= Bradley-Terry maximum-likelihood estimate on a comparison tree
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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.
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