HMMT 2015 U4

Consider the function $z(x, y)$ describing the paraboloid

$$z=(2 x-y)^{2}-2 y^{2}-3 y .$$

Archimedes and Brahmagupta are playing a game. Archimedes first chooses $x$. Afterwards, Brahmagupta chooses $y$. Archimedes wishes to minimize $z$ while Brahmagupta wishes to maximize $z$. Assuming that Brahmagupta will play optimally, what value of $x$ should Archimedes choose?