Freeflows

Strategic Game Optimization

Consider the function $ z(x,y) $ describing the paraboloid \[z = (2x - y)^2 - 2y^2 - 3y.\]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?

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$\frac{a}{b}$
.
0
=
%

$a^n$
$a^{\circ}$
$a_n$
$\sqrt{}$
$\sqrt[n]{}$
$\pi$
$\ln{}$
$\log$
$\theta$
$\sin{}$
$\cos{}$
$\tan{}$

$($
$)$
$[$
$]$
$\cap$
$\cup$
$,$
$\infty$

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