5.1 Introduction

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.

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.

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.

Example 2

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.

# Math in Society 5.1 -- Fair share when people value the pieces differently
# A $12 pizza, half pepperoni and half veggie, cut into 4 slices of each.
# A fair share is 1/N of the whole AS THAT PERSON VALUES IT -- not 1/N of the slices.
PRICE, N, SLICES = 12.0, 4, 4
# How much is a pepperoni slice worth compared to a veggie slice, per person?
ratio = {"Steve": 2.0, "Kim": 0.0, "Ana": 1.0, "Raj": 0.5}      # EDIT these

share = PRICE / N
print(f"Whole pizza ${PRICE:.2f}, split {N} ways -> a fair share is ${share:.2f}\n")
for who, r in ratio.items():
    # SLICES*x + SLICES*r*x = PRICE, where x is what one veggie slice is worth
    veg = PRICE / (SLICES * (1 + r))
    pep = r * veg
    note = ("likes them equally" if r == 1 else
            "won't touch pepperoni" if r == 0 else
            f"likes pepperoni {r:g}x as much" if r > 1 else
            f"likes veggie {1/r:g}x as much")
    print(f"{who:<6} {note:<28} veggie slice ${veg:>5.2f}   pepperoni slice ${pep:>5.2f}")
    combos = []
    for p2 in range(2 * SLICES + 1):            # count in half slices
        for v2 in range(2 * SLICES + 1):
            if abs(p2 / 2 * pep + v2 / 2 * veg - share) < 1e-7:
                combos.append((p2 + v2, p2 / 2, v2 / 2))
    combos.sort()
    if not combos:
        print(f"       no half-slice combination is worth exactly ${share:.2f}")
    for _, p, v in combos[:4]:
        parts = []
        if p: parts.append(f"{p:g} pepperoni")
        if v: parts.append(f"{v:g} veggie")
        print(f"       fair share: " + " + ".join(parts))
    if pep == 0:
        print("       (pepperoni is worth $0 here, so any amount of it changes nothing)")
    print()

print("Kim is entitled to a quarter of the pizza and one veggie slice does it.")
print("One slice out of eight looks unfair until you remember that fair division")
print("measures value, not area -- and to Kim half the pizza does not exist.")
print("Try it: set Kim's ratio to 0.1 and re-run. A tiny amount of interest in")
print("pepperoni changes what counts as her fair share.")

Show 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 $2 x dollars – twice as much$ . Four veggie slices would be worth $4 \cdot x = 4 x$ dollars, and 4 pepperoni slices would be worth $4 \cdot 2 x = 8 x$ dollars. Altogether, the eight slices would be worth $4 x + 8 x = 12 x$ dollars. Since the total value of the pizza was $$ 12$ , then $12 x equals $ 12$ . Solving we get $x equals $ 1$ ; the value of a veggie slice is $$ 1$ , and the value of a pepperoni slice is $2 x equals $ 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.

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).

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.