Current assignments:

• (due Friday, December 10): More game theory problem set.

Today: Apportionment

Proportional Representation

• Suppose that in the 2004 election, there were three parties, and in every district, Republicans got 45%, Democrats got 40%, and Greens got 15%. Then Republicans would control 100% of the Congress.
• Suppose that in the 2004 election, a Republican won all 435 Congressional district elections with 51% of the vote. Then Republicans would control 100% of the Congress.
• We might avoid the first situation by implementing some other voting system in each district, like instant runoff, or a Condorcet method. Then Democrats or even Greens might win some seats.
• But there's no way to avoid the second situation. Every voting system in each district would award the seat to the majority winner.
• The only way to avoid the second situation is to have a separate system overall (not in each district) which gives seats to other parties based on their national vote totals.
• This is proportional representation, a method of assigning several winners at a time, rather than one at a time.
• There are a few ways to get seats allotted more proportionally:
  • Cumulative voting (as discussed just before fall break, October 22).
  • Holding elections for several offices at once, and letting people vote for more than one. For example, the Philadelphia City Council chooses seven at-large seats by letting everybody vote for five candidates. Since the Democrats can only run five candidates, Republicans are guaranteed at least two at-large seats.
  • Choose the Congressional districts themselves so that some districts have large majorities for one party or the other. This often results in gerrymandering. It is common because (for example) even if Republicans thought they could somehow get 51% in every district, it makes each election expensive. If they set up the districts so that they win with 80% in some districts and Democrats win with 80% in other districts, then elections become cheaper and candidates don't have to make much of an effort.
• But what is usually meant by proportional representation is a centralized system of voting, where people have one vote for a party, rather than for a candidate.
• For example, in Israel there are currently 26 parties. Everyone in the country votes for the same parties.
• The parties choose their candidates and publish the lists, but usually the individual candidates are not considered as important as the party platform.
• Example: if there are 120 seats, and Likud gets 40% of the vote, Labor gets 35% of the vote, Shinui gets 25% of the vote, then:
  • Likud gets 40% of 120 seats, which is 48.
  • Labor gets 35% of 120 seats, which is 42.
  • Shinui gets 25% of 120 seats, which is 30.
• This is the basic idea... However...
• The percentages normally don't work out that nicely.
• Suppose Likud gets 40.4%, Labor gets 33.9%, and Shinui gets 25.7%. Then we end up with:
  • Likud gets 48.48.
  • Labor gets 40.68.
  • Shinui gets 30.84.
• It's not completely obvious how to choose the distribution.
• Everyone can agree Likud should get at least 48 seats, Labor should get at least 40, and Shinui should get at least 30. But that leaves two seats left over.
• The simplest method is the Hare method (also known as "largest remainder"), which says to give the leftover seats to the party with the largest fractional part.
• This would give Shinui and Labor the two extra seats, resulting in 48 for Likud, 41 for Labor, and 31 for Shinui.
• Easy enough...
• But as we've seen, even in simple math we can find strange paradoxes.
• This very good site has a bunch of examples based on Hong Kong's system.
• (Side note: although we don't use a national system of proportional representation in the U.S., the same issues come up when allocating the number of representatives for a state.)
• (In that context, the largest remainder method was called Hamilton's method, and was vetoed by President Washington.)
• The most well-known paradoxes are the Alabama paradox, the population paradox, and the new states paradox.
• The Alabama paradox can occur when the number of seats changes. The mathematical database illustrates a simple example.
• Suppose there are three parties: Republican, Democrat, and Green; running for four seats.
• First suppose Republicans get 56% of the vote, Democrats get 33% of the vote, and Greens get 11% of the vote.
• Republicans would get 0.56 x 4 = 2.24 seats, Democrats 1.32, and Greens 0.44 seats.
• We start off giving 2 seats to Republicans, 1 to Democrats, and 0 to Greens. We have one left over. So we look for the largest remainder, which is 0.44, and so the last seat goes to the Greens.
• Now suppose we want to add one more seat. Then Republicans get 0.56 x 5 = 2.80, Democrats get 1.65, and Greens get 0.55.
• We again give 2 seats to Republicans, 1 to Democrats, and 0 to Greens. We now have two left over.
• Since Republicans have the largest remainder (0.80), we give them one leftover seat. Since Democrats have the second largest remainder (0.65), we give them the second leftover seat.
• So now Republicans get 3 seats, Democrats get 2 seats, and the Greens get no seats.
• Greens have lost a seat even though the number of seats has increased and they didn't lose any votes.
• (Here is the original Alabama paradox, which was the first objection to Hamilton's largest-remainder method used for determining Congressional seats.)
• The population paradox occurs when there are new voters.
• "Cut the Knot" has some nice Java applications with examples.
• For example, suppose there are 25 seats.
• Suppose Republicans get 70.62% of the vote, Democrats get 18.81% of the vote, and Greens get 10.57% of the vote.
• Then Republicans will get 17.65 seats, Democrats will get 4.70 seats, and Greens will get 2.64 seats.
• We assign 17 to Republicans, 4 to Democrats, 2 to Greens. There are 2 seats left over. Democrats get one and Republicans get one.
• So 18 R, 5 D, 2 G.
• Now change the percentages.
• Suppose Republicans get 70.10% of the vote, Democrats get 19.59% of the vote, and Greens get 10.31% of the vote.
• Then Republicans will get 17.52 seats, Democrats will get 4.90 seats, and Greens will get 2.58 seats.
• We assign 17 to Republicans, 4 to Democrats, 2 to Greens. There are 2 seats left over. Democrats get one and Greens get one.
• So 17 R, 5 D, 3 G.
• Greens now have more seats, even though they lost votes.