But what about when the cake is not all the same? Maybe it’s topped with a few icing roses or artfully placed cookies. The corner pieces may have more frosting. A maraschino cherry–lover might be happy with the smallest slice if they get the cake’s only cherry. To them, that piece is more valuable than a larger slice.
For two people, I-cut-you-choose still works with a non-uniform cake. The divider cuts the cake into two pieces they view as equal; the chooser then picks their preferred piece. But add more cake eaters, each with their own preferences, and easy solutions crumble.
More eaters, more cuts
Hugo Steinhaus was one of the first mathematicians to dive into this complexity. He worked at the University of Warsaw in Poland in the 1940s. During World War II, he saw questions about fair division of land playing out on a large and violent scale. Steinhaus came up with a modified I-cut-you-choose strategy for three players.
It came to be called the lone-divider method.
Here, someone cuts the cake. Let’s call her Alice. She cuts three pieces that she values equally (each at 1/3 of the total). Then a second person, Bob, says which of the pieces he would accept. If he approves at least two pieces, then the third person, Carla, can take any piece she wants. At least one of the remaining pieces is acceptable to Bob. So he picks next. Alice gets what’s left.
If Bob and Carla both turn down the same piece, that piece goes to Alice. She valued all pieces equally, so it still seems fair to her. The remaining two pieces are recombined and shared between Bob and Carla using I-cut-you-choose.
Steinhaus described this method in a short paper published in 1948. It was one of the first serious studies in the field of cake-cutting. And it worked for three eaters.
Ariel Procaccia has thought a lot about how to cut cake over the past 15 years. That’s partly because he has three children. As a group, they’ve celebrated more than …read more
Source:: Science Explores
