Can the multiclass log score be made payable?

What it found

Extending x ln x below a cutoff ε by its second-order Taylor polynomial gives a strictly proper K-class score that equals −ln r_y whenever every r_j≥ε and has payment range exactly 1−ln ε, the same for any K. Per unit of range it has 3× Brier’s rare-class incentive at a 5% class and 63× at 0.1%, the best cutoff is the rarest class’s mass, and a verifier that zeroes out a 1% class loses a closed-form 0.005 (Brier 1.3×10⁻⁴, log unbounded). Closes the gap left open in categorical-scores.

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

• 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.
• How should a tolerance interval be paid for?Paying for a drift tolerance interval: Winkler interval score is proper for the equal-tailed pair with exact regret (finite for Cauchy) but unbounded payment; the bounded width-plus-miss loss elicits the shortest interval (22–31% narrower on lognormal drift) at a 3–48× larger detection sample for α≥0.1.

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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.