Weighted Average

A weighted average is calculated by multiplying each data point by its assigned weight, summing these products, and then dividing by the sum of all weights. The formula is: Weighted Average = (Σ (Value_i * Weight_i)) / Σ Weight_i. This contrasts with a simple average where all weights are equal to 1.

The power of a weighted average lies in the weight assignment. Weights are determined by the relative importance, frequency, or relevance of each data point. Common examples include:

• Market Capitalization: In a stock index, larger companies have greater weight.
• Time: Recent data points can be given higher weight in a moving average.
• Stake or Deposits: In DeFi governance or yield calculations, a user's influence is weighted by their stake.

A simple arithmetic mean treats every data point equally, which can be misleading if the underlying data has inherent imbalances. A weighted average corrects for this by reflecting the true proportional impact of each element. For example, calculating the average price of a token portfolio requires weighting each purchase by the number of tokens bought at that price, not simply averaging the prices.

Weighted averages are critical in decentralized systems for creating fair and resistant metrics.

• Oracle Prices: Protocols like Chainlink use time-weighted average prices (TWAP) to smooth out volatility and mitigate manipulation.
• Governance Voting: Voting power is often a token-weighted average, where one token equals one vote.
• Liquidity Pool Pricing: Automated Market Makers (AMMs) use constant function formulas that effectively calculate a weighted average price based on pool reserves.

A TWAP is a specific type of weighted average where the weight is the length of time a price persists. It is calculated by taking the average of prices at regular intervals over a specified period. This is a crucial oracle mechanism to prevent price manipulation via flash loans or wash trading, as it requires sustained price movement to significantly impact the average.

Key properties define the behavior of weighted averages:

• Linearity: The weighted average of a sum is the sum of the weighted averages.
• Boundedness: The result always lies between the minimum and maximum data values.
• Sensitivity: The average is most sensitive to changes in data points with the largest weights. A single point with a weight of 0 has no effect on the outcome.

A weighted average is crucial for determining a single, reliable price for an asset across multiple decentralized exchanges (DEXs). It accounts for varying liquidity depths.

• Mechanism: Prices from different pools are weighted by their liquidity or trading volume. A price from a pool with $10M liquidity influences the final average more than one from a $1M pool.
• Example: The Time-Weighted Average Price (TWAP) oracle uses this principle, sampling prices over time to resist manipulation.

In token-weighted voting systems, a user's voting power is a weighted average of their stake, often modified by locking duration.

• VeToken Model: Protocols like Curve Finance use vote-escrowed tokens. Tokens locked for 4 years receive maximum weight (1), while tokens locked for 1 year receive a weight of 0.25.
• Outcome: The final vote tally is not a simple sum of tokens but a weighted average of commitment, aligning long-term holders with protocol health.

DeFi portfolio trackers and crypto indices rely on weighted averages to calculate net asset value (NAV) and index prices.

• Market-Cap Weighting: An index like the DeFi Pulse Index holds tokens in proportion to their market capitalization. The index price is a weighted average of its constituents' prices.
• Portfolio Value: A wallet's total value is the sum of each asset's balance multiplied by its weighted average price, providing an accurate snapshot of wealth.

Proof-of-Stake (PoS) networks and liquidity pools use weighted averages to distribute rewards proportionally to stake or contribution.

• Validator Selection: In some PoS systems, the probability of being chosen to propose a block is a weighted average based on the validator's stake relative to the total.
• LP Rewards: In liquidity pools, yield farming rewards are often distributed based on a user's share-weighted contribution to the pool over time.

On-chain credit and risk scores are often composite metrics built using weighted averages of multiple factors.

• Creditworthiness: A score might combine transaction history (weight: 40%), asset diversification (weight: 30%), and protocol engagement (weight: 30%).
• Collateral Health Factor: In lending protocols, the health factor is a weighted average of the value of collateral assets, with different assets having distinct loan-to-value (LTV) weights.

Decentralized oracles aggregate data from multiple sources using weighted averages to produce tamper-resistant feeds for smart contracts.

