Why liquidity providers can't vote on crypto exchange rules

Automated market makers never let liquidity providers collectively choose the trading function. The authors prove why: on pools with 3+ assets, no voting rule can satisfy both fairness (weighted geometric mean) and strategy-proofness (no incentive to lie) simultaneously. The math forces a choice between median-style incentive alignment and mean-style fairness—you can't have both. With 2 assets, a fair rule exists; with more, only a single dictator works.