Ariel Procaccia has thought so much about learn how to reduce cake over the previous 15 years. That’s partly as a result of he has three kids. As a gaggle, they’ve celebrated greater than two dozen birthdays. So Procaccia is aware of what it’s like to face with a knife earlier than a tempting dessert. Candy layers of cake. Buttercream frosting and chocolate curls. And a crowd of small, keen partygoers who will immediately spot if another person will get a greater slice.
However Procaccia can be a pc scientist at Harvard College in Cambridge, Mass. There, he research the mathematical guidelines for dividing stuff up. And dessert is a useful manner to consider that. It comes right down to a deceptively easy query about equity: How will you reduce a cake to verify everybody on the occasion is pleased with what they get?
The solutions attain far past birthday events. For greater than 75 years, mathematicians have puzzled over learn how to pretty divide assets. Such questions have real-world makes use of. How can meals be divvied up between hungry communities, for instance? How ought to roommates break up up hire or chores? How can communities draw boundaries for honest voting districts.
These questions embrace greater than math, too. They have to contemplate what individuals choose and different points. In order that they turn into fascinating to scientists, economists and authorized specialists.
However cake works as a stand-in for something that may be divided, says Steven Brams. He’s a sport theorist and political scientist at New York College (NYU) in New York Metropolis. Cake-cutting concepts can simply be utilized to splitting up land, time or different restricted assets.
Recipes for honest cake-cutting
Specialists have provide you with many guidelines, referred to as algorithms, for learn how to reduce a cake pretty. (Practically all give attention to rectangular truffles. The associated however newer “pie-cutting” downside addresses round desserts or pizza.) The only guidelines present how two individuals can pretty share a cake: One particular person cuts the cake into two items that they consider to be equal in worth. The opposite particular person chooses between them. Every eater receives a bit that they really feel is at the least as priceless as the opposite’s, if not higher.
Stories of this technique date again to historic Greece.
Within the Forties, mathematicians began utilizing cake-cutting as a strategy to research equity. The “I reduce, you select” technique works for 2 individuals. What about sharing amongst three or extra? That has led to new challenges, similar to: What’s equity, precisely — and the way do you show it?
Right here’s a technique to consider equity. Perhaps every particular person is happy in the event that they really feel like their slice represents a fair proportion of the full. For 2 individuals, a fair proportion can be 1/2; for 3, it might be 1/3, and so forth. (And for some arbitrary n variety of cake eaters, a fair proportion can be 1/n.) If the cake is identical all through, all of the slices simply must be the identical measurement. That’s not too onerous to handle with a knife and a ruler or kitchen scale.
However what about when the cake isn’t all the identical? Perhaps it’s topped with just a few icing roses or artfully positioned cookies. The nook items could have extra frosting. A maraschino cherry–lover could be pleased with the smallest slice in the event that they get the cake’s solely cherry. To them, that piece is extra priceless than a bigger slice.
For 2 individuals, I-cut-you-choose nonetheless works with a non-uniform cake. The divider cuts the cake into two items they view as equal; the chooser then picks their most well-liked piece. However add extra cake eaters, every with their very own preferences, and simple options crumble.
Extra eaters, extra cuts
Hugo Steinhaus was one of many first mathematicians to dive into this complexity. He labored on the College of Warsaw in Poland within the Forties. Throughout World Conflict II, he noticed questions on honest division of land taking part in out on a big and violent scale. Steinhaus got here up with a modified I-cut-you-choose technique for 3 gamers.
It got here to be referred to as the lone-divider technique.
Right here, somebody cuts the cake. Let’s name her Alice. She cuts three items that she values equally (every at 1/3 of the full). Then a second particular person, Bob, says which of the items he would settle for. If he approves at the least two items, then the third particular person, Carla, can take any piece she needs. At the least one of many remaining items is suitable to Bob. So he picks subsequent. Alice will get what’s left.
If Bob and Carla each flip down the identical piece, that piece goes to Alice. She valued all items equally, so it nonetheless appears honest to her. The remaining two items are recombined and shared between Bob and Carla utilizing I-cut-you-choose.
Steinhaus described this technique in a brief paper revealed in 1948. It was one of many first severe research within the discipline of cake-cutting. And it labored for 3 eaters.
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In the identical paper, although, he mentioned an algorithm developed by two of his colleagues. This “last-diminisher” technique might work for any variety of cake eaters.
Right here, somebody cuts off a bit of cake they assume is a fair proportion and passes it to the following particular person. One another particular person on the occasion has a alternative. They will move the piece alongside, in the event that they agree it’s honest or lower than a fair proportion. Or, in the event that they assume it’s too huge, they will trim it. As soon as everybody has had an opportunity to trim, or “diminish” the slice, the final one that trimmed will get the piece and exits the sport.
