Divide Evenly, Again

A few months ago I did a blog on dividing evenly. The subject was cutting a equal-sized piece of pie for each person. A few days ago I did another and showed how to evenly-space 3 pictures on the wall.

In both cases the object was to get the same size for each (person or picture). Today I have a more interesting challenge from of a new book called Numbers Rule by George Szipiro.

George the mathematician/journalist says that it's impossible to evenly or fairly divide the 435 representatives in our Federal government among the 50 states. Some states are always under-represented; some states are over-represented.

Why? Because we don't have a way to impose fractional voting shares on representatives. Since we have to allocate, for example, 1 or 2 delegates to each state rather than 1.5 delegates, we have to give more to one state and take some away from another. This has been studied for hundreds of  years, using dozens of different methods of dividing and rounding. You can read all about it in his book, and in another book called Fair Representation.

There is another problem with counting votes which is related to small parties. If your country has more than two major political parties, it's impossible to allocate representatives evenly in proportion to the votes cast for them.

For example, in one district, the elections results look like this:

• Green party 1000 votes
• Yellow party 600 votes
• Orange party 599 votes

If there's only one seat, then Green is represented and the other two parties are not. If there are 22 seats, then we can give 10, 6 and 6 seats.

In another district, it's like this:

• Green  400 votes
• Yellow 801 votes
• Orange 799 votes

If there's only one seat, then Yellow is represented and the other two parties are not. If there are 20 seats, then we can give 4, 8, and 8 seats and all are represented.

If we only have the one seat per district, then Green gets one and Yellow gets one. The elections are over and Orange gets left out completely.

But if you add the two district votes, you find that the votes were split with one-third to each party.

• Green 1400
• Yellow 1401
• Orange 1398

Green and Yellow are now evenly represented in the Assembly and Orange has no representation at all despite being only 2 or 3 votes down out of 4200 votes. This is unfair to one-third of the electorate.

You might say "Too bad, tough luck" but come on, it's still unfair. This situation would not meet anyone's hope of having proportional representation for all viewpoints - one of the goals of a democratic government. In many places (including here) people get very angry. What are the alternatives?

1. Let cooperative parties pool their fractional (or excess) votes across districts to get more representatives? That's what Switzerland, Israel and some other countries do.
2. Rank the acceptable candidates then choose the one with the overall highest rating? This method is used in Australia, New Zealand, Scotland, Ireland, Malta.
3. Put a check next to everyone we could live with, and no check next to those we dislike? Then the ballots could be tallied and the overall winner chosen. This is used by the United Nations and a lot of professional societies.
4. Evaluate candidates on excellent, good, average, poor and unacceptable, then select the best people and have another go at it if necessary? That's how the latest French presidential elections were counted.
5. Have a dictatorship. This is mathematically acceptable. One viewpoint is fully represented.

Alas, mathematicians say there are no perfect solutions and there is apparently no perfect democracy. All are subject to paradoxes, inconsistencies, and manipulation behind the scenes.

If you want to see fireworks, forget the 4th of July. Wait until the 2010 Census figures are in and the next reallocation of our 435 state representatives begins!