Optimal Benchmark Design under Costly Manipulation

ArXiv ID: 2506.22142 “View on arXiv”

Authors: Ángel Hernando-Veciana

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.

Keywords: Price Benchmarks, Market Manipulation, Blockchain Contracts, Trimmed Mean, Optimization

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper presents a formal optimization model with concepts from contract theory (multitask moral hazard) and non-convex analysis, requiring advanced mathematical techniques. It is purely theoretical, with no data, backtests, or implementation details provided, focusing instead on deriving general results for benchmark design.

  flowchart TD
    A["Research Goal<br>Optimal Design for Costly Manipulation"] --> B["Model Setup<br>Blockchain-inspired Pricing Contract"]
    B --> C{"Key Manipulation Costs"}
    C --> D["Variable: Price Adjustments"]
    C --> E["Fixed: Number of Adjustments"]
    D & E --> F["Computational Optimization<br>Find Benchmark Minimizing Expected Cost"]
    F --> G{"Dominant Cost Factor?"}
    G -- Variable Only --> H["Outcome: Weighted Mean<br>Weights ∝ Marginal Costs"]
    G -- Fixed Only --> I["Outcome: Median<br>Optimal even with heterogeneous costs"]
    G -- Both Significant --> J["Outcome: Trimmed Mean<br>Trimming increases as Fixed Costs dominate"]