Fairness and Strategy-Proofness in Automated Market Makers
This result provides a theoretical foundation for the structural absence of voting mechanisms in AMMs, showing it is not an oversight but an inherent limitation.
The paper proves that for automated market makers with three or more assets, no aggregation rule for liquidity providers' preferred trading functions can be both fair and strategy-proof, establishing a fundamental impossibility that explains why no deployed AMM allows voting on the trading function.
No deployed automated market maker lets its liquidity providers vote on the trading function. We show this is structural, not an oversight. On the weighted-product family with $n \geq 3$ assets, no aggregation rule is at once fair and strategy-proof. Arrovian fairness forces a unique form, the weighted Aitchison centroid, the weighted geometric mean of the providers' preferred pools. But fairness forces mean-type aggregation and strategy-proofness forces median-type, and the only rule that is both is a single-provider dictator. The obstruction is sharp: it vanishes at $n = 2$, where a fair strategy-proof rule exists. Under the Frongillo--Papireddygari--Waggoner equivalence, the centroid is Genest's logarithmic opinion pool, and the impossibility transfers to externally Bayesian pooling.