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# **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.
---
## **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 |
---
### **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.
---
### **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.
---
### **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.
---
### **Decision Card (Result)**
```
Decision: Choose A2
Rationale:
- Expected Loss optimal
- Minimax Regret optimal
- CVaR optimal
```
All lenses agree. This represents a **fully aligned decision**.
---
## **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
---
## **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**.
---
## **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**.
---
## **Appendix E — Integration Positioning**
Decision Kernel Lite is designed to sit between analytics and action:
```text
Forecasts → Scenarios → Probabilities → Losses
↓
Decision Kernel Lite
↓
Action / Policy / Price
```
It does not replace forecasting or optimization.
It **binds them into a decision**.
---
## **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.
---
## **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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