Balinski & Young's Impossibility Theorem & Political Apportionment

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  • 0:01 Fairness and the…
  • 2:27 Quota Rule and Paradoxes
  • 3:22 Apportionment Process
  • 5:38 Lesson Summary
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Lesson Transcript
Instructor: Maria Airth

Maria has a Doctorate of Education and over 20 years of experience teaching psychology and math related courses at the university level.

The Balinski & Young Impossibility Theorem points out that there is no apportionment method that allows for the Quota Rule and does not allow any paradoxes to occur. This lesson investigates that statement.

Fairness and the Impossibility Theorem

What is fair? In a family with children, I think many people think it is fair to treat them all the same. Do you agree? Well, let's consider this family with three children. What does it really look like to treat them all the same?

Is it possible to treat all the children fairly?
family with children of different ages

What if the parents said they would have broccoli for dinner and to ensure that they are all treated the same, they all get the same amount of broccoli? Well, let's assume that one of them loves broccoli. He's happy. But the others hate it, so they are not happy at all. Was it fair for the parents to give broccoli to all three of them? They were treated the same, but the outcome was not equal for all.

Okay, so maybe that didn't work out, but what about changing it a bit. What if the parents asked the three children for their favorite foods? One child might say broccoli (we've met him already). Another might say lobster and the last might say lasagna. Would it be fair to feed them each their favorite food? It may make them all happy, but the child with lobster is getting a much more expensive item while the lasagna lover's getting an item that takes great amounts of time to prepare. Is that fair?

In this case, the parents have to decide which level of fairness they want to go with; either some get something they don't want, or some get comparatively more than the others.

Fairness is the main idea of this lesson. In particular, this lesson explains why it is impossible to have a completely fair method of apportioning voting seats in the House of Representatives.

Yes, I said that it is impossible to have a completely fair method to assign the number of voting seats for each state in the House of Representatives. Balinski and Young's Impossibility Theorem states that there does not exist an apportionment method that both satisfies the Quota Rule and is free from any paradoxes of apportionment. Basically, this is the same issue as our example with the children. Balinski and Young are saying that the governing board has to make a decision about whether they want to have possible fairness issues due to the Quota Rule or have paradoxes occurring periodically when apportioning voting seats.

But, let's take a few steps back and review the terms being discussed.

The Quota Rule and Paradoxes

The first thing to understand is that apportionment is the method used to divide voting seats in the House of Representatives between voting states.

Next we'll define the four major fairness measures in apportionment.

  • The Quota Rule states that no state shall receive more than its Upper Quota nor less than its Lower Quota of voting seats.
  • The Alabama Paradox occurs when the total number of voting seats is increased, but a state can lose seats when reapportioning occurs.
  • The New State Paradox occurs when a new state is included in the Union and the appropriate number of additional voting seats are added to the House of Representatives; other state's total number of seats are impacted by the inclusion of the new state.
  • The Population Paradox occurs when a state loses seats during reapportionment even if its total population has increased.

Apportionment Process

All methods of apportionment start with finding the Standard Divisor, which is the total population divided by the number of seats to be apportioned. Next, a Standard Quota for each state is found by dividing the State population by the Standard Divisor. This leads us to a fractional number at the fork in the road and decision about fairness to be made.

Some of the methods of apportionment are most concerned about eliminating the occurrence of paradoxes in apportionment. These methods use modified versions of quotas to arrive at final assignments of voting seats. They accept that, on occasion, the Quota Rule will be violated, meaning that sometimes a state might get more or less votes than their original Quotas called for.

These methods are Jefferson's Method, Adams' Method, Webster's Method, and the Huntington-Hill Method. Each of these methods uses a different kind of modified quota system that can sometimes result in a Quota Rule violation but does not ever result in the occurrence of a paradox.

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