Here is why mathematicians are so occupied with cake reducing

Ariel Procaccia has idea so much about reduce cake over the past 15 years. That’s in part since the Harvard pc scientist has 3 youngsters who amongst them have celebrated greater than two dozen birthdays. He is aware of what it’s like to face with a knife ahead of a layered masterpiece, frosted with buttercream and chocolate curls, whilst pressed on each side by means of small partygoers who instinctively acknowledge when anyone else will get a greater slice.

But it surely’s additionally as a result of a lot of Procaccia’s paintings makes a speciality of exploring the mathematical laws for dividing stuff up. A method to try this is to suppose abstractly about dessert. For greater than 75 years, he and different researchers looking to formalize equity were asking the deceptively easy query: What strategies for reducing a cake ensure that everybody who displays as much as the birthday party is pleased with what they get?

The solutions succeed in some distance past birthday events. Cake-cutting contemplation is a part of a sprawling mathematical subfield centered at the truthful department of sources. It has spurred a raft of algorithms informing allocate meals amongst hungry communities, break up hire or chores amongst roommates, how to attract barriers for truthful vote casting districts and extra. A mathematical downside at its middle, cake reducing connects rigorous reasoning to questions of human personal tastes and real-world problems, and so draws now not most effective mathematicians, but additionally pc scientists, economists, social scientists and prison professionals. Questions of equity (and unfairness) are decidedly common. After all, so is dessert. “It’s this very chic fashion by which you’ll in reality distill what equity is, and explanation why about it,” Procaccia says.

The cake, says Steven Brams, a recreation theorist and political scientist at New York College, is a metaphor for any divisible just right, like land or time or restricted sources. When cake-cutting insights are implemented to settling global disputes, he says, “we’re probably serving to the sector to find answers.”

Mavens have get a hold of cake-cutting algorithms — the mathematical laws for describing reduce a cake rather — again and again and in lots of guises. (The approaches nearly at all times focal point on oblong desserts. The comparable however more moderen “pie-cutting” downside addresses round cakes or pizza.) The perfect laws expose rather percentage a cake between two other people: One consumer cuts the cake into two items that they consider to be equivalent in price, and the opposite consumer choices first. Every eater receives a work that they really feel is a minimum of as precious as the opposite’s, if now not higher. Stories of this truthful department technique date again to historical Greece.

You’ll argue that equity, or the dearth thereof, is without doubt one of the maximum vital issues on the earth lately.

Steven Brams

Within the Nineteen Forties, mathematicians started taking severe hobby in a mathematical technique to equity, the use of cake reducing as an get right of entry to level. They began exploring rather percentage amongst 3 other people, since I-cut-you-choose is a two-player recreation. That led to searching for tactics to increase the ones algorithms to arbitrarily huge numbers of other people, and to asking extra nuanced questions, like what’s equity precisely, and the way do you turn out it?

Cake reducing is straightforward to formulate and simple to narrate to, says recreation theorist Bettina Klaus of the College of Lausanne in Switzerland, who research equity in real-world scenarios like college selection allocation and equivalent get right of entry to to housing. “However on the similar time, the issue is mathematically fascinating and difficult on account of its complexity as soon as the collection of brokers to percentage the cake grows.”

Contemporary years have introduced growth in figuring out the fewest collection of cuts wanted for a given collection of other people, in addition to the utmost collection of cuts, which will get ridiculously prime however a minimum of displays that cake reducing is finite. And new diversifications at the query stay rising. What when you divide a cake for more than one teams of other people as a substitute of people? Or, as explored in a paper printed final 12 months, what if cake eaters lie about their personal tastes? And what when you’re divvying up one thing that is available in discrete, indivisible items, like unopened Halloween goodies, as a substitute? By means of that specialize in exact definitions and new eventualities, mathematicians have discovered new programs and stored cake reducing at the leading edge of investigations into equity.

“You’ll argue that equity, or the dearth thereof, is without doubt one of the maximum vital issues on the earth lately,” says Brams, who over 4 many years has printed loads of works on cake reducing or equity extra usually. “And we’re having a look on the theoretical foundations of equity.”

