Complexity Pricing

How do you price uncertainty that comes from structural complexity rather than from the underlying risk itself?

The Core Problem

Standard risk pricing assumes you can estimate probability distributions for losses. But complex organizational structures create epistemic uncertainty—uncertainty about your uncertainty.

Risk Type	What You Know	How to Price
Simple risk	Distribution of outcomes	Expected loss + margin
Parameter uncertainty	Distribution family, uncertain parameters	Bayesian methods, wider intervals
Structural complexity	Unknown interactions, hidden dependencies	???

Complexity doesn’t change the expected value of losses—it changes how confident you can be in that estimate.

Approaches from Other Fields

Insurance: Loading Factors

Traditional insurance uses loading factors to account for uncertainty:

Premium = Expected Loss × (1 + Loading Factor)

Source of Loading	Typical Factor
Administrative costs	10-20%
Profit margin	5-15%
Parameter uncertainty	10-30%
Model uncertainty	20-50%
Structural complexity	50-300%+

The problem: loading factors are often set by judgment rather than systematic analysis.

Reinsurance: Credibility Theory

Credibility theory balances individual experience against class statistics:

Credibility-weighted estimate = Z × (individual estimate) + (1-Z) × (class estimate)

where Z (credibility factor) depends on:

Volume of individual data
Variance in individual experience
Variance in class experience

Application to complexity: High-complexity organizations have low credibility (Z → 0) because their structure makes historical data less predictive. They’re priced closer to worst-case class estimates.

Operations Research: Robust Optimization

Robust optimization handles uncertainty sets rather than point estimates:

Minimize worst-case cost over all scenarios in uncertainty set U

For complexity pricing:

Larger uncertainty sets for complex structures
Premium reflects worst case within the set
Complexity score determines set size

Key insight: You’re not pricing expected loss—you’re pricing the worst plausible loss given structural uncertainty.

Software Engineering: Cyclomatic Complexity

McCabe (1976) defined cyclomatic complexity as the number of linearly independent paths through code:

V(G) = E - N + 2P

where E = edges, N = nodes, P = connected components.

Higher complexity correlates with:

More defects
Harder to test
Higher maintenance costs

Analogy for organizations: Delegation structures have “paths” through authority chains. More paths = harder to predict behavior.

A Framework for Complexity Scoring

Structural Factors

Factor	Contribution	Rationale
Delegation depth	+2 per layer	Each layer adds interpretation variance
Parallel delegates	+1 per delegate	Coordination uncertainty
Shared resources	+2 per shared resource	Contention and priority conflicts
Informal relationships	+3 per relationship	Undocumented authority creates surprises
Cross-layer dependencies	+2 per dependency	Bypasses normal authority flow
Ambiguous boundaries	+3 per boundary	Unclear who’s responsible
External dependencies	+2 per dependency	Less control, less visibility

Dynamic Factors

Static structure isn’t everything. Some complexity emerges from dynamics:

Factor	Contribution	Rationale
High turnover	+2	Institutional knowledge loss
Rapid growth	+3	Structure lags reality
Recent reorganization	+2	Transition period uncertainty
Multi-jurisdictional	+2 per jurisdiction	Different rules, enforcement

Information Factors

Factor	Contribution	Rationale
Poor documentation	+3	Can’t verify claims about structure
Audit findings	+1 per finding	Evidence of hidden issues
Information asymmetry	+2	Principal can’t observe agent
Opacity of decision-making	+3	Can’t attribute outcomes to causes

From Complexity Score to Uncertainty Multiplier

Empirical Calibration Approach

If we had data on organizational failures vs. complexity scores, we could fit:

Uncertainty Multiplier = f(Complexity Score)

Proposed functional form (requires empirical validation):

σ_multiplier = e^(0.15 × complexity_score)

Complexity Score	Uncertainty Multiplier
2	1.35× (±35%)
5	2.1× (±110%)
10	4.5× (±350%)
15	9.5× (±850%)
20	20× (±1900%)

Theoretical Justification

Why exponential? Each complexity factor potentially:

Introduces new failure modes (additive)
Creates interactions with existing factors (multiplicative)

If interactions dominate, we expect multiplicative (exponential) growth in uncertainty.

Confidence Interval Pricing

Given uncertainty multiplier σ_m:

Premium = Expected Loss × σ_m × Safety Factor

The safety factor depends on insurer risk tolerance and capital requirements.

Pricing Models

Model 1: Upper Bound Pricing

Price at the upper end of the confidence interval:

Premium = E[Loss] × (1 + k × σ_multiplier)

where k reflects how many standard deviations to cover (typically 1-3).

Advantage: Simple, conservative. Disadvantage: May be too expensive for high complexity.

Model 2: Variance Loading

Load premium proportional to variance:

Premium = E[Loss] + λ × Var[Loss]

For complexity: Var[Loss] = Base_Var × σ_multiplier²

Advantage: Standard actuarial practice. Disadvantage: Requires variance estimate.

