Three-player Bradley-Terry comparison cycle ($k{r}^3+(k-1)r^2=1$)

For one win of player 1 over 2, one win of 3 over 1, and $k\ge 1$ wins of 2 over 3, the Bradley-Terry score equations force abilities proportional to $(r,1,r^2)$, where $r>0$ satisfies the displayed equation. Its left side is strictly increasing on $r>0$. For $k=1$, $r=1$ and all three estimates tie. For $k>1$, $r<1$, giving the ranking $2>1>3$.