What share of the mass should a new module get?

research/module-entry

Admitting new modules to a Hedge/LMSR router: regret certificates obey a Kraft identity Σ e^{−c_i}=1, so uniform entry 1/(n+1) is the unique minimax rule; prior-optimal shares P_j/S_j cost H(P); the price of entry is exactly ln(1/π)/η independent of the quality gap.

6 tests

What it found

When a modular system adds an expert to a Hedge/LMSR router, the regret certificate to every expert is c_i/η + ηT/8 with c_i = ln(1/π_i) + Σ_j>i ln(1/(1−π_j)), and for any share sequence Σ_i e^−c_i = 1 (a Kraft identity). So uniform entry π = 1/(n+1) is the unique minimax rule (every c_i = ln N, against 4.1–9.7 for constant shares at 4+12 modules), and against a prior on which module will be best the optimal shares are P_j/S_j with expected coefficient equal to the entropy of the prior. The price of entry is exactly ln(1/π)/η for a better arrival and ln(1/(1−π))/η for a worse one, independent of the quality gap; a recency prior cuts simulated regret from 3.11 to 2.70 nats. Known arrival rounds, fixed η. Pure-Python, 6 tests.

A figure from its own output

Figure produced by research/module-entry: catchup — module-entry/experiments/catchup.svg, written by the project's own experiment code.

Paper, code and results

More in Markets as routers

How much liquidity does a payment system need, and what do netting and gridlock resolution save?Does bilateral or multilateral netting, or gridlock resolution, save a payment system liquidity? A simulator fitted to Bech and Soramäki's Danish and Finnish bounds (34.7% and 10.1% of gross value against the published 37.2% and 10.7% for DK) shows reciprocal flows are what make netting work, and resolution cuts delay by 24–46% in the Danish-shaped system but 34–62% in the Finnish-shaped one, where the paper found about 10%.
How much longer is an electric delivery fleet's route plan once the battery and the chargers are counted, and does a charge-aware search recover it?Solomon RC208 (100 customers) with a battery: keeping the battery-blind routes and adding charging stops costs 28% more distance at 150 range units, a charge-aware search brings that to 5%; the battery is harmless above 200 units.
Markets as routers: LMSR = Hedge, quadratic potential = sparsemaxCost-function markets as expert routers: LMSR = Hedge, quadratic potential = sparsemax; regret/loss bounds verified, continual-learning simulation.
Wagering mechanisms as routers for modular, continually learning expertsSelf-financed wagering mechanisms as verifiable routers for modular, continually learning experts; shows plain WSWM locks in under recurring regimes and a fixed-share tax fixes it at the cost of subsidising free-riders.
How much should a router extremize?Pooling calibrated experts with shared training data: Bayes-optimal extremization a* = n/(1+(n−1)ρ) verified numerically, saturation at n_eff → 1/ρ, asymmetric robustness, and online learning of the exponent.
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.

All 13 Markets as routers 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.