Who drops out of a race when workers have unequal costs?

research/race-heterogeneity

Paying the first k of n finishers when costs differ: proof of the π'(1) telescoping identity, exact exit rule (slow workers idle iff c_s/c_f ≥ n_f/(n_f−k), never for k ≥ n_f), cost ratio 4 excludes slow workers for k ≤ 3 of 4 fast, convex costs make headcount buy speed.

13 tests

What it found

If only the first k of n finishers are paid, a slow worker entering at a vanishing rate wins with probability rate × E X_(k) of the fast workers, which gives an exact exit rule: workers with cost ratio c_s/c_f ≥ n_f/(n_f−k) stay out entirely, and paying k ≥ n_f never excludes anyone (checked by best-entry payoffs at thresholds 4/3, 2, 4). At cost ratio 4 with four fast workers, k ≤ 3 shuts the slow ones out and k=4..7 readmits them (share of prizes 0.21→0.46 against a fair 0.5). The note also proves the telescoping identity behind the symmetric race and shows that with convex speed cost, headcount does buy speed. Two cost types, exponential races, grid-checked equilibria.

A figure from its own output

Figure produced by research/race-heterogeneity: discouragement — race-heterogeneity/experiments/discouragement.svg, written by the project's own experiment code.

Paper, code and results

More in Paying contributors

How much accuracy can a market buy from contributors who value their privacy?Buying private data with a telescoping log-score market: payment equals information, best Gaussian-mechanism noise in closed form, participation iff κa<1 so precision is capped at λ/(cΔ²) for any number of contributors, and early arrivals over-collect (1–22% welfare loss).
Paying for improvements, not for overfittingPaying for model improvements against a hidden holdout: telescoping budget, a payment-feedback overfitting attack (paid 0.15 while true loss worsens 0.37), and the Ladder fix with its measured cost.
What does submitting copies earn under Shapley payments?Shapley and leave-one-out payments under replication: Shapley pays every extra copy even when worthless, saturating at stand-alone value ((n+2)/2× honest), naive evaluators hand over the pot, LOO pays copies 0 but distributes 1/(n+1), dedup underpays lookalikes by 45%.
How should Shapley payments be sampled and audited?Auditing sampled Shapley payments: exact estimator variances in a saturating game, permutation sampling is exactly efficient but stratified sampling has 6–8× lower variance (zero for equal contributors) at up to 7% budget error, antithetic reversal gives 1.6× lower RMSE, Hoeffding sample counts are 4–5× conservative.
How precise should a training-contest leaderboard be?Winner-take-all training contests scored on a noisy holdout: effort V a_n √m/s with a_2 = a_3 = 1/(2√π) exactly, a precision ceiling V/σ² ≤ λ_max(n) beyond which no pure equilibrium exists (below the zero-rent limit), and the resulting optimal holdout size.
What does it cost to buy decentralised compute?Buying k units of decentralised compute by reverse auction: exact VCG payment k(k+1)/(n+1) is exactly 2× winners' cost, optimal reserve is exactly v/2, first-price equilibrium matches by revenue equivalence.

All 8 Paying contributors 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.