Choose among two actors randomly and place the chosen actor at the origin. Place the other actor in the unit circle uniformly at random. Both actors move at the same speed. Both actors are said to have "found" the other actor if they are at or within a distance $\delta$ of one another. Both actors know only their position relative to where they were initially placed. Both actors also know that the other actor must be within a distance $1$ of them. That is all either actor knows. Consider two scenarios: one, the actor placed uniformly at random in the unit circle is "searching" (adopting a strategy profile to minimize time spent before finding the other actor) for the other actor while the other stays still; two, both actors are searching for one another.
Neither actor in either scenario knows whether or not they were placed at the origin. A searching actor does not know whether the other actor is searching as well (that is, a searching actor does not know which scenario it is).
What is the best response strategy profile for the searching actor(s)?
Is the first or second scenario faster (on average), or are they both equal in terms of time spent searching?
I would assume that they don't know where the origin is otherwise it would be a simple strategy for both to move there. The origin is just convenient coordinates for us to track the problem and is the initial location of the first player.
If you move to new coordinates by subtracting the movements of the first player from both them and the second player you get back to one player stands still and the other moves, but now the second play moves between zero and twice the speed available. Therefore what ever strategy they are using it would work twice as fast. Therefore the answer is both should move.