• Source Reliability: Data from reputable, high-stake nodes is given more weight.
• Manipulation Resistance: By taking a weighted median or average across many independent sources, the system minimizes the impact of any single corrupted data point, securing DeFi applications.

The integrity of a weighted average depends on the immutability and correctness of its input weights. An attacker who can manipulate these weights can skew the result. This is a critical vulnerability in systems like:

• Oracle price feeds that average data from multiple sources.
• Governance voting where voting power (weight) is based on token holdings.
• Reputation systems where scores are averaged with trust weights. A compromised or malicious data source with disproportionate weight can dictate the outcome.

A Sybil attack, where an entity creates many fake identities, can be used to inflate weights in a permissionless averaging system. For example:

• In a proof-of-stake delegation system, an attacker could create many small validators to gain disproportionate voting power in a committee's average view.
• In a decentralized oracle network, a malicious actor running numerous low-stake nodes could bias the weighted average price. Defenses include requiring costly-to-fake identities (e.g., stake, work) or using sybil-resistant reputation.

A weighted average is only as reliable as its constituent data. Security failures occur when:

• A major data source (high weight) becomes corrupted or provides malicious outliers.
• The averaging mechanism lacks robustness against such outliers (e.g., using a simple mean instead of a trimmed mean).
• There is no slashing mechanism or reputation decay to reduce the weight of faulty sources over time. This makes the system vulnerable to targeted attacks on key oracles or validators.

If weights are static or slow to change (e.g., based on historical stake or age), it can lead to centralization risks and security decay.

• Early participants or large holders may retain disproportionate influence indefinitely, reducing system adaptability.
• A compromised entity with permanent high weight becomes a persistent threat. Secure designs incorporate dynamic weight adjustments based on recent performance and weight decay mechanisms to mitigate these risks.

The on-chain or off-chain calculation logic itself must be secure. Common flaws include:

• Integer overflow/underflow when summing large weights or values.
• Rounding errors that can be exploited in financial calculations (e.g., determining interest rates).
• Lack of input validation, allowing negative weights or zero denominators.
• Front-running the publication of a weighted average result if the calculation is predictable and has value. These require rigorous auditing, use of safe math libraries, and secure commit-reveal schemes for sensitive outputs.

Some consensus mechanisms (e.g., Tendermint's weighted round-robin) or Byzantine Fault Tolerance (BFT) models use weighted averages of validator votes. Here, security depends on:

• The honest majority of weight assumption, not just a majority of nodes.
• Accurate and tamper-proof weight distribution knowledge among all participants.
• Liveness risks if high-weight validators are offline, potentially halting the network. An attacker controlling >1/3 of the total weight in a BFT system can cause safety failures.

A Volume-Weighted Average Price weights each price by the trading volume that occurred at that price. It reflects the average price a token traded at, weighted by market activity.

• Trading Benchmark: Used by traders to assess whether they bought above or below the "fair" market average for a session.
• Blockchain Context: While common in traditional finance, on-chain VWAP is complex due to fragmented liquidity across DEXs, but is used in some advanced analytics platforms.

In Proof-of-Stake (PoS) consensus, a validator's influence is often a weighted average of their staked capital and other factors.

• Voting Power: A validator's vote weight is directly proportional to the amount of tokens they have staked.
• Slashing & Rewards: Penalties (slashing) and block rewards are typically calculated based on this staked weight, making it a critical economic parameter for network security.

Token-based governance systems use weighted averages to aggregate voter sentiment, where each vote's power is proportional to the voter's token balance.

• Quadratic Voting: A more complex form of weighting where voting power increases with the square root of tokens committed, designed to reduce whale dominance.
• Delegation: Voters can delegate their voting weight to others, creating weighted representative systems like those in Compound or Uniswap governance.

In Automated Market Makers (AMMs), a liquidity provider's share of the pool's fees and assets is a weighted average based on their contributed liquidity relative to the total.

• LP Tokens: Represent a weighted claim on the pool's reserves.
• Impermanent Loss Calculation: The change in value of an LP position is effectively a weighted average of the price movements of the two assets in the pool.