Any bits trimmed off are added again to the remaining cake, and the method begins once more with the remaining gamers. When solely two gamers are left, they use I-cut-you-choose.
The last-diminisher technique ensures that everybody judges their very own piece to be at the least a fair proportion. But it surely’s not excellent. That’s as a result of it doesn’t account for envy. In each the lone-divider and last-diminisher approaches, an individual who will get an early slice might even see a later one they need that they had as a substitute — though that they had at first thought their piece was honest.
Final-diminisher has one other flaw, too. If a number of individuals trim, the cake could also be decreased to crumbs in later rounds. And that may really feel unfair regardless of how huge your pile is.
Can cake reducing be freed from envy?
After the lone diminisher, mathematicians continued to seek for envy-free methods to divide one thing. Within the Nineteen Sixties, John Conway and John Selfridge every got here up with the identical thought for a way three individuals can every really feel they received a fair proportion, with no envy by others.
Of their plan, Alice first splits the cake into three items. She believes every are of equal worth. Then, Bob can trim one piece — at most — to create a two-way tie for essentially the most priceless. (Any trimmings are put aside.) Carla is left to decide on among the many three. Then the order reverses. If Carla didn’t select the trimmed piece, Bob takes it. Alice will get the one that continues to be. Then the eaters flip to the trimmings. They observe the same means of reducing, trimming and selecting.
In 1995, one other crew confirmed learn how to prolong this strategy to any variety of individuals. Brams at NYU labored with Alan D. Taylor of Union School in Schenectady, N.Y. Their technique utilized the “trimming” thought utilizing all potential pairs of cake diners. “That was thought-about a breakthrough of types,” Brams says.
The strategy nonetheless had its limits, nevertheless. There was no assure of what number of cuts it’d take. “We confirmed typically that you possibly can require three cuts or 3 million cuts,” Brams says. Or much more.
The cake-cutting downside endures
The algorithms mentioned up to now assume that each one eaters play honest. That’s, all attempt to obtain items that may really feel like a fair proportion to everybody else. However individuals aren’t all the time sincere.
That is one thing Biaoshuai Tao acknowledges. Tao is a pc scientist at Shanghai Jiao Tong College in China. And he studied what occurs if you attempt to account for dishonest cake eaters. “If everybody is aware of how the cake is allotted, then I ought to get extra if I inform the reality,” he says.
However in some instances, dishonesty may give a bonus. Say Alice and Bob are going to separate a cake. If Alice knew that Bob all the time most well-liked chocolate, she may reduce the cake unequally on function so the smaller piece contained extra chocolate. Then, if Bob selected in keeping with his choice, Alice would get the bigger slice.
In a 2010 paper, Procaccia requested a curious query: Can there be a strategy to reduce cake that ensures honesty and equity? Greater than 10 years later, the reply appears to be: no.
Tao used math to point out it’s not possible to chop cake in a manner that guarantees truthfulness and equity, with no envy. He offered this at a July 2022 assembly in Boulder, Colo. It was held by the Affiliation for Computing Equipment Convention on Economics and Computation.
Past cake
Cake-cutting is simple to narrate to, says Bettina Klaus. And it has a number of sensible makes use of. On the College of Lausanne in Switzerland, Klaus research equity in real-world conditions. Dividing issues pretty “is mathematically fascinating and difficult,” she says. And its complexity grows with the quantity of people that need to share.
One instance she research is college alternative. Right here, a faculty district has restricted seats in sure colleges. “Up to now, colleges have been simply assigned [to students] … with out asking individuals what they need,” Klaus says. Extra lately, she notes, colleges have tried to position college students the place they need to go (or the place their mother and father need them to go). In addition they should observe guidelines set by the college board. Deciding learn how to pretty assign these seats means balancing the priorities of those teams.
Different real-world purposes present up nearly in all places you look.
Brams has used concepts from cake reducing to check honest voting procedures. (To elect their leaders, at the least 4 scientific societies adopted an algorithm he developed. The Mathematical Affiliation of America is one.)
In 2014, Procaccia was a part of a crew that designed a web-based instrument referred to as Spliddit. Based mostly on customers’ preferences, it produced mathematically honest methods to divide something. It could be hire amongst roommates and even possessions amongst divorcees.
Even after a long time of research, cake reducing defies a easy resolution. Certainly, the extra researchers discover it, the extra there appears to be to discover.
“I’m fascinated by it no longer solely as a result of it’s lovely in math,” Tao says, but additionally as a result of “I nonetheless consider there’s so much to be achieved.”
