The only aggregation rule satisfying same-scale normalization, recursive consistency, and marginal Elo-strength consistency converts ratings to strengths, takes their weighted arithmetic mean, and converts back.
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Aggregating Elo Ratings: An Axiomatization
The only aggregation rule satisfying same-scale normalization, recursive consistency, and marginal Elo-strength consistency converts ratings to strengths, takes their weighted arithmetic mean, and converts back.