Bradley-Terry model ($\mathbb{P}(i\text{ beats }j)=\frac{{\theta }_i}{{\theta }_i+{\theta }_j}$)

The Bradley-Terry model assigns positive abilities ${\theta }_i$ and the displayed pairwise winning probabilities. Multiplying all abilities by the same positive scalar leaves the probabilities unchanged, so identifiability requires a normalization. In log abilities ${\beta }_i=\log {\theta }_i$, the model has comparison probabilities obtained from the logistic function. Either parametrization induces the same ordering of players.

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.

For a connected tree of comparisons with every observed win fraction strictly between zero and one, the Bradley-Terry model can fit every empirical edge probability exactly. Fix one positive strength to remove scaling ambiguity, then propagate strength ratios $p/(1-p)$ along tree edges. Edge log-ratios are independent real coordinates and each binomial log-likelihood term is strictly concave, proving existence and uniqueness of the normalized estimate. Cycles would impose extra compatibility relations.

For observed neighbor win fractions $p_0,\dots ,p_{n-1}\in (0,1)$ on the path graph $0,1,\dots ,n$, the normalized Bradley-Terry model estimate with ${\hat{\theta }}_n=1$ is ${\hat{\theta }}_i={\prod }_{k=i}^{n-1}{p}_k/(1-p_k)$. This is backward propagation of empirical odds along the path. Every edge proportion is fitted exactly, and strict concavity in edge log-ratios proves that this is the unique global maximum.

Orient the edges of a comparison tree and set ${\eta }_{ij}=\log {\theta }_i-\log {\theta }_j$. After anchoring one vertex strength, these edge quantities are unconstrained coordinates. The Bradley-Terry model probability is the logistic function of ${\eta }_{ij}$, and its binomial log-likelihood contribution is $m[p{\eta }_{ij}-\log (1+e^{{\eta }_{ij}})]$. Differentiating gives the fitted odds $p/(1-p)$.

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 $2>1>3$.

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.