Long ago, in Iowa, Occupy Math taught a course entitled “Introduction to Mathematical Concepts”. It was the course for people who might not have mastered arithmetic yet, but had a major that required a math course. During this course, there were only two days (other than examinations) with nearly complete attendance. The first day, because people wanted to know what was going to happen, and the day Occupy Math used a fair division algorithm to figure out the property division of Donald Trump’s (first) divorce. Short version: Ivana did not do well. In this post, Occupy Math examines two kinds of fair division. The simple one is dividing cake and the hard one is dividing property. We will also discuss why, with fair and equitable methods of dividing things, why we still spend piles of money on things like divorce lawyers.

**Mathematical formulas exist to divide almost anything as close to fairly as possible — why then all the squabbles?
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The easy fair division problem is splitting a piece of cake between two people. Most of Occupy Math’s readers are probably already familiar with a good technique: one person cuts and the other one chooses one of the two pieces. The person that cuts makes the pieces equal so that a larger piece is not available to the chooser and the chooser picks the piece they think is better. This means that everyone thinks they got at least their fair share, which is what is meant by *fair*. In more complex situations its possible to achieve a fair division, but still have someone think someone else got *more* than their fair share. That situation is called *envy*. There is a substantial mathematical literature on fair division and “fair” and “envy” are technical terms in that literature.

What if we have more than two people dividing a cake? The moving knife procedure works for two people — and is more symmetric than the cut-and-choose method — but it can be generalized to any number of people. One person moves the knife along the cake. If anyone shouts “cut” the knife-wielder cuts at the current position and gives the person who shouted that part of the cake. This continues with each person shouting “cut” until the last remainder is given to the last person. This method is fair — you don’t shout until the current piece looks fair to you — but with three or more people this method can result in envy.

An envy-free method of cutting cake has been devised but, for five people, it requires as many as 120 cuts, so it is not a method you would use in practice. You might say its a crumby method. There is a factor that makes cake division much more difficult: suppose the cake has decorations or different flavors. This means that different people can place different values on the same slice of cake. The technical term for this is that different people place a different *measure* on the cake. The moving knife method is enough to be going on with, though Occupy Math notes that a completely different solution to this kind of problem is to have more than enough cake.

**What if we are dividing items we cannot cut just anywhere?**

Cake can be cut anywhere. A house, a car, furniture, and so on cannot. This means that if two parties are dividing property, a very different method of dividing property is needed. The adjusted winner algorithm works in this situation. The algorithm works like this. Each party is given points, usually 100, to divide among the items. Initially items are assigned to the person that bid the most on them. This often leads to a situation in which one person got more. The items are put in order by the ratio of points assigned to them by the person who is ahead divided by the points assigned to them by the person who is behind. The item with the smallest ratio is transferred to the person who is behind until the next transfer would put them ahead. This is the most even possible division of property *if* people were honest about their point assignments.

If there are divisible assets, like money or a case of wine, then there are additional procedures for evening up the score with these assets. Assets like money and cake are *divisible* or *continuous* while assets like a car or couch are *discrete*. The adjusted winner algorithm is implemented in software and so could permit two people to divide property. If they agree that an asymmetric division is fair, then even that can be managed by giving one party more points. This brings us to the question that yielded the title of today’s post: why is acrimonious property division still so common? A fairly technical article looks at how to cheat if you know the other person’s point assignments, so that’s a thing, but in general:

**The problem is that people dividing property also often want revenge!**

Uncontested or consensual divorce is easier, much cheaper, and quite a bit faster. A couple that agrees to such a divorce would find the adjusted winner algorithm useful, maybe to do the division or possibly as a way to generate a starting point for discussion. An angry divorce, on the other hand, doesn’t have much use for fair, objective mathematics. Both sides are using the law and whatever else they can find to “get” the other person. There is often very real betrayal or even criminal conduct in such divorces. Occupy Math’s point is that reason and mathematics can only go so far before you have to deal with people.

This conundrum of hate and anger causing waste appears in many places. Lingering hatred and suspicion can poison many sorts of endeavors. Israel has figured out how to grow enough crops to feed itself in the desert by a well-planned combination of conservation and making fresh water from the salt water of the Mediterranean sea. Iran is about to collapse from a prolonged water shortage caused by a water policy driven by politics without any planning. Iran has a longstanding hatred of Israel — which is the only nation in the area that has the technology to save them from economic ruin. Much as a vengeful or spiteful spouse who would rather give the couple’s joint worth to lawyers rather than have their opposite number get it, Iran is likely to displace whole cities rather than ask Israel for help — which still begs the question of Israel’s willingness to help.

Mathematics, in the form of planning, could have prevented the crisis in Iran and mathematics, in forms like the prisoner’s dilemma, indicate that asking for and granting help is the superior course of action for both parties. When will this chaos end? As the Immortal Bard says, “Not till God make men of some other metal than earth.” If this sort of change happens, mathematics is ready to help.

**Yes, splitting up dessert or your worldly goods is math!**

Fair division, both of cakes and property, is an interesting area of math. It is also an area of math that many people are not aware of. Partly this is because professional dividers of property make more money when there is a fight, but mostly it is because the math we choose to teach is the prelude to being an engineer, not a prelude to being an effective citizen. People that know *how* to divide property fairly and without envy are less likely to get into a fight. Again, Occupy Math notes that **Math is the Right of All Free People**.

Occupy Math hopes this post has stretched your ideas about what math includes and helped you think about the issue of how to divide things. Its also worth mentioning again that having more than enough cake is a viable approach. When Occupy Math has his students over for dinner, this is his solution. There are also non-adversarial situations, like business partners who need to dissolve a business because one of them is moving, where fair division might be natural. Fair division is a big part of the discipline of conflict resolution. Occupy Math would like to know if you have situations where fair division might apply or situations where it looks to you like there ought to be helpful math. Please comment or tweet!

I hope to see you here again,

Daniel Ashlock,

University of Guelph,

Department of Mathematics and Statistics

Hi,

Here’s an article I wrote on dividing pizza a while ago http://www.statisticool.com/pizzamath.htm

Cheers,

J

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