Math in… Fair Division
Fair division is the problem of splitting resources in a way that makes all parties feel they got at least their fair share. When it comes to dividing a sandwich, “fair share” could mean a big piece, or a small piece with lots of toppings, or a piece without crusts. But how do we measure fairness?
One way to measure fairness is in terms of envy. If two people split something, we say the division is envy-free if neither person prefers the other’s portion over their own. They could strongly prefer their own portions or think the portions are equally good, but either way they wouldn’t want to switch.
If three friends want to share a sandwich with no envy, each could think of two places to slice, so they would be fine taking any of the three slices.
All three then place a toothpick where their leftmost cut would be.
Whoever’s toothpick is furthest to the left then places a second toothpick where their rightmost cut would be.
At this point, one of three things can happen:
1
If the other two friends prefer the middle section, the toothpicking friend takes the toothpicks and moves them inward along the middle section, keeping the growing endpieces equally desirable to themself.
When one of the endpieces looks as desirable as the middle piece to one of the other friends, they shout “stop!” and the toothpicks get planted.
The sandwich is then cut, with the silent friend taking the (now smaller) middle, the shouter taking the newly desirable endpiece, and the toothpicker taking the remaining endpiece.
2
If the other two friends both prefer the same endpiece (e.g., right), the toothpicker moves the toothpicks toward that end, keeping the other endpiece and middle piece equally desirable to themself.
When one of the other pieces looks as desirable as the right piece to one of the other friends, they shout “stop!” and the toothpicks get planted.
The sandwich is then cut with the silent friend taking the (now smaller) right piece, the shouter taking the newly desirable left or middle, and the toothpicker taking the remaining piece.
3
If the other two friends each prefer different sections, the sandwich is cut with the two friends taking their preferred slices while the toothpicker takes the remaining one.
