This tool can be used to distribute a pot of money (such as an estate) equitably among a set of people based on their income levels. The less you have, the more you get. This page explains the fair distribution process using mathematical equations. The goal is to distribute an estate among a group of individuals based on their respective incomes while considering fairness. To read more on why this was created, you can read my blog post on the subject here.

Let \( n \) be the number of individuals (size of the "incomes" array).

Let \( \text{incomes}[i] \) represent the income of the \( i \)th individual, where \( i \) ranges from 0 to \( n-1 \).

The total income, \( \text{totalIncome} \), is the sum of all individual incomes:

\[ \text{totalIncome} = \sum_{i=0}^{n-1} \text{incomes}[i] \]

The income share, \( \text{incomeShares}[i] \), of the \( i \)th individual is calculated as:

\[ \text{incomeShares}[i] = \frac{\text{incomes}[i]}{\text{totalIncome}} \]

Let \( \text{fairnessAdjustment} \) be a constant value used for fairness adjustment. This will range from 0-∞, with 0 being equal amounts going to every person, and the higher you go the more unequal it gets. The slider in the tool is logarithmic, to account for the more significant changes the closer to 0 you are.

The adjusted income share, \( \text{adjustedIncomeShares}[i] \), of the \( i \)th individual is calculated as:

\[ \text{adjustedIncomeShares}[i] = (1 - \text{incomeShares}[i])^{\text{fairnessAdjustment}} \]

The sum of all adjusted income shares after fairness adjustment is:

\[ \text{sumAdjustedIncomeShares} = \sum_{i=0}^{n-1} \text{adjustedIncomeShares}[i] \]

The normalized share, \( \text{normalizedShares}[i] \), of the \( i \)th individual is calculated as:

\[ \text{normalizedShares}[i] = \frac{\text{adjustedIncomeShares}[i]}{\text{sumAdjustedIncomeShares}} \]

Let \( \text{estateAmount} \) be the total amount to be distributed among the individuals.

The individual allocation, \( \text{individualAllocations}[i] \), of the \( i \)th individual is calculated as:

\[ \text{individualAllocations}[i] = \text{estateAmount} \times \text{normalizedShares}[i] \]

**Putting it all together:**

$$ \text{individualAllocations}[i] = \text{estateAmount} \times \frac{(1 - \frac{\text{incomes}[i]}{\sum_{j=0}^{n-1}\text{incomes}[j]})^{\text{fairnessAdjustment}}}{\sum_{i=0}^{n-1} (1 - \frac{\text{incomes}[i]}{\sum_{j=0}^{n-1}\text{incomes}[j]})^{\text{fairnessAdjustment}}} $$

In this equation:

- \( n \) is the number of individuals (size of the "incomes" array).
- \( \text{incomes}[i] \) represents the income of the \( i \)th individual, where \( i \) ranges from 0 to \( n-1 \).
- \( \sum_{{j=0}}^{n-1}\text{incomes}[j] \) is the total income, which is the sum of all individual incomes.
- \( \text{fairnessAdjustment} \) is a constant value used for fairness adjustment.
- The equation calculates the fair allocation, \( \text{individualAllocations}[i] \), for each individual, taking into account their income share, the fairness adjustment, and the normalization step to ensure that the sum of individual allocations equals the \( \text{estateAmount} \).

This equation represents the fair distribution of the "estateAmount" among the individuals based on their incomes and the fairness adjustment applied. The normalization step ensures that the sum of individual allocations is equal to the "estateAmount," creating a balanced distribution that takes into account income disparities and fairness considerations.