{if } a_i = a_{-i} \\ a_i + 2 & \text{if } a_i < a_{-i} \\ a_{-i} - 2 & \text{otherwise} \end{cases} \end{aligned}\]

Optimal Policy

Most people tend to put down between $\$ 97$ and $\$ 100$. However, somewhat counter-intuitively, there is a unique Nash equilibrium of only $\$ 2$.