What does a constant-product pool concede to informed traders?

research/cfmm-markets

Constant-product AMMs as cost-function prediction markets: worst-case loss C0 never attained, informed-trader profit exactly C0(1−n·GM(π)) (ratio to LMSR → ln n, crossover at n=e), polynomial price tails (p=0.999 costs 3.4× LMSR), exact regret identity, no forgiveness gain under regime switches.

10 tests

What it found

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.

A figure from its own output

Figure produced by research/cfmm-markets: push_cost — cfmm-markets/experiments/push_cost.svg, written by the project's own experiment code.

Paper, code and results

More in Prediction-market design

What does not knowing the horizon cost a market maker?LMSR with time-varying liquidity as anytime Hedge: b_t = c*√t costs exactly √2 over the horizon-aware bound, doubling is 1.70× worse, and self-tuning liquidity makes the subsidy equal the accumulated mixability gap (1.3 vs 102 on easy data).
The subsidy cost of privacy in prediction marketsDifferentially private LMSR via the binary-tree mechanism: path-wise loss bound, and the accuracy/subsidy cost of privacy.
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Proper scoring & elicitation28 Effort & peer prediction9 Verification games & slashing19 Detection & tolerances14 Markets as routers13 Prediction-market design9 Paying contributors8 Decentralised training systems23 Digital twins & sim-to-real164 LLM chip design loops40 Robot manufacturing loops53 Ambient AI assistant46 Self-replicating manufacturing34 Simulation-first design64 Engineered clothing45 Task-specific AI accelerators40 Algorithmic chip-design loops42 Open-source purpose-built robots44 The software-engineering loop43 Adversarial human imitation41 Open-source residential construction41 Autonomous multi-angle capture34 Sensing, imaging and 3D31 Energy supply and dense energy storage39 Robot societies among people31 Agentic co-design of low-cost robots27 Laundry-hanging robots25 Machine learning on qualitative data23 Robots on the building site27 Civilizations & group psychology1 History & 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.