What do risk-averse verifiers report under proper scoring?

What it found

Proper scoring rules are truthful only for risk-neutral agents. For a CARA verifier the log score yields reported odds exactly equal to true odds^(1/(1+k)), k = αb; the Brier score yields logit p = logit r + k(2r−1), which the principal can invert exactly. Summing reported logits under-extremises (log-loss 0.332 vs 0.214 at k=2); correcting with the mean k under heterogeneous verifiers is worse than not correcting; a log-score report needs true belief 0.964 to clear a 0.9 accept threshold at k=0.5. A binarised lottery rule is truthful for every increasing utility at a 2.6–27% certainty-equivalent cost. Stylised.

A figure from its own output

Paper, code and results

• README and method
• White paper
• Experiments and results
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More in Proper scoring & elicitation

• What does a verifier's silence tell you?Selective reporting under proper scoring: abstention iff |p−α| < √c (Brier), silence is evidence unless the anchor is a symmetric prior, ignoring it costs 30× the log-loss at n=12, and a stale anchor makes silence point at the wrong state.
• How do you price the tail of benign drift?Jointly eliciting VaR and expected shortfall of benign drift with a scale-free Rockafellar–Uryasev score: ES alone is not elicitable, misreport cost is exactly ln r + 1/r − 1, quantile-only tolerances leave a/(a−1) of the Pareto tail uncovered, and auditing tail reports breaks down at index ≤ 2.
• Should you pay verifiers for their calibration?Paying verifiers for resolution, not calibration: raw Brier/log excess is exactly E[(r-p)²]/E[KL(p‖r)], Brier is bounded but log is not, log-score selection reverses for overconfident high-resolution verifiers (a ≥ 2.49), and n-sample recalibration costs tr(HI⁻¹)/2n (0.264/n).
• 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).

All 28 Proper scoring & elicitation projects →

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.