What does it cost to pay for a ranking instead of probabilities?

research/rank-elicitation

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.

7 tests

What it found

A verifier who reports only the order of K classes can be paid with a score linear in the outcome indicator. It is strictly proper, and swapping two classes at adjacent ranks costs exactly (p_i−p_j)(w_r−w_r+1), linear in the gap. Per unit of payment range that beats a Brier-derived ranking whenever the gap is below 1/(K−1) (2.5× at gap 0.1, 250× at 0.001, K=5). Equal spacing maximises the smallest incentive, ranking everything costs K−1 times the mode, a top-k set costs the same as the mode, and the sample size to detect a swap is identical to Brier’s.

A figure from its own output

Figure produced by research/rank-elicitation: ratio — rank-elicitation/experiments/ratio.svg, written by the project's own experiment code.

Paper, code and results

More in Proper scoring & elicitation

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.
How should a verifier's vector report be scored?Scoring a verifier's vector report: the precision score Σ⁻¹ is uniformly most detectable among quadratic Bregman scores (Cauchy–Schwarz), an isotropic score can need (1+κ)²/(4κ) times more tasks (25× at κ=100, power 0.29 vs 0.80), diagonal normalisation barely helps, estimated precision reaches 0.61 efficiency at m=2d draws.

All 28 Proper scoring & elicitation projects →

Other research topics

Proper scoring & elicitation28 Effort & peer prediction9 Verification games & slashing19 Detection & tolerances14 Markets as routers13 Prediction-market design9 Paying contributors8 Decentralised training systems23 Digital twins & sim-to-real164 LLM chip design loops40 Robot manufacturing loops53 Ambient AI assistant46 Self-replicating manufacturing34 Simulation-first design64 Engineered clothing45 Task-specific AI accelerators40 Algorithmic chip-design loops42 Open-source purpose-built robots44 The software-engineering loop43 Adversarial human imitation41 Open-source residential construction41 Autonomous multi-angle capture34 Sensing, imaging and 3D31 Energy supply and dense energy storage39 Robot societies among people31 Agentic co-design of low-cost robots27 Laundry-hanging robots25 Machine learning on qualitative data23 Robots on the building site27 Civilizations & group psychology1 History & 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.