Scoring Rules as Least-Squares Estimators

Satoru Fujishige and Satoshi Nakada

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Abstract: Kawada (2018) proved that every scoring rule is equivalent to the corresponding cosine similarity rule. The original proof relies on a direct analysis of the cosine similarity optimization problem. In this note, we present an alternative, simpler proof based on a basic least-squares characterization. Our argument shows that the arithmetic mean of the score vectors is the unique minimizer of the total squared Euclidean distance and that the cosine similarity formulation is an immediate consequence of this optimization property. This result provides a transparent geometric interpretation of scoring rules and clarifies why the cosine similarity rule necessarily coincides with the corresponding scoring rule.

Date: 2026-07
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