Bradley-Terry score equation
= Bradley-Terry score equation
{c}
{title2=$W_i=\sum_{j\ne i}n_{ij}\dfrac{\theta_i}{\theta_i+\theta_j}$}
Let $W_i$ count observed wins and $n_{ij}$ count comparisons of players $i,j$. Differentiating the <log-likelihood> with respect to $\log\theta_i$ gives observed wins minus model-expected wins. Vanishing derivatives therefore give the displayed equations. One equation is redundant because abilities are identifiable only up to common scaling.