From Means to Medians: Optimal Benchmark Design

Manipulation of price benchmarks has cost hundreds of millions of dollars, making manipulation-resistant design a first-order question. A puzzling empirical pattern offers a starting point: benchmarks in traditional finance are typically means, those in decentralised finance medians. I rationalise this divide with a variable cost rising in each price distortion's size and a fixed cost per manipulated price. The optimal benchmark is a mean when fixed costs are negligible and variable costs convex, a median when variable costs are negligible, and otherwise a trimmed mean, with weights reflecting cost heterogeneity.