arXiv:2605.08417 [q-fin.TR]
LMSR Curvature Under Adversarial Flow:
A Live Benchmark for Prediction-Market Makers
Chicago & New York
May 2026
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Abstract
We study automated market making in binary prediction markets when liquidity is parameterized by a single curvature constant. Logarithmic market scoring rules (LMSR) offer closed-form prices but expose makers to adverse selection whenever flow becomes correlated with private information. This note presents a synthetic benchmark in which regime shifts compress or expand curvature, and we visualize price impact, book depth, and a toxicity proxy under live order flow.
The figures below are interactive. They run continuously — scroll through each section to inspect a different facet of the mechanism. No parameters are tuned to any particular venue; the goal is structural intuition before we attach real tape.
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1. Regime-switching LMSR
Price follows p(q) = (1 + e−q/b)−1. The liquidity parameter b toggles between calm and shock regimes every fourteen seconds. Position q executes against the curve as synthetic trades arrive.
Figure 1. Live LMSR with alternating calm (b ≈ 36) and shock (b ≈ 8) regimes. Ghost curve shows the inactive regime.
2. Curvature and marginal impact
Drag the slider to sweep b. Compare against the reference curve (b = 36) and watch marginal slippage at q = 0 grow as liquidity thins.
Low b → steep curve, thin liquidity. High b → flat curve, deep book.
Figure 2. Marginal price impact at q = 0 (¢ per share).
3. Limit-order book analogue
For contrast, a classical central-limit book with discrete depth levels. Spread, mid, and last trade update on a fixed cadence — useful baseline when evaluating how much structure LMSR hides inside a single scalar.
Figure 3. Synthetic CLOB — asks above mid, bids below. Depth bars encode size at each level.
4. Flow toxicity proxy
A coarse index built from signed volume imbalance. Spikes correspond to intervals where informed flow would dominate a passive maker's inventory risk.
Figure 4. Simulated toxicity proxy — order-flow imbalance under informed trading pressure.
5. Discussion
The open benchmark asks whether a single-parameter AMM can survive adversarial flow without manual intervention. Early simulations suggest that static curvature is insufficient: makers must either widen proactively or accept inventory drift during shock regimes. We leave optimal dynamic b policies to future work.
Acknowledgments. Compute supported by PredictIt. All errors are our own; all prices are synthetic.