Recipes for truthful cake reducing

Documented experiments to find truthful tactics to separate stuff up cross long ago, a minimum of to Hesiod’s poem Theogony, written some 2,700 years in the past. In a single tale within the poem, gods and mortals clashed in Mecone, a legendary Greek town. As a sacrifice to soothe the gods, Prometheus, who used to be each a god and humankind’s largest benefactor, divided a not too long ago slaughtered ox into two piles, one containing unappealing naked bones lined with a layer of fats and the opposite containing the fascinating meat hid underneath an unappealing phase of abdomen. Prometheus invited Zeus to take his select. Zeus, seduced by means of the glossy fats, selected the unappetizing bones.

On this historical tale, Prometheus infuses the vintage I-cut-you-choose technique — the most straightforward model of cake reducing — with deception. But if I-cut-you-choose is finished in pursuit of equity, it will have to ensure the pleasure of everybody concerned.

The end result is proportional, that means that each and every participant seems like their slice represents a fair proportion of the whole. So for 2 avid gamers, a participant would price their very own slice at 1/2; for 3, a fair proportion could be 1/3. (And for some arbitrary n collection of cake eaters, a fair proportion could be 1/n.) If the cake is identical during, proportionality is similar to all of the slices being the similar dimension.

However cake reducing isn’t an enchanting mathematical downside if the cake is all of the similar. Extraordinary department and a kitchen scale may just readily separate a slab of uniform chocolate sponge into any collection of proportional items. The issue turns into extra sophisticated when you think that the cake is heterogeneous — if it’s erratically frosted, for instance, or comprises sections of various flavors and toppings.

A maraschino cherry–lover may favor the smallest slice and really feel glad in the event that they get the cake’s most effective cherry. On this case, what mathematicians name the “serendipity of war of words” provides upward push to wealthy math. Essentially the most fascinating math arises when there are differing evaluations.

A two-person I-cut-you-choose situation nonetheless works right here. The divider divides the cake into two items of equivalent price of their view and might be pleased with both; the chooser chooses their most well-liked piece. However building up the collection of cake eaters, each and every with specific personal tastes, and there’s no simple answer.

Hugo Steinhaus of the College of Warsaw used to be one of the crucial first mathematicians to dive into this complexity. Right through International Conflict II, as questions of truthful department of land had been enjoying out on a big and violent scale, Steinhaus evolved a changed I-cut-you-choose technique for 3 avid gamers. It got here to be known as the lone-divider way.

On this means, one consumer, let’s name her Alice, cuts the cake into 3 items that she values similarly (each and every at 1/3 of the whole). Then a 2nd consumer, Bob, signifies which of the items could be appropriate to him. If he approves a minimum of two items, then the 3rd consumer, Carla, can take any piece she desires, adopted by means of Bob (who has a minimum of one appropriate piece to be had). Alice will get the person who’s left.

But when both Bob or Carla disapprove of the similar piece, then that piece is going to Alice (who valued all items similarly). The remainder two items (which Bob and Carla should price at 2/3 or extra of the whole) are recombined and shared between Bob and Carla the use of I-cut-you-choose.

Steinhaus described this set of rules in a brief paper printed in 1948 in Econometrica. It represented one of the crucial first rigorous investigations within the box of cake reducing. “The rule of thumb for the primary spouse,” Steinhaus wrote, “lets in him to chop the thing — it can be a cake — as he pleases.”

Steinhaus’ way labored for most effective 3 eaters, however in the similar paper, he reported that two colleagues had evolved an set of rules that would succeed in proportionality for any collection of cake eaters. The process is referred to as the last-diminisher way, and it is going like this: One consumer cuts off a work of cake they deem to constitute a fair proportion and passes that piece alongside to the following consumer. Every last participant has an opportunity to both trim the cake (if they believe it represents greater than a proportional percentage) or go (if they believe it’s proportionally truthful or lower than truthful). 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.

The trim is then recombined with the rest cake, and the method starts once more with the rest avid gamers. When most effective two avid gamers are left, they use I-cut-you-choose.

Brams has known as the last-diminisher way a sublime answer, and it promises that everybody judges their very own piece to be a minimum of as precious as a fair proportion. But it surely’s now not best possible as it doesn’t take envy into consideration. In each the lone-divider and last-diminisher approaches, an individual who exits the sport early would possibly finally end up coveting a work this is reduce later within the recreation — even supposing they idea their piece used to be proportional. Those algorithms aren’t what mathematicians name “envy-free,” which is differently to take into consideration equity.

