Announcements

John Allen Paulos will be coming to speak at Penn on Tuesday, April 22, at 7:00 pm, in Meyerson Hall.

He is well-known for applying mathematics to real-world situations, especially politics, in his articles and books (including Innumeracy and A Mathematician Reads the Newspaper).

Mark your calendars: attending will be part of your grade. (If you absolutely cannot attend due to a time conflict, a makeup assignment will be given to you.)

Current assignments:

• Problem set on voting systems
• Read Sections 5.1--5.3 of Mathematics and Politics.

Alternative voting systems

• Consider the Presidential election of 2000.
• For simplicity, we will just consider the election in Florida.
• Approximate the vote totals as follows:
  • Bush: 49%
  • Gore: 48%
  • Nader: 3%
  (Actually the vote was a statistical tie, according to John Allen Paulos.)
• In the plurality system, Bush wins since he gets the largest percentage (even though it is not more than 50%).
• Some people feel this result does not reflect majority will. People consider the winner to have less legitimacy.
• If there were a runoff election, the two top vote-getters would face off with no other candidates.
• If all the Nader voters preferred Gore to Bush, then in the runoff the results would be
  • Bush: 49%
  • Gore: 51%
  The runoff would ensure that the winner got a majority vote. Many countries do this.
• In some sense the runoff better reflects democratic preferences.
• But runoff elections are costly and often get much lower turnout than general elections.
• So the question is:

  How do we determine a winner in a single election with more than two candidates, in such a way that everyone agrees the choice is fair and democratic?

• Instead of each person voting for their top choice, each person gives their preferences in order.
• So Republicans might give their preferences as
  • Bush, Gore, Nader
  or
  • Bush, Nader, Gore
  and similarly for Democrats and Greens.
• Let us assume for simplicity that the preferences look like:
  • Bush, Nader, Gore: 49%
  • Gore, Nader, Bush: 48%
  • Nader, Gore, Bush: 3%
  (We are basically assuming everybody is voting for the "lesser of two evils," i.e. the Bush voters are mainly voting against Gore and would prefer anyone over Gore. This is being pretty generous to Nader.)
• One popular alternative: instant runoff (also called the Hare system or single transferable vote).
• (Advocated by the Center for Voting and Democracy and many minor parties.)
• Instant runoff:
  1. If any candidate has over 50% of the vote, we are done. (not in this case).
  2. Find the person with the fewest first-place votes (in this case, Nader).
  3. Eliminate this person from all preference lists. (in this case, we get:
     • Bush, Gore: 49%
     • Gore, Bush: 48%
     • Gore, Bush: 3%
  4. Now see if any candidate has over 50% of the vote (in this case, Gore does). If not, go back to Step 1 and eliminate the next candidate.
• A more obscure voting system involves studying all possible pairwise contests first (without eliminating any candidate).
• There are many ways to do this (one in the book is called sequential pairwise voting).
• Again let's assume the preferences look like this:
  • Bush, Nader, Gore: 49%
  • Gore, Nader, Bush: 48%
  • Nader, Gore, Bush: 3%
• Now look at each of the possible two-person contests:
  • Bush vs. Nader: 49% prefer Bush over Nader, 51% prefer Nader over Bush
  • Bush vs. Gore: 49% prefer Bush over Gore, 51% prefer Gore over Bush
  • Nader vs. Gore: 52% prefer Nader over Gore, 48% prefer Gore over Nader
• Notice: Nader beats both Bush and Gore in pairwise contests.
• When one candidate can beat all others in pairwise contests, that candidate is called a Condorcet winner.
• Methods which pick a Condorcet winner are advocated on this web site, for example.
• Gore is not a Condorcet winner (he beats Bush but not Nader).
• Bush is not a Condorcet winner (he loses to both Gore and Bush).
• The idea behind sequential pairwise voting is to make sure the Condorcet winner really wins the election.
• Sequential pairwise voting
  1. Set up an agenda, an arbitrary ordering of the candidates (e.g. alphabetical: Bush, Gore, Nader)
  2. See who wins the pairwise race between the first two candidates (Bush and Gore: Bush loses).
  3. Eliminate the loser (so what's left is Gore, Nader)
  4. See who wins the pairwise race between the next two candidates (Gore and Nader: Gore loses).
  5. Whoever is left is the winner (Nader).
• This is not entirely unreasonable: polling data suggests Nader was the Condorcet winner. (Did Ralph Nader elect George W. Bush?)
• Questions:
  • Does this always elect a Condorcet winner? Yes. (Exercise 11).
  • Does the order of the agenda matter? Yes, if there is no Condorcet winner! (Exercise 9).
• So we have three different voting systems (plurality, instant runoff/Hare, and sequential pairwise/Condorcet), which generate three different winners in the exact same election.
• Which one is the most democratic?
• Depends what is most important to you:
  • The most people strongly in favor of the winner, even with minor candidates (plurality)
  • A majority favoring the winner, with minor candidates eliminated (instant runoff)
  • A winner that majorities agree is better than any alternative (Condorcet)
• We will find that no voting system is perfect!