= Solution
A feasible point lies in the <probability simplex>, so $r^Tx=\sum_i x_ir_i$ is a <convex combination> of the coordinates of $r$. Hence $r^Tx\le r_{\max}:=\max_i r_i$. Taking a coordinate vector $e_j$ with $r_j=r_{\max}$ attains equality. Therefore
$$
\boxed{v^*=\max_i r_i,\qquad x^*=e_j\ \text{for any }j\in\operatorname{argmax}_i r_i.}
$$
More generally, all optimal solutions are exactly the probability vectors supported on the maximizing indices: equality requires $x_i(r_{\max}-r_i)=0$ for every $i$. Thus ties give an entire optimal face, while a unique maximizing index gives a unique optimizer.
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