Should you pay verifiers for their calibration?

What it found

A verifier reporting expit(aL+b) for the true posterior logit L is overpaid-for-error by exactly E[(r−p)²] under Brier and E[KL(p‖r)] under log score. Brier is bounded (a hard 0/1 report still beats the prior at μ=1) but log is not: raw log score ranks a higher-resolution verifier first only for slope a in (0.264, 2.49), so at a=4 a weaker calibrated verifier wins, while recalibrated scores never reverse. Recalibrating on n outcomes costs about tr(HI⁻¹)/2n excess Brier (0.264/n, simulation within ~10% for n≥100), beating raw pay beyond n≈20–80. Quadrature and seeded simulation. Stylised.

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

• Should a verifier's continuous drift report be scored by CRPS or log score?CRPS vs log score for continuous drift reports: exact Gaussian closed forms, overconfidence capped at 0.234σ under CRPS (unbounded under log), 1-Lipschitz outcome sensitivity, plug-in cost 5σ/(8√π n), and finite score variance for tails ν>2 vs ν>4.
• Can a verifier be paid for reporting drift variance?Eliciting drift variance from two independent re-executions: D=(y1−y2)²/2 unbiased with Var=σ⁴(κ+1)/2, exact Brier and scale-free misreport costs (1/λ+ln λ−1), detection needs ≈2z²(κ+1)/(λ−1)² tasks, disjoint pairs 1.8× worse than the sample variance, heavy tails break m^{−1/2} concentration.
• What can be elicited from several independent draws of benign drift?Eliciting drift *variance* from pairs of independent draws: exact pair score (excess loss (r−σ²)²), degree table (mean² 2, third moment 3, σ⁴ 4), audit cost ((z_α+z_β)/ρ)²(κ+1)/2 pairs, robust Gini score beats it for heavy tails, all-pairs pooling saves (κ+1)/(κ−1), correlated draws bias the target to σ²(1−ρ).
• Can the log score be made payable?Bounded-payment log score by quadratic tangent extension: strictly proper, log-identical on [ε,1−ε], worst-case payment exactly ln((1−ε)/ε)+1/(1−ε), 63× Brier's curvature per range at p=10⁻³, crossover p(1−p)=1/(2R), best cutoff ε=p.
• Which proper score should verifiers face with K outcomes?K-ary proper scores: exact worst-case regrets (1−2p_min+‖p‖² Brier, ‖p‖−p_min spherical), spherical beats Brier by exactly √K per payment range at a uniform truth, and both bounded scores nearly ignore a dropped rare class (1e-4 vs 0.15 for log).
• 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.

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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.