Prediction-market design

An LMSR with liquidity b routes like Hedge with rate 1/b, but the tuned b needs the horizon T. A b_t = c√t schedule with c* = 1/(2√ln N) pays exactly √2 over the horizon-aware bound (1.36 rising to 1.41 measured); the doubling trick is 1.70× worse than the continuous schedule against a price-reactive adversary; and self-tuning liquidity b_t = Δ_t−1/ln N makes the subsidy equal the accumulated mixability gap (1.3 against 102 on easy data) with regret at most 2Δ. Regret analysis only. Stylised.

Differentially private LMSR via the binary-tree mechanism: error scales as 1/(bε), tree noise beats per-release noise ~15×, and privacy noise is exploitable so market-maker loss far exceeds b ln 2. Stylised.

An uninformed manipulator pushing an LMSR verification market from q to τ loses exactly b·KL(q‖τ), which equals to the digit the extra rent it hands to informed traders; the accuracy change is μ[e₁(1−q) − e₀q − (1−2q)], positive only in a liquidity band (0.107, 0.413 in the baseline): below it nobody enters and manipulation costs 1−2q accuracy, above b = Nc/(H(q)−ln(1/(1−ε))) entry is certain and manipulation is neutral. Matches an executable LMSR Monte Carlo. Stylised.

A searcher who buys before and sells after a victim's LMSR trade extracts exactly the victim's slippage tolerance times its fair cost (εC₀, error 1.4×10⁻¹⁴ over 216 instances), with a closed-form front-run of about εb/(1−p) shares, so return on capital falls as 1/b. A proportional fee of about v(1−p)/(2b) makes it unprofitable at any tolerance; the victim's optimal tolerance grows only like √log of the stakes; pro-rata batching keeps 0.989–0.999 of the extraction and uniformly random ordering cuts it exactly 3×. Stylised.

A Glosten–Milgrom market maker facing an informed share μ with accuracy q moves its posterior log-odds by exactly 2·atanh(μ(2q−1)) per net buy, so its price path is an LMSR with b* = 1/(2 atanh m) (verified to 2×10⁻¹⁶). A wrong b costs log loss only in a transient whose peak is scale-free in m and 1.8–2.5× worse when overconfident; the LMSR subsidy tends to b* ln 2, exactly the spread the GM maker charges. Stylised.

A proportional fee f on an LMSR verification market gives every trader an exact no-trade band [q/(1+f), (q+f)/(1+f)] of width f/(1+f), so an informed trader leaves the price at q/(1+f). At price ½ a signal of log-likelihood ratio λ moves the price iff f < tanh(λ/2) = 2a−1, above which the market is dead; fee revenue on a monotone path is exactly fb ln((1−p₀)/(1−p_T)). With ten 0.65-accurate traders, the break-even fee (≈0.11) raises Brier error from 0.115 to about 0.16 and revenue is hump-shaped in f. Naive traders, synthetic signals.

In a sequential LMSR verification market the verifier in position k earns in expectation b·I(y; s_k | s_<k), a conditional mutual information, so seat rents telescope to the maker’s expected loss (at most b ln 2); an explicit cost-function simulation agrees to three decimals. Rents fall with position, the successive ratio approaching e^−C (Chernoff information) from below; order only redistributes a fixed total (a 0.9-accurate verifier gets 0.368 of 0.407 first, 0.303 last). With entry cost c, accuracy improves roughly as 1/b: 11 verifiers at b=0.25 (92%), 41 at b=5 (99.6%). Myopic traders, symmetric signals, monotonicity numerical.

Traders who each move prices within a subspace of securities make a cost-function market a randomized subspace descent. The market maker’s expected loss telescopes to ½(e₀′Ae₀ − e_T′Ae_T) whoever trades, so accuracy is bought once. Correlated securities slow single-security traders in closed form (5 securities: 31 rounds to reach 10⁻³ of the error uncorrelated, 356 at correlation 0.95), while offering A-conjugate bundles restores the exact rate 1−1/d at any correlation. With noisy beliefs the fraction κ a trader moves acts as an SGD step size: floor dκσ²/(2−κ), and a per-security running-mean step beats the best constant step (error 0.26 vs 0.47 at 100 rounds; full steps stay at 5). A simulated 4-outcome LMSR matches the local rate predictions to within 0.01 for two of three bundle designs and is faster than predicted for the third, for a reason not isolated. Quadratic-cost theory with i.i.d. trader arrival and Gaussian belief noise.

A Uniswap-style pool with invariant ∏r_i = C₀ⁿ is a cost-function prediction market: its worst-case loss is C₀ and is never attained, and a trader who moves the price to belief π earns exactly C₀(1 − n·GM(π)) with GM the geometric mean, against C₀(1 − H(π)/ln n) for LMSR at equal worst-case loss. Near uniform beliefs the ratio is exactly ln n, so the pool is cheaper than LMSR for two outcomes and dearer from three. Its price tails are polynomial, (C₀/gap)ⁿ, so pushing a binary price to 0.999 costs 3.4× LMSR. As an expert router it satisfies an exact regret identity (settlement term ≤ C₀ plus Bregman terms), but under regime switches it shows no advantage over Hedge (negative result). Symmetric pools, no fees. Pure-Python, 10 tests.