Three-player Bradley-Terry comparison cycle
= Three-player Bradley-Terry comparison cycle
{c}
{title2=$kr^3+(k-1)r^2=1$}
For one win of player 1 over 2, one win of 3 over 1, and $k\geq1$ 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 $2>1>3$.