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Delegation Risk

Euler Allocation: The Fundamental Aggregation Problem

Euler Allocation: The Fundamental Aggregation Problem

Section titled “Euler Allocation: The Fundamental Aggregation Problem”
flowchart TB
    TR["Total Risk (R)"]
    TR -->|decompose| C1["Component 1: RC1"]
    TR -->|decompose| C2["Component 2: RC2"]
    TR -->|decompose| C3["Component 3: RC3"]
    C1 --> Sum["Sum: RC1 + RC2 + RC3 = R"]
    C2 --> Sum
    C3 --> Sum

Each component’s risk contribution is: RC_i = x_i × (∂R/∂x_i) — weight times marginal impact.

The mathematical foundation for hierarchical risk decomposition comes from financial risk budgeting, where Euler’s theorem for homogeneous functions enables perfect additive decomposition of total risk into component contributions. For any risk measure R(x) that is homogeneous of degree 1 in portfolio weights, total risk equals the sum of each component’s marginal contribution multiplied by its weight: R(x) = Σᵢ xᵢ · ∂R(x)/∂xᵢ.

Practical Implementation

Section titled “Practical Implementation”

The practical implementation uses Component VaR (CoVaR) for Value at Risk or Expected Shortfall contributions. Component VaR for position i equals CoVaR_i = x_i · β_i · VaR, where β_i is the position’s beta to the portfolio. The crucial property: Σᵢ CoVaR_i = Total VaR exactly, accounting for diversification benefits. Banks cascade these budgets from board-level risk appetite through business units to trading desks, with each level receiving explicit risk limits derived from the aggregate. The Financial Stability Board’s 2013 Risk Appetite Framework established this as regulatory standard, requiring quantitative risk tolerances that flow hierarchically through organizations.

Budget Cascade Example

Section titled “Budget Cascade Example”

Here’s how a $10M annual risk budget might flow through an AI research organization:

flowchart TB
    BOARD["🏛️ Board Level<br/>Total Risk Budget: $10M/year<br/>(System-wide harm tolerance)"]

    BOARD --> DIV1["Research Division<br/>$4M (40%)"]
    BOARD --> DIV2["Engineering Division<br/>$4M (40%)"]
    BOARD --> DIV3["Operations Division<br/>$2M (20%)"]

    DIV1 --> T1["Hypothesis Generation<br/>$1.5M"]
    DIV1 --> T2["Literature Analysis<br/>$1.5M"]
    DIV1 --> T3["Experiment Planning<br/>$1M"]

    DIV2 --> T4["Code Generation<br/>$2M"]
    DIV2 --> T5["Code Review<br/>$1M"]
    DIV2 --> T6["Deployment<br/>$1M"]

    T4 --> C1["Coordinator<br/>$500K"]
    T4 --> C2["Generator<br/>$800K"]
    T4 --> C3["Verifier<br/>$400K"]
    T4 --> C4["Human Gate<br/>$300K"]

    style BOARD fill:#e1f5fe
    style C1 fill:#fff3e0
    style C2 fill:#fff3e0
    style C3 fill:#fff3e0
    style C4 fill:#fff3e0

Key properties of this cascade:

  • Full allocation: 500K+500K + 800K + 400K+400K + 300K = $2M (Code Generation budget)
  • Marginal contribution: Generator gets more budget because it has higher impact potential
  • Diversification: Independent components can use their full budgets simultaneously
  • Accountability: Each level is responsible for staying within its allocation

Application to AI Safety

Section titled “Application to AI Safety”

This approach directly transfers to AI safety: if system-level harm probability is the risk measure, and subsystem contributions can be computed via marginal impacts, Euler allocation provides mathematically principled decomposition. The challenge lies in defining appropriate homogeneous risk measures for AI systems—current approaches like Anthropic’s AI Safety Levels provide tiers but lack the continuous differentiability Euler allocation requires.

Key Properties

Section titled “Key Properties”
  • Full allocation: Component risks sum exactly to total
  • Marginal contribution: Each component’s share reflects its impact on total risk
  • Diversification accounting: Correlations between components are handled correctly
  • Hierarchical cascading: Budgets flow from top-level to components

Key Takeaways

Section titled “Key Takeaways”
  1. Budgets can sum exactly — Euler’s theorem guarantees full allocation when risk measures are homogeneous
  2. Marginal impact matters — Each component’s share reflects its actual contribution to total risk
  3. Diversification is captured — Correlations between components are handled mathematically
  4. Hierarchy works — Budgets cascade cleanly from board level to individual components

Next Steps

Section titled “Next Steps”