A score curve evaluated at the mean error misstates the mean score

Fact

For a nonlinear scoring function f(error), the score at the average error is not the average score: f(E[error]) ≠ E[f(error)]. With a decaying curve such as 1000 · exp(−d² / 2σ²), the near misses in the error distribution dominate the mean. Scoring uniformly random answers in RGB space at their mean distance (≈169) gives ≈20 points, but random answers actually average ≈110 — over five times more.

The degenerate strategy does even better: always answering the midpoint minimises expected distance and averages ≈180 under the same curve.

Why it matters

Balance claims written from the mean error ("guessing earns almost nothing") end up in rules text, tutorials and test comments, and are wrong by a large factor. The floor that a lazy or random player gets is set by the whole distribution, not by its centre, so a curve tuned this way leaks more points to non-skill play than intended.

How to apply

• Calibrate by Monte Carlo over the real error distribution: sample answers, score each, average the scores. A few hundred thousand samples take milliseconds.
• Always measure the baselines, not just random: a constant answer (midpoint, most common value) is often the strongest no-skill strategy.
• When documenting a curve, quote scores at named errors ("off by 20 on every channel ≈ 850") and measured averages for strategies — never a score at the mean error presented as a typical result.