I like this approach, but I am having trouble following it to the right answer. The floating passenger will pick from the free seats with equal probability whenever he is evicted from someone else’s seat, but if, at any such event, he picks his own seat, then the evictions stop and you are guaranteed your seat. In the case where the person before you evicted the floating passenger, there is an equal probability of him taking your seat or his (the only two remaining), but what about the possibility that he ended up in his own seat earlier in the process?
Update: I have the answer to my own question: The passenger before you will also find his own seat unoccupied if, at any point, the floating passenger picked your seat. Therefore, we have three cases: 1) At some point, the floating passenger picked his own seat; 2) At some point, the floating passenger picked your seat; 3) the floating passenger ends up in the seat of the passenger before you. The corresponding probabilities of you getting your seat are 1, 0, and 0.5 respectively. Cases 1 and 2 are equally likely, so we can see, without knowing what that probability actually is, that the overall probability of getting your seat is 0.5.
So now I can see that the same argument, applied to your own seat, simply has cases 1 and 2 — neat!