= Solution
The bold black curve is the <pointwise minimum> of the two opposing-column <expected payoff> functions. It is the row player's <security level payoff> at the specified <mixed strategy>:
$$
g(p)=\min(3p-1,1-4p)=
\begin{cases}3p-1,&0\le p\le2/7,\\1-4p,&2/7\le p\le1.\end{cases}
$$
The lines meet where $3p-1=1-4p$, hence at $p=2/7$. The first branch increases with slope $3$ and the second decreases with slope $-4$, so their intersection is the unique global <maximum> of the lower envelope:
$$
\boxed{\operatorname*{arg\,max}_{0\le p\le1}g(p)=\frac27,\qquad \max g=-\frac17.}
$$
The endpoint values are $g(0)=-1$ and $g(1)=-3$, both smaller. Since an opposing <mixed strategy> forms a <convex combination> of the two pure-column <expected payoffs>, it cannot make the row player's <expected payoff> lower than this envelope.
\Image[/past-exam-of-the-msc-in-advanced-computer-science-course-of-the-university-of-oxford/2021/computational-game-theory-row-security.png]
{title=The row player's lower payoff envelope and its maximum at p = 2/7}
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