Optimal Benchmark Design under Costly Manipulation

TL;DR

This paper investigates how to design optimal market price benchmarks that minimize manipulation incentives, considering costs that depend on the magnitude and number of price adjustments, with solutions varying based on cost structures.

Contribution

It introduces a model for benchmark design under manipulation costs, identifying when weighted means, medians, or trimmed means are optimal depending on cost factors.

Findings

Weighted mean is optimal with negligible fixed costs.

Median is optimal with negligible variable costs.

Trimmed mean is optimal when both fixed and variable costs are significant.

Abstract

Price benchmarks are used to incorporate market price trends into contracts, but their use can create opportunities for manipulation by parties involved in the contract. This paper examines this issue using a realistic and tractable model inspired by smart contracts on blockchains like Ethereum. In our model, manipulation costs depend on two factors: the magnitude of adjustments to individual prices (variable costs) and the number of prices adjusted (fixed costs). We find that a weighted mean is the optimal benchmark when fixed costs are negligible, while the median is optimal when variable costs are negligible. In cases where both fixed and variable costs are significant, the optimal benchmark can be implemented as a trimmed mean, with the degree of trimming increasing as fixed costs become more important relative to variable costs. Furthermore, we show that the optimal weights for a mean-based benchmark are proportional to the marginal manipulation costs, whereas the median remains optimal without weighting, even when fixed costs differ across prices.