"Assassin" - The Escape Game

Imagine Agent Don Eppes is chasing a fugitive on separate subway trains traveling in the same direction. There are only two stops remaining, A and B. If Don and the fugitive exit at the same stop, Don will capture the fugitive. If they exit at different stops, the fugitive escapes.

We can model this using a payoff matrix, where the rows represent Don's choices and the columns represent the fugitive's choices:

$\begin{array}{c|cc} & \text{Fugitive Stop A} & \text{Fugitive Stop B} \\\hline \text{Don Stop A} & \text{Caught} & \text{Free} \\ \text{Don Stop B} & \text{Free} & \text{Caught} \end{array}$

If both players act purely at random, we want to determine how often the fugitive escapes.