Mixed equilibrium in a distance game on [0,1]

Players 1 and 2 each secretly choose a real number in $[0,1]$ at the same time. Player 1 receives, and Player 2 pays, the absolute difference between the two numbers. This zero-sum game has an equilibrium in mixed strategies. Under optimal play, what is Player 1's expected payoff?

$1/2$. Player 2 can guarantee losing at most $\tfrac12$ by always choosing $\tfrac12$. Player 1 can guarantee $\tfrac12$ by choosing $0$ or $1$ with probability $\tfrac12$ each, since against any $y$ the expected distance is $\tfrac{y+(1-y)}{2}=\tfrac12$. So the game's value is $\tfrac12$.