docs/Appendix.md

Appendix — Decision Kernel Lite

Purpose

This appendix consolidates worked examples, edge cases, and interpretation notes supporting Decision Kernel Lite.

It is intended for:

• reviewers and auditors
• advanced users
• downstream system integrators

This appendix is reference-only and complements the Executive and Technical Briefs.

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Appendix A — Worked Example (Baseline Case)

Inputs

Actions: A1, A2, A3 Scenarios: Low, Medium, High Probabilities: 0.30, 0.40, 0.30

Loss Matrix

Action	Low	Medium	High
A1	10	5	1
A2	6	4	6
A3	2	6	12

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Expected Loss

[ \text{EL}(A1)=5.3,\quad \text{EL}(A2)=5.2,\quad \text{EL}(A3)=6.6 ]

Optimal: A2

Interpretation: A2 minimizes average loss given the stated probabilities.

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Regret Matrix

Action	Low	Medium	High	Max Regret
A1	8	1	0	8
A2	4	0	5	5
A3	0	2	11	11

Optimal (Minimax Regret): A2

Interpretation: A2 minimizes worst-case hindsight regret.

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CVaR @ 0.8

With three discrete scenarios, CVaR selects outcomes in the worst 20% tail, which collapses to the worst scenario.

Action	CVaR@0.8
A1	10
A2	6
A3	12

Optimal (CVaR): A2

Interpretation: A2 has the lowest average loss conditional on being in the tail.

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Decision Card (Result)

Decision: Choose A2

Rationale:
- Expected Loss optimal
- Minimax Regret optimal
- CVaR optimal

All lenses agree. This represents a fully aligned decision.

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Appendix B — When Decision Lenses Disagree

Disagreement between lenses is expected and informative.

Situation	Expected Loss	Minimax Regret	CVaR
Aggressive upside bet	Favors	Rejects	Rejects
Conservative safety choice	Rejects	Neutral	Favors
High accountability / political risk	Neutral	Favors	Neutral

Guidance:

• Do not average lenses
• Select the rule that matches the risk posture
• Document the choice explicitly

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Appendix C — CVaR in Discrete Scenario Settings

In small discrete scenario sets:

• CVaR approximates worst-case average
• This behavior is correct by definition

As the number of scenarios increases:

• CVaR becomes smoother
• Tail behavior is better resolved

Decision Kernel Lite intentionally operates in the discrete regime.

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Appendix D — Probability Misspecification

When probabilities are uncertain or contested:

• Expected Loss becomes fragile
• Minimax Regret remains valid
• CVaR protects against catastrophic misestimation

Operational rule: If probabilities are debated → prefer Regret or CVaR.

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Appendix E — Integration Positioning

Decision Kernel Lite is designed to sit between analytics and action:

Forecasts → Scenarios → Probabilities → Losses
                           ↓
                    Decision Kernel Lite
                           ↓
                    Action / Policy / Price

It does not replace forecasting or optimization. It binds them into a decision.

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Appendix F — Design Exclusions (Intentional)

Decision Kernel Lite deliberately excludes:

• forecasting models
• probability estimation
• optimization solvers
• learning or calibration

Rationale:

• forecasting belongs upstream
• optimization belongs downstream
• decision justification belongs here

This separation preserves clarity, auditability, and governance.

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Appendix G — Audit & Governance Notes

• Deterministic computations
• Explicit assumptions
• No hidden state
• Copy/paste Decision Card output

This makes the kernel suitable for:

• executive review
• governance committees
• post-decision audits

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