Markets as routers: LMSR = Hedge, quadratic potential = sparsemax

What it found

Cost-function markets as online learners for modular, continual-learning expert routing: verified equivalence and loss/regret bounds, and a switching-environment simulation where the quadratic market stays sparse and low-regret at liquidity levels where LMSR degrades. Small-scale; limitations stated.

Paper, code and results

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

More in Markets as routers

• 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.
• 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.

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.