Past exam of the Probability, statistics and options II course of the University of Oxford / 2007 / S / 3 / d / Solution

Use four vertices, with lengths $a_{12}=a_{24}=0$, $a_{23}=a_{32}=-1$, and every other non-loop arc of length $100$. The two arcs between $2$ and $3$ form a negative-length cycle in a graph. For $i=1$, $j=4$, $k=3$,The recurrence would give $-2$, by concatenating $1\to 2\to 3$ and $3\to 2\to 4$. That walk repeats $2$, so it is not a simple graph path. Every simple graph path from $1$ to $4$ still has length at least zero, attained by $1\to 2\to 4$. Thusa strict inequality in part (a).