How much should a router extremize?

What it found

Pooling calibrated experts with shared training data: the Bayes-optimal logit exponent a* = n/(1+(n−1)ρ) (matched to a numerical argmin), extra experts saturating at 1/ρ effective experts, naive independence pooling doing worse than chance at ρ=0.9, and an online learner that finds a*. Stylised.

A figure from its own output

Paper, code and results

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

More in Markets as routers

• What does late feedback cost a market router?Hedge/LMSR routing with delayed feedback: regret ≤ ln N/η + ηT/8 + ηD/2 (mean delay inflates horizon by 1+4d̄), 12× regret from ignoring delay against a reactive adversary, delay-tuned rate too cautious on oblivious data, LMSR subsidy ~20× at mean delay 5.
• Is load balancing a congestion price?Capacity-constrained expert routing as a congestion-priced market: softmax with dual prices, verifier-computable duality-gap certificate from prices alone, sign-update balancing has a ≈2.5/√B sampling floor plus a step-size limit cycle, closed-form two-expert price of balance, Hoeffding load audit.
• Is risk tolerance the learning rate of a wealth market?Fractional-Kelly wealth markets as expert routers: risk-tolerance-weighted price, ln(1/s)/λ regret bound, growth-optimal λ*=(π−p)/(q−p), manipulation cost V·χ² vs LMSR b·KL (7.5× stiffer at p=0.99), and a switching-truth trade-off where λ≈0.1–0.2 beats full Kelly 3–7×.
• How should a router pick k distinct experts?Top-k MoE routing as a capped Hedge market: price min(1, λ w_i) with exact KL projection, regret vs the best fixed k-set held under k ln(N/k)/η+ηTk/2 in every run (41–58% of it), plain normalisation prices experts above 1 (282/300 rounds, loss understated 3.2×), and systematic sampling gives exactly k distinct experts with 8–14× lower variance.
• How should a router treat modules that are usually asleep?A self-financed wealth ledger for specialist modules that sleep: total wealth conserved exactly, sleeping wealth frozen (one private ledger per regime), regret ≤ ln(1/π_i)/η + η·n_i in the module's *own* awake time, entry of a module with wealth ε costs incumbents ln(1/(1−ε))/η; matches the regime oracle (0.1004 vs 0.1000) where forced-to-report Hedge sits at 0.30 and fixed share is worse.
• Can a module game a ledger by choosing when to sleep?When may a module choose to sleep? The ledger pays exactly leave-one-out savings ηW_A(1−p_i)S_i; sleeping *after* seeing losses gains wealth ηp_Bθ(1−θ)/2 (best 1/8) and takes the market in ~130 rounds with the router's loss unchanged; committed *before* serving it is a real saving; the deterring tax ηp_B/2 is a fixed share that halves honest specialists in 6 rounds.

All 13 Markets as routers 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.