Graphs of Piecewise Functions

If $0 \le p \le 1$ and $0 \le q \le 1$, define $F(p, q)$ by $$F(p, q) = -2pq + 3p(1-q) + 3(1-p)q - 4(1-p)(1-q).$$ Define $G(p)$ to be the maximum of $F(p, q)$ over all $q$ (in the interval $0 \le q \le 1$). What is the value of $p$ (in the interval $0 \le p \le 1$) that minimizes $G(p)$?

(Hint: For any fixed value of $p$, the function $F(p,q)$ is linear in $q$, so it is maximized at one of the two endpoints, $0$ or $1$.)

Source: Math Prize for Girls