Antagonistic game (Zero-sum game)

Let there be two players. Each of them chooses any real number and writes it down on a piece of paper. Then players show those pieces of paper to each other. The player, who chose the number closer to the mean between two chosen numbers divided by 2, wins a dollar from the other player. If numbers are equally close, then it is a draw and nobody gets anything. So, if the first player chooses number $x$ and the second player chooses number $y$, then the first player wins if

$|\dfrac{x+y}{4} - x| < |\dfrac{x+y}{4} - y|$

and the second player wins if

$|\dfrac{x+y}{4} - y| < |\dfrac{x+y}{4} - x|.$

Otherwise, it's a draw.