Model 3: Expected Shortfall

Use Expected Shortfall (CVaR) at a high confidence level:

Premium = ES_α(Loss) where α depends on complexity

Complexity Score	α Level	Interpretation
Low (< 5)	95%	Standard tail risk
Medium (5-10)	99%	More conservative
High (10-15)	99.9%	Near worst-case
Very High (> 15)	99.99%	Extreme caution

Advantage: Coherent risk measure, tail-focused. Disadvantage: Requires distribution assumptions.

Model 4: Ambiguity Pricing

Use robust optimization over uncertainty sets:

Premium = max_{θ ∈ Θ(complexity)} E_θ[Loss]

where Θ(complexity) is the set of plausible models given complexity level.

Advantage: No distribution assumptions. Disadvantage: Defining Θ is challenging.

Practical Implementation

Step 1: Assess Complexity Score

Structured questionnaire covering:

Organization chart analysis
Process documentation review
Interview key personnel
Audit report review
External dependency mapping

Step 2: Map to Uncertainty

Use calibrated function (initially judgment-based, refined with data):

if complexity_score < 5:
    uncertainty = "low"
    multiplier = 1.5
elif complexity_score < 10:
    uncertainty = "medium"
    multiplier = 3.0
elif complexity_score < 15:
    uncertainty = "high"
    multiplier = 7.0
else:
    uncertainty = "very high"
    multiplier = 15.0+

Step 3: Price with Multiplier

Premium = Base_Premium × multiplier

where Base_Premium is the rate for a simple, well-understood structure.

Step 4: Offer Complexity Reduction Incentives

Show client how premium changes with structural changes:

Change	Complexity Δ	Premium Δ
Document informal relationships	-3	-15%
Eliminate shared resources	-2	-10%
Add clear authority boundaries	-3	-15%
Reduce to 2 layers	-2	-10%

Validation Challenges

Limited Data

Organizational failures linked to complexity are rare and idiosyncratic. Building a training dataset is difficult.

Survivorship Bias

We observe organizations that haven’t catastrophically failed. High-complexity survivors may have unobserved mitigating factors.

Confounders

Complex organizations may differ from simple ones in ways that affect risk independently of complexity.

Proposed Validation Approaches

Historical case studies: Analyze past organizational failures for complexity factors
Simulation: Agent-based models of delegation chains under stress
Expert elicitation: Structured surveys of risk professionals
Regulatory data: Insurance claim data by organizational characteristics
Near-miss analysis: Study incidents that almost became failures

Connection to Other Frameworks

Delegation Accounting

Complexity pricing extends the delegation accounting framework by quantifying the uncertainty term:

Net Delegation Value = Receivable - (Exposure × σ_multiplier) - Costs

The complexity multiplier captures how much to discount the exposure estimate based on structural uncertainty.

Pattern Interconnection

Pattern interconnection creates complexity when defense patterns share failure modes. Complexity pricing provides a method to quantify this “correlation tax.”

Compositional Risk Measures

Compositional risk measures assume you can decompose system risk. Complexity represents the degree to which this decomposition fails—interactions that can’t be captured by analyzing components separately.

Open Questions

Optimal complexity scoring: Which factors matter most? How should they be weighted?
Functional form: Is the uncertainty multiplier truly exponential? Could it be polynomial, logistic, or stepped?
Industry variation: Do complexity factors have different impacts in different domains?
Dynamic complexity: How do you price complexity that changes over time?
Complexity reduction ROI: What’s the most cost-effective way to reduce complexity for insurance purposes?
AI-specific factors: What additional complexity factors apply to AI systems (emergent behavior, capability uncertainty, alignment uncertainty)?

Key Citations

Insurance and Actuarial

Bühlmann & Gisler (2005) - A Course in Credibility Theory
Klugman, Panjer, Willmot (2012) - Loss Models
Daykin, Pentikäinen, Pesonen (1994) - Practical Risk Theory for Actuaries

Robust Optimization

Ben-Tal, El Ghaoui, Nemirovski (2009) - Robust Optimization
Bertsimas & Brown (2009) - “Constructing Uncertainty Sets for Robust Linear Optimization”

Software Complexity

McCabe (1976) - “A Complexity Measure” - IEEE Transactions on Software Engineering
Halstead (1977) - Elements of Software Science

Organizational Complexity

Perrow (1984) - Normal Accidents
Reason (1997) - Managing the Risks of Organizational Accidents
Leveson (2011) - Engineering a Safer World

Ambiguity and Model Uncertainty

Ellsberg (1961) - “Risk, Ambiguity, and the Savage Axioms”
Gilboa & Schmeidler (1989) - “Maxmin Expected Utility”
Hansen & Sargent (2001) - “Robust Control and Model Uncertainty”