Which proper score should verifiers face with K outcomes?

What it found

For K-ary reports the Brier, spherical and log scores have exact regrets and exact worst cases (1−2p_min+‖p‖² for Brier, ‖p‖−p_min for spherical). Per unit of payment range the spherical score gives exactly √K times Brier’s incentive at a uniform truth, but both bounded scores nearly ignore a dropped rare class: at a 1% class the loss is 1.3×10⁻⁴ (Brier) and 1.0×10⁻⁴ (spherical) against 0.15 for log. Multiclass follow-up to tangent-log.

A figure from its own output

Paper, code and results

• README and method
• White paper
• Experiments and results
• Code
• Tests

More in Proper scoring & elicitation

• Can the multiclass log score be made payable?Bounded multiclass log score by quadratic tangent extension of x ln x: log-identical when all r_j≥ε, payment range exactly 1−ln ε for any K, 63× Brier's rare-class incentive per range at ρ=10⁻³ with best cutoff ε=p_min, and a closed-form bounded cost for dropping a rare class.
• What does it cost to pay for a ranking instead of probabilities?Eliciting class rankings with linear rank-weight scores: strictly proper, adjacent-swap regret exactly (p_i−p_j)(w_r−w_{r+1}), beats Brier-derived rankings per unit range below gap 1/(K−1) (250× at gap 10⁻³), equal spacing is optimal, top-k set costs the same as the mode, detection sample size unchanged.
• Should severity reports be scored with the order in mind?Ranked probability score for ordered severities: exact regret Σ(R_k−P_k)², shift over distance d costs dε² (Brier 2ε²), pooling adjacent classes costs half of Brier's, threshold errors ≤ √regret, and per range RPS never beats Brier's incentive.
• Can the tail of benign drift be elicited with a proper score?Jointly eliciting VaR and expected shortfall of benign drift: 0-homogeneous Fissler–Ziegel score minimised at (VaR, ES) with profile ln ES, ES alone non-elicitable (mixture 12.5 vs 10), exact regret formulas, ES-understatement of 30% detectable with ~1k samples at τ=0.95 while pinball is blind.
• Can we elicit the mode of a mixture of honest and cheating drift?Eliciting the modal drift with the window loss 1[|y−r|>α]: exact window-mass regret, best width α=σ (normal) / γ/√3 (Cauchy), closed-form critical width α* above which bimodal drift has one α-mode, and 0 bias against contamination where the mean moves 5ε.
• Can a verifier be paid for a tolerance interval, not a point?Companion to interval-elicitation: Winkler interval score for drift-tolerance intervals: strictly proper for the α/2 and 1−α/2 quantiles, closed-form regret, optimal expected score 4sφ(z)/α, width-misreport costs whose asymmetry flips with the level, and Cauchy drift where scores have no mean but paired differences are bounded.

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Other research topics

Proper scoring & elicitation28Effort & peer prediction9Verification games & slashing19Detection & tolerances14Markets as routers13Prediction-market design9Paying contributors8Decentralised training systems23Digital twins & sim-to-real164LLM chip design loops40Robot manufacturing loops53Ambient AI assistant46Self-replicating manufacturing34Simulation-first design64Engineered clothing45Task-specific AI accelerators40Algorithmic chip-design loops42Open-source purpose-built robots44The software-engineering loop43Adversarial human imitation41Open-source residential construction41Autonomous multi-angle capture34Sensing, imaging and 3D31Energy supply and dense energy storage39Robot societies among people31Agentic co-design of low-cost robots27Laundry-hanging robots25Machine learning on qualitative data23Robots on the building site27Civilizations & group psychology1History & human sciences1

Produced with AI assistance (Claude). Results are from the code in this repository and are simulated or stylised unless the abstract says otherwise; nothing here implies affiliation with or endorsement by any lab, researcher or company cited. MIT licensed.