Mathematics Puzzle: Round-trip On A Treadmill

Alice and Basil are bored while waiting in an airport. They decide to play a game with one of the nearby “moving sidewalks” (treadmills).

They decide that Alice will walk on the treadmill at constant speed $v$ (with respect to the treadmill) for a distance $y$, and then turn around and walk back to the beginning at the same constant speed (with respect to the treadmill). Meanwhile, Basil will walk on the floor next to the treadmill at a constant speed $v$ (with respect to the floor) for a distance $y$, and then turn around and come back to the starting point at the same constant speed (with respect to the floor).

The treadmill is much longer than the distance $y$. And the treadmill moves at a constant speed $x$.

Who wins this race?

(My apologies for not giving the source of the problem yet; I shall do so once I post some solutions in a day or two. Update: I have posted a solution and discussion here.)

(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 21 January 2021.)