There’s any other sensible limitation to the last-diminisher way: With sufficient avid gamers, the cake that is still in later rounds may finally end up damaged aside by means of numerous cutting — and even diminished to crumbs. It’s simple to peer how a partygoer may now not price that as extremely as a complete piece.

Can cake reducing be freed from envy?

Because the debut of the last-diminisher way, cake reducing has fueled a small however mighty frame of great arithmetic.

The Nineteen Sixties introduced a big step ahead when mathematicians John Conway and John Selfridge, independently of one another, got here up with a brand new reducing set of rules for 3 other people. In contrast to the paintings by means of Steinhaus and associates, the brand new recipe accomplished each perceived proportionality and have shyed away from any envy a number of the recipients.

An envy-free answer, by which nobody covets someone else’s piece, is straightforward to succeed in, issues out pc scientist Haris Aziz of the College of New South Wales in Sydney. Simply throw all of the cake away. “If you happen to don’t give anything else to any one, that’s envy-free,” he says.

But when the cake lands within the garbage bin, nobody is excited. In Conway’s and Selfridge’s extra pleasant scheme, Alice first divides the cake into 3 items she believes are of equivalent price. Then, Bob can trim one piece — at maximum — to create a two-way tie for probably the most precious. (The trappings are put aside.) Carla is left to select a number of the 3. Then the order reverses, and if Carla didn’t favor the trimmed piece, Bob takes it. Alice will get the person who stays. The eaters then flip to the trappings, following a an identical iterative protocol of reducing, trimming and opting for.

But for many years extra, an envy-free cake-cutting answer for any arbitrary collection of eaters remained elusive. Within the overdue Eighties, on his PBS tv display For All Sensible Functions: Creation to Fresh Arithmetic, mathematician Sol Garfunkel featured the unsolved cake-cutting downside and comparable questions of truthful department.

However the issue wouldn’t cross unsolved for for much longer. In 1995, Brams at NYU and Alan D. Taylor of Union Faculty in Schenectady, N.Y., devised a brand new process that cuts cake for 4 other people with nobody envying someone else. “That used to be regarded as a step forward of types,” Brams says. It constructed at the “trimming” means of Conway and Selfridge, working a an identical process on all conceivable pairs of cake eaters. Brams and Taylor described how the process may well be prolonged to any collection of other people.

The means nonetheless had obstacles. There used to be no ensure of what number of cuts it would take. “We confirmed typically that it’s essential require 3 cuts or 3 million cuts,” Brams says. Or many, many extra.

A couple of years later, mathematicians Jack Robertson and William Webb of Washington State College in Pullman described an invaluable computation fashion that may be used to research what number of steps, together with cuts and reviews, are required by means of an set of rules. Its calculations showed, for instance, that no most collection of cuts may well be predicted for any set of rules identified at the moment that divided cake proportionally and with out envy for any arbitrary collection of avid gamers.

Over the following couple of many years, many mathematicians got here to wonder if an higher sure for envy-free cake reducing even existed. If now not, in idea, cake reducing may just cross on ceaselessly. What’s extra, Procaccia says, nobody had found out the minimal, both.

What mathematicians name the “serendipity of war of words” provides upward push to wealthy math.

Is cake reducing countless?

Procaccia by no means in fact got down to find out about cake reducing. In 2008, he used to be instructing a path at the mathematical foundations of synthetic intelligence. Someday, strolling house after turning in a lecture on useful resource allocation and the Robertson-Webb fashion, he discovered how he may just discover a decrease sure — a minimal collection of steps, together with cuts — for envy-free cake reducing for any collection of other people. The decrease sure he discovered used to be someplace round n² steps, the place n is the collection of cake eaters.

That may result in his first paper on cake. Procaccia has a knack for giving mathematical papers catchy titles. The lower-bound paper, printed in 2009, used to be titled “Thou shalt covet thy neighbor’s cake.” In 2010, he coauthored one known as “Fact, justice, and cake reducing,” which offered the query of truthfulness — along with making certain proportionality and getting rid of envy. If an individual hides their personal tastes all through the reducing, anyone would possibly finally end up with an unequal percentage. It’s “mathematically interesting,” Procaccia says.

As Procaccia endured within the box, he started pondering extra about helpful algorithms that would put insights from cake reducing — and the speculation of truthful department typically — to just right use. One instance: dividing hire.

