= Bradley-Terry likelihood Hessian
{c}
{title2=$-z^{\mathsf T}\nabla^2\ell(\beta)z=\sum_{i<j}n_{ij}p_{ij}(1-p_{ij})(z_i-z_j)^2$}
In log abilities, the negative <Hessian matrix> is a weighted <Graph Laplacian> of the comparison graph. If that comparison graph is a <connected graph> and the abilities are finite, the quadratic form vanishes only for constant vectors. Thus the <log-likelihood> is strictly concave after fixing the common additive constant. If the directed graph of observed wins is a <strongly connected directed graph>, letting contrasts diverge forces at least one observed-win probability to zero, so the log-likelihood tends to negative infinity. A finite maximizer exists and is unique up to common scaling of abilities.
Back to article page
