Fairness and Strategy-Proofness in Automated Market Makers

Abstract

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.