💰 quantitative finance

From Means to Medians: Optimal Benchmark Design

This paper rationalizes the empirical divide between mean-based benchmarks in traditional finance and median-based benchmarks in decentralized finance by demonstrating that the optimal design depends on the relative magnitudes of fixed and variable manipulation costs, ranging from means to medians or weighted trimmed means accordingly.

Original authors: Ángel Hernando-Veciana

Published 2026-06-09

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ángel Hernando-Veciana

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out the "true" price of a rare item, like a vintage watch. You ask 100 different people what they think it's worth, and you need a single number to represent the group's opinion. This single number is called a benchmark.

However, there's a problem: some people are liars. They might try to lie about the price to make money. If you are a borrower, you want to lie and say the watch is worth more so you can borrow more money against it. If you are a lender, you might want to lie and say it's worth less so you can seize the watch from someone else.

This paper asks a simple but crucial question: What is the best way to calculate that single number so that lying is as expensive and difficult as possible?

The author, Ángel Hernando-Veciana, discovers that the answer depends entirely on how much it costs to tell a lie. He identifies two types of "costs" for lying:

The "Gas Fee" Cost (Fixed Cost): This is the cost of just starting the lie. In the world of digital finance (DeFi), this is like paying a transaction fee to a blockchain just to submit a lie, no matter how big or small the lie is. It's like paying an entry fee to a game.
The "Effort" Cost (Variable Cost): This is the cost that goes up the bigger the lie is. If you want to lie that the watch is worth $1,000 instead of $100, it takes more effort (and money) than lying it's worth $110.

Here is how the paper explains the best "lie-detector" formulas based on these costs, using some creative analogies:

1. The "Average" (The Mean)

When to use it: When the "Entry Fee" is zero, but the "Effort" cost gets harder the bigger the lie gets.

The Analogy: Imagine a classroom where the teacher asks for the average height. If lying about your height costs nothing to start, but getting taller costs you more and more energy (like stretching a rubber band that gets tighter), a liar would try to cheat by making everyone in the room slightly taller by the same tiny amount.
Why it works: If you calculate the Average, and everyone is shifted up by the same tiny bit, the average goes up exactly by that bit. The liar can't cheat by just changing a few people; they have to change everyone. Because the "Effort" cost gets steep quickly (convex), it becomes too expensive to shift everyone.
Real World: This is why traditional finance (like stock markets) often uses the Average. It assumes that manipulating a price requires a lot of trading volume, which gets expensive very fast.

2. The "Middle" (The Median)

When to use it: When the "Effort" cost is zero, but the "Entry Fee" is high.

The Analogy: Imagine a line of people. The Median is just the person standing exactly in the middle. If lying about your height costs nothing to do, but you have to pay a heavy "Entry Fee" just to get into the line to lie, a liar will try to be efficient. They won't try to move everyone. Instead, they will bribe just enough people to push the middle person to a new spot.
Why it works: To move the Median, you only need to move half the people. If the "Entry Fee" is high, the liar will try to move the minimum number of people possible (just over 50%) to flip the result. The Median is the most resistant to this because it ignores the extreme outliers (the very tall or very short liars) and only cares about the middle.
Real World: This is why Decentralized Finance (DeFi) often uses the Median. In crypto, you can use "flash loans" to borrow huge amounts of money instantly, making the "Effort" cost of moving a price almost zero. However, you still have to pay the blockchain "Entry Fee" (gas) for every single transaction. So, it's cheaper to bribe a few people to move the middle than to try to move everyone.

3. The "Trimmed Average" (The Compromise)

When to use it: When you have both an Entry Fee and an Effort Cost.

