# 5.1: Introduction

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Whether it is two kids sharing a candy bar or a couple splitting assets during a divorce, there are times in life where items of value need to be divided between two or more parties. While some cases can be handled through mutual agreement or mediation, in others the parties are adversarial or cannot reach a decision all feel is fair. In these cases, fair division methods can be utilized.

A fair division method is a procedure that can be followed that will result in a division of items in a way so that each party feels they have received their fair share. For these methods to work, we have to make a few assumptions:

- The parties are non-cooperative, so the method must operate without communication between the parties.
- The parties have no knowledge of what the other players like (their valuations).
- The parties act rationally, meaning they act in their best interest, and do not make emotional decisions.
- The method should allow the parties to make a fair division without requiring an outside arbitrator or other intervention.

With these methods, each party will be entitled to some **fair share**. When there are *N *parties equally dividing something, that fair share would be 1/*N*. For example, if there were 4 parties, each would be entitled to a fair share of \(\frac{1}{4}=25\%\) of the whole. More specifically, they are entitled to a share that *they* value as \(25\%\) of the whole.

When \(N\) parties divide something equally, each party’s **fair share** is the amount they entitled to. As a fraction, it will be \(\frac{1}{N}\)

It should be noted that a fair division method simply needs to guarantee that each party will receive a share they view as fair. A basic fair division does not need to be envy free; an **envy-free** division is one in which no party would prefer another party’s share over their own. A basic fair division also does not need to be Pareto optimal; a **Pareto optimal** division is one in which no other division would make a participant better off without making someone else worse off. Nor does fair division have to be **equitable**; an equitable division is one in which the proportion of the whole each party receives, judged by their own valuation, is the same. Basically, a simple fair division doesn’t have to be the best possible division – it just has to give each party their fair share.

Suppose that 4 classmates are splitting equally a \(\$12\) pizza that is half pepperoni, half veggie that someone else bought them. What is each person’s fair share?

**Solution**

Since they all are splitting the pizza equally, each person’s fair share is \(\$3\), or pieces they value as \(25\%\) of the pizza.

It is important to keep in mind that each party might value portions of the whole differently. For example, a vegetarian would probably put zero value on the pepperoni half of the pizza.

Suppose that 4 classmates are splitting equally a \(\$12\) pizza that is half pepperoni, half veggie. Steve likes pepperoni twice as much as veggie. Describe a fair share for Steve.

**Solution**

He would value the veggie half as being worth \(\$4\) and the pepperoni half as \(\$8\), twice as much. If the pizza was divided up into 4 pepperoni slices and 4 veggie slices, he would value a pepperoni slice as being worth \(\$2\), and a veggie slice as being worth \(\$1\).

If we weren’t able to guess the values, we could take a more algebraic approach. If Steve values a veggie slice as worth x dollars, then he’d value a pepperoni slice as worth \(2x\text{ dollars }–\text{ twice as much}\). Four veggie slices would be worth \(4 \cdot x = 4x\) dollars, and 4 pepperoni slices would be worth \(4 \cdot 2x = 8x\) dollars. Altogether, the eight slices would be worth \(4x + 8x = 12x\) dollars. Since the total value of the pizza was \(\$12\), then \(12x = \$12\). Solving we get \(x = \$1\); the value of a veggie slice is \(\$1\), and the value of a pepperoni slice is \(2x = \$2\).

A fair share for Steve would be one pepperoni slice and one veggie slice (\(\$2 + \$1 = \$3\) value), \(1\frac{1}{2}\) pepperoni slices (\(1\frac{1}{2} \cdot \$2 = \$3\) value), 3 veggie slices (\(3 \cdot \$1 = \$3\) value), or a variety of more complicated possibilities.

Suppose Kim is another classmate splitting the pizza, but Kim is vegetarian, so won’t eat pepperoni. Describe a fair share for Kim.

**Answer**-
Kim will value the veggie half of the pizza at the full value, \(\$12\), and the pepperoni half as worth \(\$0\). Since a fair share is \(25\%\), a fair share for Kim is one slice of veggie, which she’ll value at \(\$3\). Of course, Kim only getting one slice doesn’t really seem very fair, but if every player had the same valuation as Kim, this would be the only fair outcome. Luckily, if the classmates splitting the pizza are friends, they are probably cooperative, and will talk about what kind of pizza they like.

You will find that many examples and exercises in this topic involve dividing food – dividing candy, cutting cakes, sharing pizza, etc. This may make this topic seem somewhat trivial, but instead of cutting a cake, we might be drawing borders dividing Germany after WWII. Instead of splitting a bag of candy, siblings might be dividing belongings from an inheritance. Mathematicians often characterize very important and contentious issues in terms of simple items like cake to separate any emotional influences from the mathematical method.

Because of this, our requirement that the players not communicate about their preferences can seem silly. After all, why wouldn’t four classmates talk about what kind of pizza they like if they’re splitting a pizza? Just remember that in issues of politics, business, finance, divorce settlements, etc. the players are usually less cooperative and more concerned about the other players trying to get more than their fair share.

There are two broad classifications of fair division methods: those that apply to **continuously divisible** items, and those that apply to **discretely divisible** items. Continuously divisible items are things that can be divided into pieces of any size, like dividing a candy bar into two pieces or drawing borders to split a piece of land into smaller plots. Discretely divisible items are when you are dividing several items that cannot be broken apart easily, such as assets in a divorce (house, car, furniture, etc).