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title: Untitled

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## Toy Example

- Exposure & Utility

| Rank | 1   | 2   | 3   |
| --- | --- | --- | --- |
| Exposure ($v$) | 1   | 0.5 | 0   |
| Utility Coefficient ($u$) | 1   | 0.2 | 0   |

- Relevances

| Query | q1  | q2  | q3  | q4  |
| --- | --- | --- | --- | --- |
| $d_1$ | 1   | 1   | 1   | 0.1 |
| $d_2$ | 1   | 1   | 1   | 0.1 |
| $d_3$ | 2/3 | 1/3 | 0   | 0.11 |

- Rank without intervention

| Query | q1  | q2  | q3  | q4  |
| --- | --- | --- | --- | --- |
| $d_3$ | 3   | 3   | 3   | 1   |

Exposure of $d_3$: 1

- Exposure Constraint of $d_3$
  
  Setpoint: $2$
  

### P-Controller

1. $\lambda \lt \frac{2}{3}$

###### Error

| Query | q1  | q2  | q3  | q4  |
| --- | --- | --- | --- | --- |
| $d_3$ | 0.5 | 1   | 1.5 | 1   |

- Relevances

| Query | q1  | q2  | q3  | q4  |
| --- | --- | --- | --- | --- |
| $d_1$ | 1   | 1   | 1   | 0.1 |
| $d_2$ | 1   | 1   | 1   | 0.1 |
| $d_3$ | $2/3 +\lambda/2$ | $1/3 +\lambda$ | $3\lambda/2$ | 0.11+$\lambda$ |

###### Rank

| Query | q1  | q2  | q3  | q4  |
| --- | --- | --- | --- | --- |
| $d_3$ | 3   | 3   | 3   | 1   |

Does not satisfy the constraint!

2. $\lambda > 2$

###### Error

| Query | q1  | q2  | q3  | q4  |
| --- | --- | --- | --- | --- |
| $d_3$ | 0.5 | 0   | 0.5 | 1   |

- Relevances

| Query | q1  | q2  | q3  | q4  |
| --- | --- | --- | --- | --- |
| $d_1$ | 1   | 1   | 1   | 0.1 |
| $d_2$ | 1   | 1   | 1   | 0.1 |
| $d_3$ | $2/3 +\lambda/2$ | $1/3$ | $\lambda/2$ | 0.11+$\lambda$ |

###### Rank

| Query | q1  | q2  | q3  | q4  |
| --- | --- | --- | --- | --- |
| $d_3$ | 1   | 3   | 1   | 1   |

3. $2 >\lambda > \frac{2}{3}$

###### Error

| Query | q1  | q2  | q3  | q4  |
| --- | --- | --- | --- | --- |
| $d_3$ | 0.5 | 0   | 0.5 | 1   |

- Relevances

| Query | q1  | q2  | q3  | q4  |
| --- | --- | --- | --- | --- |
| $d_1$ | 1   | 1   | 1   | 0.1 |
| $d_2$ | 1   | 1   | 1   | 0.1 |
| $d_3$ | $2/3 +\lambda/2$ | $1/3$ | $\lambda/2$ | 0.11+$\lambda$ |

###### Rank

| Query | q1  | q2  | q3  | q4  |
| --- | --- | --- | --- | --- |
| $d_3$ | 1   | 3   | 3   | 1   |

### Online-Learning Lambda
###### Error
learning_rate = 1

| Query | q1  | q2  | q3  | q4  |
| --- | --- | --- | --- | --- |
| $d_3$ | 0.5  | 0.16  | 0.5 |  1.  |
| $\lambda$ | 0  | 0.5 | 0.66 | 1.16   |

- Relevances

| Query | q1  | q2  | q3  | q4  |
| --- | --- | --- | --- | --- |
| $d_1$ | 1   | 1   | 1   | 0.1 |
| $d_2$ | 1   | 1   | 1   | 0.1 |
| $d_3$ | $2/3$ | $1/3$+$1/2$ | $0$ + $2/3$ | $0.11$ + $1.16$ |

###### Rank

| Query | q1  | q2  | q3  | q4  |
| --- | --- | --- | --- | --- |
| $d_3$ | 3   | 1   | 2   |  2  |


### SMPC

Rank

| Query | q1  | q2  | q3  | q4  |
| --- | --- | --- | --- | --- |
| $d_3$ | 2   | 3   | 2   | 1   |