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# **Technical Brief — Decision Kernel Lite**

## **Purpose**

This document specifies the **mathematical definitions**, **algorithms**, and **implementation choices** used in Decision Kernel Lite.

The goal is not theoretical novelty, but **correct, transparent, and reproducible decision logic**.

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## **Formal Problem Definition**

Let:

* Actions: ( a \in {1,\dots,A} )
* Scenarios: ( s \in {1,\dots,S} )
* Scenario probabilities: ( p_s \ge 0,\ \sum_s p_s = 1 )
* Loss matrix: ( L_{a,s} \in \mathbb{R}_{\ge 0} )

A decision consists of selecting one action ( a^* ) under uncertainty about which scenario ( s ) will occur.

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## **Decision Lenses**

Decision Kernel Lite evaluates each action using three lenses.

### **1. Expected Loss**

[
\text{EL}(a) = \sum_{s=1}^{S} p_s \cdot L_{a,s}
]

**Interpretation**

* Risk-neutral criterion
* Minimizes long-run average loss

**Implementation**

```python
expected_loss = loss_matrix @ probabilities
```

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### **2. Regret and Minimax Regret**

**Regret matrix**

[
R_{a,s} = L_{a,s} - \min_{a'} L_{a',s}
]

**Max regret per action**

[
\text{MR}(a) = \max_s R_{a,s}
]

**Decision rule**

[
a^* = \arg\min_a \text{MR}(a)
]

**Interpretation**

* Robust to probability misspecification
* Optimizes post-hoc defensibility

**Implementation**

```python
regret = loss - loss.min(axis=0, keepdims=True)
max_regret = regret.max(axis=1)
```

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### **3. CVaR (Conditional Value at Risk)**

CVaR evaluates **average loss in the worst tail of outcomes**.

#### **Procedure (discrete)**

1. Sort losses ( L_{a,s} ) in ascending order
2. Sort probabilities accordingly
3. Compute cumulative probability
4. Select outcomes where cumulative probability ≥ ( \alpha )
5. Compute probability-weighted average over the tail

[
\text{CVaR}_\alpha(a)
=====================

\mathbb{E}[L_{a,s} \mid \text{worst } (1-\alpha) \text{ mass}]
]

**Interpretation**

* Tail-risk-aware
* Penalizes catastrophic downside

**Implementation**

```python
order = np.argsort(losses)
cum = np.cumsum(probs[order])
tail = cum >= alpha
cvar = (losses[order][tail] * probs[order][tail]).sum() / probs[order][tail].sum()
```

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## **Probability Normalization**

To guarantee numerical and logical safety:

* Negative probabilities are clipped to zero
* Zero-sum vectors default to uniform probabilities
* All probabilities are normalized to sum to 1

```python
p = np.clip(p, 0, None)
p = p / p.sum() if p.sum() > 0 else np.ones_like(p) / len(p)
```

This ensures the kernel **never fails due to malformed inputs**.

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## **Rule Selection Heuristic**

Decision Kernel Lite includes an **advisory heuristic**:

[
\text{Tail Ratio} =
\frac{\min_a \text{CVaR}_\alpha(a)}{\min_a \text{EL}(a)}
]

* If Tail Ratio ≥ 1.5 → recommend **CVaR**
* Otherwise → recommend **Expected Loss**

**Properties**

* Transparent
* Non-binding
* Overridable by the user

This preserves governance while guiding non-technical users.

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## **Decision Logic Flow**

```text
Inputs
  ↓
Normalize probabilities
  ↓
Compute:
  - Expected Loss
  - Regret matrix
  - Max Regret
  - CVaR
  ↓
Select primary rule
  ↓
Choose action
  ↓
Generate Decision Card
```

All computations are deterministic and stateless.

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## **System Architecture**

* **Core kernel**: pure NumPy (testable, embeddable)
* **UI layer**: Streamlit (input + presentation only)
* **Deployment**: Docker / Hugging Face Spaces

There is no persistence, no external API, and no hidden state.

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## **Design Constraints (Intentional)**

The system deliberately avoids:

* forecasting
* probability estimation
* optimization solvers
* learning or calibration

These belong to **adjacent layers**, not the decision kernel.

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## **Computational Complexity**

Let ( A ) = number of actions, ( S ) = number of scenarios.

* Expected Loss: ( O(A \cdot S) )
* Regret: ( O(A \cdot S) )
* CVaR: ( O(A \cdot S \log S) )

The system is effectively instantaneous for realistic decision sizes.

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## **Reproducibility & Auditability**

* Deterministic outputs
* Explicit assumptions
* Full transparency of trade-offs
* Copy/paste Decision Card

This makes the kernel suitable for:

* executive decisions
* governance reviews
* post-decision audits

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## **Summary**

Decision Kernel Lite implements **classical, defensible decision theory** in a minimal, production-ready form.

It transforms uncertainty from an excuse into a **structured input**.

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