11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem

The Apportionment Problem

In the new democracy of Imaginaria, there are four states: Fictionville, Pretendstead, Illusionham, and Mythbury. Each state will have representatives in the Imaginarian Legislature. You might now have an agreement on which voting method your citizens will use to elect representatives. However, before that process can even begin, you must decided on how many representatives each state will receive. This decision will present its own challenges.

When sharing your birthday cake, it’s only fair that everyone gets the same portion size, right? You were portioning the cake by dividing it up equally and giving everyone a slice. A great thing about cake is that you can slice it any way you want, but how do you apportion, or divide and distribute, items that can't be sliced? Suppose that you have a box of 16 Ring Pops™, gem-shaped lollipops on a plastic ring. You are going to share the box with four other kids. Dividing the 16 Ring Pops™ among the group of five leads to a problem; after each person in the group gets three Ring Pops™, there is still one left! Who gets the last one? The apportionment problem is how to fairly divide, or apportion, available resources that must be distributed to the recipients in whole, not fractional, parts.

The apportionment problem applies to many aspects of life, including the representatives in the Imaginarian legislature. The table below provides a short list of examples of resources that must be apportioned in whole parts, and the recipients of those resources.

Fair division of a resource is not necessarily equal division of the resource like when distributing cake slices. When distributing airport terminals amongst airlines, there are many factors to consider such as the size of the airline, the number and types of aircraft they have, and the demand for the service. In most cases, fairness is defined as being proportional; two quantities are proportional if they have the same relative size. In the case of the Covid-19 vaccine, the expectation would be that countries with larger populations get more doses of the vaccine. In the Imaginarian legislature, the expectation may be that the states with larger populations will receive the larger number of representatives. This concept is referred to as a part-to-part ratio.

Suppose that a supermarket has a special on pies, two for $5. The first customer purchases four pies for $10, and the second customer purchases eight pies for $20. The dollar to pie ratio for the first customer is $\frac{10\text{dollars}}{4\text{pies}}=2.5\text{dollars per pie}$ and the dollar to pie ratio for the second customer is $\frac{20\text{dollars}}{5\text{pies}}=2.5\text{dollars per pie}$. So, the dollar to pie ratio is constant. Although the customers do not spend the same amount of money, the amount each spent was proportional to the number of pies purchased.

Now suppose that the supermarket changed the special to $5 for the first pie, and $2 for each additional pie. In that case, four pies would cost $\text{\$}5+3\left(\text{\$}2\right)=\text{\$}11$, while 8 pies would cost $\text{\$}5+7\left(\text{\$}2\right)=\text{\$}19$. The dollar to pie ratios would be $\frac{11\text{dollars}}{4\text{pies}}=2.75\text{dollars per pie}$ and $\frac{19\text{dollars}}{8\text{pies}}=2.375\text{dollars per pie}$, respectively. This special does not result in a constant part to part ratio. The dollars spent are not proportional to the number of pies purchased.

There are some useful relationships between quantities that are proportional to each other. When there is a constant ratio between two quantities, the one quantity can be found by multiplying the other by that ratio. Remember the supermarket special on pies, 2 pies for $5? The ratio of dollars to pies is $\frac{5\text{dollars}}{2\text{pies}}=2.5\text{dollars per pie}$ and the ratio of pies to dollars is$\frac{2\text{pies}}{5\text{dollars}}=0.4\text{pies per dollar}$. These two values are reciprocals of each other, $\frac{1}{2.5}=0.4$ and $\frac{1}{0.4}=2.5$. This means that multiplying by one has the same effect as dividing by the other. This also means that knowing either constant ratio allows us to calculate the price given the number of pies. To find the cost of 20 pies, multiply by the ratio of dollars to pies or divide by the ratio of pies to dollars.

• $20\text{pies}\times 2.50\text{dollars per pie}=50\text{dollars}$
• $20\text{pies}÷0.4\text{pies per dollar}=50\text{dollars}$

These patterns are true in general.

The apportionment application that will be important to the founders of Imaginaria occurs in representative democracies in which elected persons represent a group. The United Kingdom, France, and India each have a parliament, and the United States has a Congress, just as Imaginaria will have a legislature! The citizens of a country must decide what portion of the representatives each group, such as a state or province or even a political party, will have. A larger portion of representatives means greater influence over policy.

You might be wondering why the ratio doesn't appear to be quite the same depending on the rounding of the values. We will see that the key to this variation lies in the fractions. Just like the five children sharing 16 Ring Pops™, there are going to be leftovers and there are many methods for deciding what to do with those leftovers.

The Standard Divisor

There are two houses of congress in the United States: the Senate and the House of Representatives. Each state has two senators, but the number of representatives depends on the population of the state. The number of representative seats in the U.S. House of Representatives is currently set by law to be 435. In order to distribute the seats fairly to each state, the ratio of the population of the U.S. to the number of representative seats must be calculated. The ratio of the total population to the house size is called the standard divisor, and it is the number of members of the total population represented by one seat.

Although apportionment applies to many other scenarios, such as the pencil distribution during the SAT, the terminology of apportionment is based on the House of Representatives scenario. Thus, several government-related terms take on a more general meaning. The states are the recipients of the apportioned resource, the seats are the units of the resource being apportioned, the house size is the total number of seats to be apportioned, the state population is the measurement of the state's size, and the total population is the sum of the state populations.

Whether the standard divisor is less than, equal to, or greater than 1 depends on the ratio of the population to the number of seats.

• The standard divisor will be equal to 1 if the total population is equal to the number of seats. This would mean that each member of the population is allocated their own personal seat.
• The standard divisor will be a number between 0 and 1 when the total population is less than the number of seats. This means that each member of the population is allocated more than one seat.
• The standard divisor will be a number greater than 1when the total population is greater than the number of seats. This means that a certain number of members of the population will share 1 seat.

If the population is five children and the house consists of five pieces of candy, the standard divisor is $\frac{5\text{children}}{5\text{candies}}=1\text{child per candy}$ meaning each child gets one candy. If the population is five children and ten pieces of candy, the standard divisor is $\frac{5\text{children}}{10\text{candies}}=0.5\text{child per candy}$ meaning that each child gets more than one candy. If the population is five children and four pieces of candy, the standard divisor is $\frac{5\text{children}}{4\text{candies}}=1.25\text{child per candy}$ meaning that each child gets less than one candy.

If the seats in the Imaginarian legislature are distributed to the states based on population, then the house size will be less than the population and we should expect the standard divisor to be a number greater than 1.

The Standard Quota

Once the standard divisor for the Imaginarian legislature is calculated, the next task is to determine the number of seats that each state should receive, which is referred to as the state’s standard quota. Unless all the states have the same population, each state will receive a different number of seats because the quantities will be proportionate to the state populations. To determine those amounts, we will use an idea we learned earlier. Recall that, when the number of units of item $A$ is proportionate to the number of units of item $B$, we have: $\text{units of}A=\frac{\text{units of}B}{\text{ratio of}B\text{'s to}A\text{'s}}$

In this case, we are trying to calculate the number of seats a state should be apportioned, the state’s standard quota. So $A$ So $A$ would refer to seats allocated to a particular state, while $B$ would refer to the state population. This means that the ratio of $B$ to $A$ is the ratio of the total population to house size, which is the standard divisor. So in apportionment terms, we have the following formula.