The Analogy: Imagine a contest where you have to pay an entry fee to join, and then you pay for every step you walk. If both costs exist, the best strategy isn't to look at everyone (Average) or just the middle person (Median). Instead, you cut off the top 10% and bottom 10% of the crowd (the most extreme liars) and take the average of the rest.
Why it works: This is called a Trimmed Mean. If the Entry Fee is high, the liar won't want to bribe too many people. If they try to bribe a few people to move the average, the "Trimmed Mean" ignores those extreme liars. If they try to bribe everyone, the "Effort" cost becomes too high. The Trimmed Mean finds the sweet spot.
The Twist: The paper shows that if the Entry Fee is very high, you should trim more (move closer to the Median). If the Entry Fee is low, you should trim less (move closer to the Average).

The Big Picture: Why Traditional Finance vs. Crypto are Different

The paper solves a mystery: Why does the old financial world use Averages, while the new crypto world uses Medians?

Traditional Finance: To manipulate a price here, you have to buy and sell a lot of stock. The more you buy, the more expensive it gets (High Effort Cost). Since the "Entry Fee" is low, the Average is the best defense.
Crypto (DeFi): You can borrow money instantly to move prices, so the "Effort" cost is almost zero. But, every time you make a move, you have to pay a blockchain transaction fee (High Entry Fee). Since it's cheap to move a price but expensive to do it many times, the Median is the best defense.

Summary

The paper argues that there is no single "perfect" formula for calculating prices. The best formula depends on the cost structure of lying:

If lying is hard to do big, use the Average.
If lying is cheap to do big but expensive to start, use the Median.
If both costs exist, use a Trimmed Average (cutting off the extremes).

By understanding these costs, we can design financial systems that are much harder to cheat.

Technical Summary: From Means to Medians: Optimal Benchmark Design

1. Problem Statement

The paper addresses the critical design of price benchmarks in financial markets, specifically focusing on their resistance to manipulation. Benchmarks, which summarize prevailing market prices, underpin derivatives, loans, and portfolio valuations. The notional value of these contracts creates incentives for agents to distort prices to their advantage. While manipulation in traditional finance has resulted in hundreds of millions of dollars in losses, the advent of Decentralized Finance (DeFi) has amplified these concerns due to automated execution, large contract values, and limited legal enforceability.

A puzzling empirical pattern motivates the inquiry: benchmarks in traditional finance (e.g., LIBOR, EURIBOR, the 4pm Fix) typically use the mean, whereas benchmarks in DeFi (e.g., MakerDAO, Tellor oracles) predominantly use the median. The paper seeks to determine the optimal benchmark formula that maximizes the cost of manipulation, thereby explaining this divide and providing a design rule for robust benchmarks.

2. Methodology and Model

The author employs a game-theoretic framework modeled as an optimal contracting problem with multi-task moral hazard. The manipulator acts as an agent who chooses how to distort a distribution of prices to induce a specific bias ( $\Delta$ ) in the benchmark, subject to a cost function.

Key Model Components:

Price Distribution: The unmanipulated prices are drawn from a distribution $F$ , assumed to be symmetric and strictly single-peaked around a point $P$ . In some results, the distribution is assumed to have no dispersion (degenerate at $P$ ).
Manipulation Strategy: The manipulator shifts prices from an initial distribution $F$ to a manipulated distribution $G$ . The strategy is defined by conditional distribution functions $G_c(\cdot|p)$ .
Cost Structure: The cost of manipulation is the sum of two components:
1. Variable Costs ( $c(\cdot)$ ): A function of the magnitude of the price distortion ( $q$ ). The paper considers both convex and concave forms. Notably, Appendix B derives that for Constant-Product Market Makers (CPMMs), variable costs are concave.
2. Fixed Costs ( $k$ ): A cost incurred per manipulated price, independent of the distortion size. This is particularly relevant in blockchain environments where securing transaction ordering (e.g., via MEV auctions or gas fees) incurs a cost per transaction.
Benchmark Definition: A benchmark is a location functional. The paper focuses on symmetric benchmarks (those equal to the point of symmetry for symmetric distributions), including the mean, median, and $\tau$ -trimmed means.
Optimality Criterion: A benchmark is optimal if it maximizes the minimum cost required for a manipulator to induce a given bias $\Delta$ . The analysis relies on Proposition 1, which states that if a symmetric, cost-minimizing manipulation strategy exists for a benchmark, that benchmark is optimal.