One of the simplest ways, after all, is to divide the whole due by means of the collection of population. However that ignores the “serendipity of war of words.” One consumer may need a window, any other may favor the larger closet. In 2014, Procaccia and associates designed a web based software known as Spliddit that gathered customers’ personal tastes and produced mathematically truthful tactics to divide anything else, from hire amongst roommates to possessions amongst divorcees.

The largest contemporary step forward in cake reducing, Procaccia says, got here from Aziz and pc scientist Simon Mackenzie, primarily based in Sydney, who decided an higher sure on envy-free, proportional cake reducing. First, in 2015, the pair tackled the issue of percentage cake amongst 4 other people. By means of borrowing concepts from Conway and Selfridge and from Brams and Taylor, the staff devised an set of rules that produced an higher sure of 203 steps, which might come with nearly as many cuts. That’s so much however now not too unreasonable.

A 12 months later, the staff prolonged the technique to an arbitrary collection of other people, reporting an set of rules with a finite collection of cuts for envy-free, proportional cake reducing. It used to be a probably astronomical collection of cuts, however it used to be finite — answering a long-standing query within the box.

Cake reducing for n other people, Aziz and Mackenzie reported, may just require as many as n^n^n^n^n^n operations. That’s a unconditionally unreasonable quantity. The utmost collection of steps for 5 other people could be round 2 x 10^2,180. That suggests 5 other people reducing the cake billions of occasions according to 2nd for 100 trillion years may slightly be getting began.

Then again, Aziz says the set of rules will also be tailored to a extra affordable, regardless that nonetheless in reality large higher sure if the partygoers, for instance, permit for just a little cake to be left over. And it’s nonetheless conceivable that mathematicians may just carry that higher sure decrease at some point.

The cake-cutting downside endures

Explorations into the query of rather reduce a cake aren’t over. Impressed by means of Procaccia’s 2010 paper on truthfulness, pc scientist Biaoshuai Tao of Shanghai Jiao Tong College investigated what occurs whilst you attempt to account for cheating cake eaters. “If we all know how the cake is allotted, then I will have to get extra if I inform the reality,” he says.

However in some circumstances, dishonesty can yield a bonus. If Alice and Bob had been to divide a cake, and Alice knew that Bob at all times most well-liked chocolate, she may knowingly divide the cake unequally so the smaller piece contained extra chocolate. Then Bob would favor in keeping with his choice, and Alice would get the bigger piece.

In his paintings, introduced in July 2022 on the Affiliation for Computing Equipment Convention on Economics and Computation, in Boulder, Colo., Tao discovered that truthfulness and proportionality are incompatible, making it unimaginable to build a cake-cutting set of rules that strictly promises truthfulness, proportionality and no envy.

Sensible programs for cake reducing additionally proceed to abound. Klaus, in Lausanne, issues to university selection for example.

A district with restricted seats in positive colleges has to stability the varsity board priorities — ratings on front tests or geographic distribution, for instance — with the personal tastes of oldsters to take a look at to discover a proportional answer with a good price. “Prior to now, colleges had been simply assigned … with out asking other people what they would like,” Klaus says. “The college selection comes from the truth that the personal tastes of the fogeys or the youngsters could be taken into consideration.”

Through the years, cake reducing has advanced into one of those mathematical sandbox, a positive playground that brings in combination summary proofs and intuitive programs.

And there are many different real-world programs for questions of truthful department. Brams has used concepts from cake reducing to check truthful vote casting procedures. (To elect their leaders, a minimum of 4 medical societies, together with the Mathematical Affiliation of The us, followed an set of rules evolved by means of him.)

Procaccia has implemented truthful department algorithms to fashion meals allocation. Aziz is exploring programs starting from divvy up chores or different duties that may’t be divided to highest agenda docs’ shifts in hospitals.

Others are finding out truthful allocation of products that may’t be cleanly divided. After a divorce, for example, former companions may come to settlement on a good break up provided that some pieces are taken out of attention. Those investigations come with approaches which can be just about envy-free if now not mathematically best possible.

Even after many years of investigation, cake reducing isn’t like a easy jigsaw puzzle with a well-defined answer. As a substitute, through the years, it has advanced into one of those mathematical sandbox, a positive playground that brings in combination summary proofs and intuitive programs. The extra researchers discover it, the extra there’s to discover.

“I’m occupied with it no longer most effective as it’s gorgeous in math,” Tao says, “however I nonetheless consider there’s so much to be performed.”