# **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 |

---

### **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:

```text
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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