What does submitting copies earn under Shapley payments?

What it found

Paying contributors of data or gradients by Shapley value rewards replication even when copies add zero information: in a Bayesian mean-estimation game the replicator's payment rises with every copy to its stand-alone value, exactly (n+2)/2 times the honest payment against n unit-precision others (5.5× at n=9). If the evaluator treats copies as independent the replicator's pot share tends to 100%; leave-one-out pays true copies exactly zero but distributes only 1/(n+1) of the pot; collapsing duplicates fixes it but underpays independent look-alikes by 45%. Exact computation checked against permutation brute force. Stylised.

A figure from its own output

Paper, code and results

• README and method
• White paper
• Experiments and results
• Code
• Tests

More in Paying contributors

• 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.
• What does it cost to make workers race?Paying the first k of n finishers makes speed a contest: π'(1)=(1−k/n)(H_n−H_{n−k}), equilibrium speed R(n−k)(H_n−H_{n−k})/(nc), exact round time nc/((n−k)R); paying all finishers buys no speed, a budget favours winner-take-all until an accuracy floor k_min binds, and recruiting more workers barely speeds the round.
• Who drops out of a race when workers have unequal costs?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.
• 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).

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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.