Three roommates rent a flat for $3,000 a month. One room has a private bathroom, one is a good size and one is small. Splitting $1,000 each is easy and quietly unfair: the person in the small room subsidises the one with the bathroom. This page shows a method where nobody can say that.
Each person privately spreads the $3,000 across the three rooms. More on one room means less on the others. The invented numbers:
| Person | Master with bath | Big bedroom | Small bedroom |
|---|---|---|---|
| Ana | $1,400 | $900 | $700 |
| Ben | $900 | $1,200 | $900 |
| Cleo | $1,000 | $1,000 | $1,000 |
Of the six ways to hand out three rooms, the highest total value is Ana in the master ($1,400), Ben in the big bedroom ($1,200) and Cleo in the small one ($1,000): $3,600 of value for $3,000 of rent, so $600 of surplus to share out.
The prices must add up to $3,000, and at those prices nobody should prefer another room. Among all such prices, the tool picks the ones where the least-happy person is as happy as possible. Here the surplus divides evenly:
| Person | Room | Rent | Their value | Surplus |
|---|---|---|---|---|
| Ana | Master with bath | $1,200 | $1,400 | $200 |
| Ben | Big bedroom | $1,000 | $1,200 | $200 |
| Cleo | Small bedroom | $800 | $1,000 | $200 |
Check it: Ana would pay $1,000 for the big room and get $900 of value, a $100 loss against her $200 gain. Ben in the master at $1,200 would get $900 of value, a $300 loss. Cleo would be happy anywhere at a fair price but pays the least. Nobody gains by swapping.
Because everyone has the same total to spread, exaggerating a favourite room means undervaluing the others, and an inflated room costs more. Compare notes after, not before.
Square feet ignore what people actually value. A 90 square foot room with its own bathroom and a window onto the street can be worth more than a 140 square foot room that is a walk-through. Splitting by area gives the wrong rent to whoever's tastes differ from the average.
Have everyone privately split the total rent across the rooms to show what each room is worth to them, then assign rooms to maximise total value and set prices so nobody prefers another room. That is envy-free rent division, and the tool does the arithmetic.
Rarely, when everyone dislikes a room strongly enough that someone has to be paid to take it. The tool shows the figure as it is rather than hiding it.