3. Key Contributions and Results

A. The Polar Cases (Homogeneous Market)

The paper establishes a correspondence between the cost structure and the optimal benchmark:

Negligible Fixed Costs ( $k=0$ ) with Convex Variable Costs: The Mean is optimal. When variable costs are convex, the cheapest manipulation strategy is to spread the bias uniformly across all prices. The mean is maximally robust to such uniform shifts, whereas the median is highly sensitive to them.
Negligible Variable Costs ( $c(\cdot)=0$ ): The Median is optimal. When only fixed costs matter, the manipulator minimizes cost by manipulating the smallest possible mass of prices required to shift the median. The median requires manipulating exactly half the mass, whereas the mean would require manipulating all prices. Thus, the median maximizes the cost of manipulation in this regime.

B. The General Case (Fixed and Variable Costs)

When both cost components are present, the optimal benchmark is a Trimmed Mean.

Convex Variable Costs: If fixed costs are large relative to the target bias ( $\delta_{min} > 2\Delta$ ), the optimal manipulation of the mean would concentrate on less than half the prices, breaking symmetry. A $\tau$ -trimmed mean counteracts this by discarding extreme prices, forcing the manipulator to trade off concentration against higher average costs. The optimal trimming parameter $\tau^*$ increases as fixed costs rise.
Concave Variable Costs: This case is empirically relevant for DeFi CPMMs. Even without fixed costs, concavity implies that the optimal manipulation concentrates mass at the extremes. The optimal benchmark remains a trimmed mean, with the degree of trimming increasing in the size of fixed costs.
Comparative Statics: The optimal degree of trimming decreases with market liquidity. In deeper pools (where marginal manipulation costs are higher), the benchmark approaches the mean; in shallower pools, it approaches the median.

C. Heterogeneous Markets

The paper extends the analysis to markets where prices come from subpopulations with different manipulation costs (e.g., varying liquidity or gas fees over time).

Purely Variable Costs: The optimal benchmark is a Weighted Mean, where weights are proportional to the marginal manipulation costs ( $c'_i(\Delta)$ ). Prices that are cheaper to manipulate at the margin are downweighted.
Purely Fixed Costs: The optimal benchmark is a Weighted Median (in the spirit of Edgeworth), where weights are proportional to the fixed costs of manipulation ( $k_i$ ).
Proportional Costs: If fixed and variable costs are proportional across subpopulations, the optimal benchmark is a Weighted Trimmed Mean, with weights and trimming parameters reflecting the combined cost structure.

4. Significance and Claims

The paper claims to provide a theoretical rationalization for the observed divergence between traditional finance and DeFi benchmark designs:

Traditional Finance: Characterized by sustained trading and convex variable costs (with negligible fixed costs per price observation), leading to the optimality of the mean.
Decentralized Finance: Characterized by the availability of tools like flash loans (making variable costs negligible) but high fixed costs for transaction ordering (MEV/gas), leading to the optimality of the median.

The author emphasizes that the results are derived under specific assumptions (symmetric distributions, specific cost structures) and that extending the analysis to asymmetric distributions or general heterogeneous cost structures remains a challenge. The paper positions its manipulation-resistance criterion as complementary to, rather than competing with, the statistical efficiency criteria found in robust statistics (L-estimators). By identifying means, medians, and trimmed means as manipulation-optimal, the paper suggests these standard estimators can be disciplined along both efficiency and robustness dimensions.

The work does not propose new experimental protocols or specific implementation details for smart contracts beyond the theoretical aggregation rules. It focuses on the fundamental trade-off between cost structures and aggregation formulas to determine the most manipulation-resistant benchmark.