Paying contributors

A model-improvement market paying holdout-loss decreases: the budget telescopes, but naive payment feedback lets a data-less adversary be paid while the published model's true loss worsens; a Ladder-style threshold removes the attack at a measured cost to honest small contributors. Stylised.

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.

In a saturating contribution game v(S)=1−exp(−Σw_i) every estimator variance is exact. Permutation sampling always pays out exactly v(N) but is noisy; size-stratified sampling has 6–8× lower variance (exactly zero for equal contributors) yet misses the pot by up to 7% in a run; reversing each permutation cuts RMSE 1.6× at no cost; Hoeffding sample counts are 4–5× larger than needed, and stratification finds the top-3 payees with about 4× fewer samples. Stylised, n≤10.

In a winner-take-all contest ranked on a noisy holdout of m samples, equilibrium effort is V·a_n√m/s with a₂=a₃=1/(2√π) exactly, but the formula stops being an equilibrium once V/σ² passes a ceiling (12.35, 7.78, 6.11, 5.60, 5.80 for n=2, 3, 5, 10, 20) that sits below the zero-rent limit and falls to half of it at n=20. The organiser's optimal holdout is min(m_cap, (wVa_n/2sκ)²): when evaluation is cheap the ceiling, not cost, binds. Quadrature checked against Monte Carlo and brute-force best replies. Stylised.

Reverse auction for k compute units from n devices with private costs: the truthful threshold auction pays exactly k(k+1)/(n+1), twice the winners’ true cost for every n and k (also with a reserve), the buyer-optimal reserve is exactly v/2 (worth up to +1.03 utility in thin markets, nothing once v≥2), and the pay-as-bid equilibrium has closed form and matches revenue (1.3334 vs 1.3333). Stylised.

If a round pays the first k of n finishers a prize R and workers buy speed at cost c per unit rate, the win probability’s slope at the symmetric point telescopes to (1−k/n)(H_n−H_n−k), so equilibrium speed is R(n−k)(H_n−H_n−k)/(nc) and the round time is exactly nc/((n−k)R). Paying every finisher buys no speed at all; for a fixed budget, speed alone wants winner-take-all (0.10 at k=1 versus 3.2 at k=16 for n=32), and only the statistical floor on k justifies more winners. Recruiting more workers barely helps: total speed bought saturates at kR/c. Exponential races, linear cost, symmetric risk-neutral workers; equilibrium checked on a grid.

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 market pays each contributor the log-score gain of a public Gaussian belief for releasing a statistic plus Gaussian-mechanism noise, so the expected payment is exactly the information released, ½λln(1+τ/a). The best noise is a closed-form quadratic root, and participation is possible if and only if κa < 1 with κ = cΔ²/λ: precision is capped at λ/(cΔ²) however many contributors arrive, approached geometrically at rate 2σ²/(2σ²+κ), and total payout is at most ½λln(1/(κa₀)). Early arrivals add too little noise (1.5–5×; 1–22% welfare loss against a planner), and admitting small data holders first lifts final precision from 28.0 to 34.2. Checked against grid search and 10⁶-run Monte Carlo. Stylised.