What is a Delegation?

Paper reference: Section 1, "The delegation setup"

A delegation is the simplest unit of governed autonomy. It has three parts:

An agent (\(B\)) that produces outputs
A corrector (\(C\)) that reviews a subset and fixes errors
A principal (\(A\)) that decides how much authority to grant

Every multi-agent pipeline is made of delegations chained together.

Two signals, not one

The theory tracks two distinct quality measures at every delegation boundary:

Signal	What it measures	How it's observed
\(\sigma_\text{raw}\)	Raw competence — agent's uncorrected success rate	Pre-correction outcomes
\(\sigma_\text{corr}\)	Corrected quality — what the system actually ships	Post-correction outcomes

Why both matter: If you only track \(\sigma_\text{corr}\) (the output), you can't tell whether quality comes from the agent being good or the corrector catching errors. This distinction is the foundation of the entire framework.

from minimal_oversight.models import Node

node = Node(
    "code_generator",
    sigma_skill=0.55,   # true competence (if known)
    sigma_raw=0.46,     # observed pre-correction success rate
    sigma_corr=0.84,    # observed post-correction success rate
    catch_rate=0.70,    # corrector catches 70% of errors
)

The fixed point

Given an observation rate \(\eta\) and a decay rate \(\delta\), the agent's measured competence converges to:

\[\sigma_\text{raw}^* = \frac{\eta \cdot \sigma_\text{skill}}{\eta + \delta}\]

This is the Return Operator's fixed point (Equation 5). The corrected quality is:

\[\sigma_\text{corr}^* = \sigma_\text{raw}^* + (1 - \sigma_\text{raw}^*) \times c\]

where \(c\) is the corrector's catch rate (Equation 6).

Worked example: With \(\sigma_\text{skill} = 0.80\), \(\eta = 10\), \(\delta = 2\), \(c = 0.70\):

\(\sigma_\text{raw}^* = 10 \times 0.80 / 12 = 0.667\)
\(\sigma_\text{corr}^* = 0.667 + 0.333 \times 0.70 = 0.900\)

The system ships at 90% quality, but the agent is only 67% competent. The 23-point gap is the corrector doing its job — and hiding the agent's weakness.

Verify the math

from minimal_oversight._formulae import sigma_raw_fixed_point, sigma_corr_fixed_point

sr = sigma_raw_fixed_point(0.80, eta=10, delta=2)  # 0.667
sc = sigma_corr_fixed_point(sr, catch_rate=0.70)    # 0.900

The Axiom of Minimal Oversight

Paper reference: Section 1, "The Axiom of Minimal Oversight (AMO)"

The AMO says: minimize the total cost of oversight, subject to meeting a quality target. The cost is measured in Fisher information geometry — a principled way to weight governance effort by how informative it is.

The result is a water-filling allocation: spend more oversight where the agent is moderately competent (\(\sigma \approx 0.75\)), less where it's very weak (review is wasted) or very strong (review finds nothing). This parallels Shannon's power allocation across channels.

\[\alpha^*(x) = \min\left(\alpha_\text{max}, \frac{\lambda}{2} \sigma_\text{raw}(x) \sqrt{\sigma_\text{raw}(x)(1 - \sigma_\text{raw}(x))}\right)\]

The key insight: oversight is not a uniform slider. It's a resource that should be allocated where it produces the most quality improvement per unit of governance cost.