Markets as routers

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.

Weighted-score wagering (Lambert et al.) as a budget-balanced, replayable mixture-of-experts gate under recurring regime drift: plain wagering locks in, fixed-share wealth tax repairs it but subsidises free-riders.

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.

Hedge/LMSR routing of modular experts when losses arrive after device-dependent delays: regret ≤ ln N/η + ηT/8 + ηD/2 with D the total outstanding rounds, so mean delay d̄ inflates the horizon by 1+4d̄; ignoring delay costs 12× regret against a reactive adversary at delay 100, the worst-case tuned rate is too cautious on oblivious data, an observable-count adaptive rate recovers most of it, and LMSR subsidy grows about 20× at mean delay 5. Stylised.

Capacity-constrained expert routing is a softmax with a per-expert price, the Lagrange multiplier of its capacity, and the dual is smooth and concave. Any price vector gives a computable upper bound and, after repair toward the uniform assignment, a feasible lower bound, so a verifier certifies near-optimality from prices alone (gap below 10⁻⁴ at the optimum, always above the true suboptimality). Sign-update balancing has a sampling floor of about 2.5/√B in max relative overload that no step size removes, plus a step-proportional limit cycle on a fixed batch (22.6% at γ=0.2, 1.2% at 0.01). Two-expert Gaussian price of balance is closed form with price shift equal to the median gap; a Hoeffding audit of ln(2N/δ)/(2ε²) tokens covered 400 of 400 trials. Synthetic scores only. Stylised.

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.

Sparse mixture-of-experts routing as a capped Hedge market: expert prices are min(1, λ w_i) with a single shadow price λ and an exact KL projection. Regret against the best fixed k-set stayed within k ln(N/k)/η + ηTk/2 in every run (41–58% of it, not proved here); plain normalisation prices an expert above 1 in 282 of 300 rounds and understates loss 3.2×; systematic sampling realises exactly k distinct experts with 8–14× lower loss variance. Oblivious losses only.

If only awake modules report and each awake module’s wealth is multiplied by 1+η(ℓ̂−ℓ_i), the ledger is exactly budget balanced, sleeping wealth never moves (each regime keeps a private ledger), and a module’s regret against the market on its awake rounds is at most ln(1/π_i)/η + ηn_i in its own awake time (the exponential variant pays ηT/8 and is not budget balanced). Admitting a module with wealth ε costs each incumbent ln(1/(1−ε))/η in its bound. With recurring regimes the ledger matches the per-round oracle (0.1004 vs 0.1000, regret about 2.7 whatever the number of specialists) while a forced-to-report Hedge sits at the generalist (0.30) and fixed share is worse (0.34–0.36). The bound is loose (measured regret 2.7 against a bound near 10³); losses bounded, wake sets exogenous.

Yes, if it declares after seeing its loss. The wealth ledger pays each awake module exactly its leave-one-out saving, ηW_A(1−p_i)S_i, but a module that sleeps whenever its own loss exceeds θ earns ηp_Bθ(1−θ)/2 per unit wealth against an equal-skill honest rival (best 1/8 at θ=1/2) and takes the market from a 0.1 share in about 130 rounds, while the router’s realised loss stays at 0.5. Declared before the router serves, the same wealth is a real saving that falls from 0.125 to 0.011 as the module’s signal about its own loss gets noisier. A sleeping tax that stops the exploit must be at least ηp_B/2, a fixed-share rate that halves an honest specialist asleep 90% of the time in 6 rounds, so commitment, not taxation, is the fix. Two modules, uniform losses, threshold strategies; mean-field predictions 3–8% slow.

Markets that forecast each model’s loss if chosen only ever see the chosen arm. An inverse-propensity Brier payment is exactly proper even when the router reads the report (without the weight a trader shades its forecast, or hides), and its price is exact: E[1/π] = 1+(K−1)e^ω2/2τ2, so payment variance blows up as the router cools (a probit router has infinite variance for τ ≤ ω; Monte Carlo at τ=0.3 sees 29 against the exact 195). Exploration regret is at most 0.2785τ. Equal owner stakes cancel in the decision, a stake gap shifts the reported gap by a closed-form fixed point, and the loss it causes is second order, ∝(ΔB/k)². Gaussian, one-shot, stylised.

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 simulated RTGS day (Zipf-concentrated banks, lognormal payment values, a share of payments followed by a return payment) with the paper's upper bound, lower bound and bilateral netting, a FIFO queue, and the Bech–Soramäki (2001) gridlock resolution algorithm, checked by brute force to return the largest feasible set of queue prefixes. Inputs from the paper: participants and payments per day (DK 146 and 925, FI 13 and 1,428), top-bank concentration and the Table 1 bounds. Independent payments need four times the published lower bound; with reciprocity fitted (ρ = 0.8 DK, 0.9 FI) the Danish-shaped system needs 34.7 % of gross value to settle on arrival and 10.1 % by close (published 37.2 and 10.7), the Finnish-shaped one 23.3 and 4.2 (27.4 and 4.3). Bilateral netting leaves 15.6 % (DK) and 7.2 % (FI) to fund. Gridlock resolution cuts mean settlement delay by 24–46 % in the Danish-shaped system (paper: up to 50 %) and 34–62 % in the Finnish-shaped one, where the paper found about 10 %: reported as an unexplained discrepancy. Assumed or fitted, not sourced: payment-value spread (swept), reciprocity (fitted to the lower bound, so the bound match is calibration), uniform timing, and my reading of the concentration shares. Simulated payments only; not financial advice.

The real Solomon RC208 VRPTW instance is solved battery-blind with PyVRP (4 routes, 778.93, equal to the published best-known distance as I recall it) and then given an electric fleet: range Q, full recharge at time g per unit, charging at the depot or at 2, 5 or 10 stations placed by k-means of the customers. Route-first, charge-second (the exact cheapest charging plan per route by a label-setting search, cutting routes that have none) costs 0 % extra distance at Q = 300, 0.4 % at 200, 28 % at 150, 36 % at 100 and 46 % at 75. An iterated local search whose moves are scored by the exact charging plan, 60 s from that start, cuts these to 4.9, 10.2 and 18.4 % (three seeds, best shown). Slower charging (g = 1) and fewer stations (2) leave 11 % and 12 % even after search; ten stations give 3.4 %. Not from a source: the battery range and recharge rate (swept, loosely anchored on one e-van’s range from memory), the station sites, full-charge-only recharging. The Schneider EVRPTW instances were unreachable, so this is RC208 with my own EV additions, one instance only; the search is short and seeds differ by up to 3 %. No tariff, so no time-of-use cost.