Is risk tolerance the learning rate of a wealth market?

What it found

When agents bet a fraction λ of wealth Kelly-style, the clearing price is the λ·wealth-weighted mean belief, total wealth is conserved, and λ=1 is exactly the Bayes mixture. Market log-loss exceeds expert i’s by at most ln(1/s_i)/λ_i on any sequence (loose on i.i.d. data), the growth-optimal fraction is (π−p)/(q−p), and moving the price against calibrated others costs Vχ²(p₀‖p): equal to LMSR b·KL at b=2V for small shifts but 7.5× stiffer at p=0.99. On a truth that switches regimes, full Kelly is wrong about half of every segment (regret about 1630 at T=4000) while λ of 0.1–0.2 cuts it 3–7×. Binary events, static experts. 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

• 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.
• What does it cost to route between models with conditional markets, and what does a stake do to the answer?Conditional markets that route between models: inverse-propensity Brier is proper even when the router reads the report, exploration costs exactly 1+(K−1)e^{ω²/2τ²} in payment variance (probit: infinite for τ ≤ ω), and a stake gap distorts the reported gap by a closed-form fixed point with second-order harm.
• What share of the mass should a new module get?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.
• 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%.

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.