Let $M$ be some metric space. We will also say that $M$ is a measurable space.
How do you find a probability measure $P$ on $M$ that maximizes the expected distance between two points? (That is, $P$ maximizes $E[d(X,Y)]$, where $X$ and $Y$ are independent random variables with distribution $P$.)
If $M$ is a discrete metric, then $P$ will the uniform distribution, for example.
Note that we can think of this as a game, and $P$ as a mixed strategy. You would have two players, which are both trying to maximize the distance. This is not a purely game-theoretical situation though, since we need to require the players to pick the same strategy. There is something called superrationality that might apply, but it is not well studied though. Maybe techniques similar to those used in game theory could be used though.