## 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 |