Source Clarifications: Axioms for Learning from Pairwise Comparisons

Lemma 2.3 distinct alternatives

• Positive directed counts for every pair → every distinct pair. Diagonal counts are zero in the paper’s pairwise-comparison model, so the literal all-pairs premise cannot hold.

Counterexample proofs for Theorems 5.3 and 6.3

• Continuity of the maximizer under small perturbations → a compact likelihood-gap argument. A uniform score bound restricts the search to a compact set; a strict base-likelihood gap separates the bad-order set. An explicit positive rational perturbation is chosen whose bounded likelihood contribution cannot close that gap. The construction retains a strict score reversal under the source assumptions.