Mathematical Models for Wardley Mapping

This repository contains research and theoretical frameworks for formalizing Wardley Mapping as a mathematical model. The goal is to transform Wardley Maps from a purely visual and qualitative strategic tool into a quantitative framework that enables computation, simulation, and data-driven strategic decision-making.

What is Wardley Mapping?

Wardley Mapping, created by Simon Wardley, is a visual technique for mapping business strategy. A Wardley Map plots components on two axes:

• Y-axis (Visibility): How close a component is to the end user
• X-axis (Evolution): How mature/commoditized a component is, from Genesis → Custom → Product → Commodity

Components are connected by dependency relationships forming a value chain.

The Mathematical Approach

This repository explores how to formalize these concepts mathematically. A Wardley Map becomes:

$$\mathcal{M} = (V, E, U, \nu, \varepsilon, t)$$

Where:

• $V$ = set of components
• $E \subseteq V \times V$ = directed dependency edges
• $U \subseteq V$ = anchor set (one or more user-need nodes)
• $\nu: V \to [0,1]$ = visibility function (Y-axis)
• $\varepsilon: V \to [0,1]$ = evolution function (X-axis)
• $t$ = time parameter for dynamics

This formalization enables:

• Generating maps from data
• Validating maps with constraints
• Quantifying strategic positions
• Simulating evolution over time
• Computing decision metrics (differentiation pressure, commodity leverage, dependency risk)

Recommended Reading Order

The repo has 20+ docs. Not all are on the critical path. If you're new:

Start here (the core):

1. Part 1 — Core Mathematical Model. Read this first. Everything else extends it.
2. Part 2 — "Map Evolution, Not Maturity". Deepens the evolution axis interpretation.
3. Part 3 — Tea Shop worked example. Grounds the math in a concrete map.
4. Part 4 — Simple evolution scoring via ubiquity + certainty.
5. Part 5 — Layer-based visibility and sigmoid evolution (refinements).
6. Part 6 — Canonical cheat-sheet scoring (19 rows). Supersedes Part 4's 2-factor method.

Then pick the extensions relevant to your problem:

• Inertia — if you're modelling why components get stuck (structured drag).
• Multi-Wave Evolution — if your horizon spans multiple technology generations.
• Component Types — if you want to distinguish Activities / Practices / Data / Knowledge.

Then the strategy layer:

• Gameplay Catalogue — reference list of the 61 plays with math-model effects.
• Doctrine — the 40 universal principles and how they constrain the model.
• Strategic Mastery and Mathematical Models for Wardley Mapping Gameplay — older, longer companion treatments (applied and formal respectively) to the Gameplay Catalogue.

Specialised applications (read if the topic applies):

• Weak Signals & Evolution — detecting when evolution is about to happen.
• Wardley Strategy Cycle — formalising the OODA-loop-style strategy cycle.

Not on the main path:

• The Mathematical Framework — long encyclopedic reference (1200+ lines). Browse for specific techniques, don't read front-to-back.

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Documents

All docs live under docs/ organised by role.

Core — the canonical progressive series (docs/core/)

Document	Description
Part 1 — Core Mathematical Model	The formal tuple model $\mathcal{M} = (V, E, U, \nu, \varepsilon, t)$ with computed visibility, evolution scoring, and derived metrics
Part 2 — Evolution, Not Maturity	Refining the evolution axis interpretation
Part 3 — The Tea Shop Worked Example	Applying the model to a classic Wardley Map
Part 4 — Single-Component Evolution	Methods for computing evolution scores
Part 5 — Layer Visibility & Sigmoid Evolution	Discrete dependency layers for visibility and a sigmoid logistic curve for evolution
Part 6 — Cheat-Sheet Evolution Scoring	Wardley's canonical 19-row cheat sheet with a formal scoring procedure that produces $\varepsilon(v)$ with uncertainty
Mathematical Framework	Long encyclopedic reference (1200+ lines). Browse for specific techniques, don't read front-to-back

Extensions — additions to the Part-1 tuple (docs/extensions/)

Document	Description
Inertia	Wardley's 17 forms of inertia (14 consumer + 3 supplier) with a structured drag term $c_v(t) = \sum \lambda_i \iota_i$ replacing the single scalar
Multi-Wave Evolution	Replaces single-logistic dynamics with multiple diffusion curves per component (generations + chasms)
Component Types	Extends the tuple with $\tau: V \to {A, P, D, K}$ (Activities, Practices, Data, Knowledge) and type-dependent evolution rates

Catalogues — reference tables Wardley published (docs/catalogues/)

Document	Description
Gameplay	Wardley's 61 gameplays across 12 categories, each mapped to a structured effect on $r_v, c_v, \nu, \varepsilon$ or $E$
Doctrine	Wardley's 40 doctrine principles (4 phases × 6 categories) with math-model readings of each

Strategy — cycles, signals, older gameplay treatments (docs/strategy/)

Document	Description
Strategy Cycle — Core	Modeling strategic cycles
Strategy Cycle — Example	Applied example of strategy cycles
Weak Signals — Core	Detecting and modeling weak signals
Weak Signals — Example	Applied example of weak signal detection
Strategic Mastery	Older companion treatment of gameplay selection (predates catalogues/gameplay.md)
Gameplay Math Models	Older quantitative treatment of plays (predates catalogues/gameplay.md)

Tools & Prompts

Document	Description
Wardley Map Generator Prompt	AI prompt for generating Wardley Maps in OWM format compatible with create.wardleymaps.ai
wardley-map Claude Code skill	Portable skill package — copy to ~/.claude/skills/ and invoke /wardley-map <scenario>. SKILL.md + references/ (7 files: climatic-patterns, doctrine, evolution-stages, gameplay-patterns, inertia, mapping-examples, mathematical-models)

Key Formulas

Visibility from graph distance:

$$\nu(v) = \frac{1}{1 + d(v)}$$

where $d(v)$ is the shortest path length from user to component.

Evolution dynamics (logistic S-curve):

$$\frac{d\varepsilon}{dt} = r(t) \cdot \varepsilon(t) \cdot (1 - \varepsilon(t))$$

where $r(t)$ incorporates market forces and strategic actions.

Caveat. Wardley's climatic patterns state "you cannot measure evolution over time or adoption." The ODE above is a stylized extension for simulation and scenario exploration, not a Wardley-endorsed forecast model.

Evolution stages (canonical names, quartile bands are conventional):

• Genesis: $[0, 0.25)$
• Custom Built: $[0.25, 0.50)$
• Product (+rental): $[0.50, 0.75)$
• Commodity (+utility): $[0.75, 1.0]$

Decision metrics (heuristics proposed in this repo — not canonical Wardley concepts):

• Differentiation pressure: $D(v) = \nu(v) \cdot (1 - \varepsilon(v))$
• Commodity leverage: $K(v) = (1 - \nu(v)) \cdot \varepsilon(v)$
• Dependency risk: $R(a,b) = \nu(a) \cdot (1 - \varepsilon(b))$

License

This repository contains theoretical research and documentation.

Acknowledgments

Based on the Wardley Mapping framework created by Simon Wardley.