On Cornell Engineering Week: Dividing up a bundle of items fairly can be very tricky, especially for families.
Paul Gölz, assistant professor of operations research and information engineering, looks to mathematics for help.
Paul Gölz is an assistant professor at Cornell University. He studies the algorithms and mathematics of democracy and fairness, and how these fields can inform AI development.
Picture three siblings dividing an inheritance. One is fond of a piece of art, the others could both use the car, everyone wants the dining table, … and suddenly they’re fighting over how to split the goods. Whether you’re splitting an estate or Halloween candy, mathematics can help by making precise when a division is “fair” and by creating procedures that find one.
Ideally, we’d like a division that is envy-free, which means that no sibling prefers another sibling’s bundle of items over their own. But perfect envy-freeness isn’t always possible—say if everyone wants the car.
So, researchers in the area of fair division developed a more flexible standard of fairness, which allows for some envy, but only a small amount: your envy for another’s bundle should disappear if you could remove just one item from it. This axiom works well because it’s always achievable when dividing items across individuals.
But here is a twist: what if the siblings have spouses? A division that seems fair to the siblings could still leave a spouse thinking their household got shortchanged, based on how the spouse values the items. Can we divide them so that the siblings and their spouses find the allocation fair?
For two couples, we prove that the answer is yes; such a division always exists. But add another couple, and sometimes no division will satisfy everyone. That’s the bad news.
The good news? We developed an algorithm that works for any number of couples, which guarantees a different fairness axiom called proportionality: every person feels that their group received at least their fair share of the total value, give or take a few items. This algorithm offers a mathematically fair way to divide goods among households or other groups of people—and hopefully to avoid an argument.
