Mathematics and politics
Lecture notes, 4/8/03
Announcements
- The movie A Beautiful Mind, about the life
of mathematician John Nash, will be shown tomorrow night (Wednesday) at 8:00 pm in
Logan B17. Refreshments (pizza and soda) served at 7:30 pm.
-
John Allen Paulos will be coming to
speak at Penn on Tuesday, April 22, at 7:00 pm, in Meyerson Hall.
Current assignments:
Today: More alternative voting systems
Grading and end-of-semester plan
- Grading for the course will be determined approximately by the following scheme:
- 5%: Office hour visit
- 5%: Introductory essay
- 10%: Math in society essay
- 10%: Class participation
- 15%: Homework
- 15%: Escalation project
- 20%: Midterm
- 10%: Voting system quiz
- 10%: Election reform essay
I reserve the right to make small modifications to this weighting.
- There will be a homework assignment on voting systems given out on April 15, to be due April 22.
- There will be an in-class quiz April 24 on voting systems, and the John Allen Paulos talk.
- On April 22, the assignment for the final paper will be posted.
- Two weeks later, May 6, is the due date for this paper.
- The final paper will involve the following (more details to follow):
- Study a proposed reform of the political system (e.g. proportional representation, abolishing electoral college, etc.)
- Summarize the arguments in favor and opposed to such reforms, from a mathematical point of view.
- What current problem does the reform correct? What new problems could the reform create?
- Give explicit examples of how it would work.
- Finally, give your own opinion on it.
Alternative voting systems
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?
- Review of systems:
- Plurality or "first-past-the-post" (the current system in the U.S.; candidate with the highest number of first-place votes wins, even if less than 50%)
- Hare or instant runoff or single transferable vote (minor candidates eliminated from lists until one candidate gets
over 50% of the vote)
- Sequential pairwise voting, a method of picking the Condorcet winner (pairwise contests between candidates, in some arbitrary order (the agenda)
- Borda count (give numerical values to candidates based on how high they are in each voter's list)
- Dictatorship (a particular voter chooses the winner)
- Borda count:
For each voter:
- The candidate at the bottom of the preference list gets 0 points.
- The next highest candidate gets 1 point.
- Next highest gets 2 points.
- Continue like this...
- The number of points for the candidate at the top of the list is the number of candidates below him/her on the preference list.
Add up all the points for each candidate. Whoever gets the highest number of points is the winner.
- Example:
- Bush, Nader, Gore: 49
- Gore, Nader, Bush: 48
- Nader, Gore, Bush: 3
- Borda count:
- From the 49 voters who put "B, N, G": Gore gets 0 points. Nader gets 1 point. Bush gets 2 points.
- From the 48 voters who put "G, N, B": Bush gets 0 points. Nader gets 1 point. Gore gets 2 points.
- From the 3 voters who put "N, G, B": Bush gets 0 points. Gore gets 1 point. Nader gets 2 points.
- Totals:
- Bush gets a total of
2 × 49 + 0 × 48 + 0 × 3 = 98
- Gore gets a total of
0 × 49 + 2 × 48 + 1 × 3 = 99
- Nader gets a total of
1 × 49 + 1 × 48 + 2 × 3 = 103
- So Nader is the winner in the Borda count.
- Dictatorship is a real voting system, but it's not terribly fair.
- We include it in the list because it has very special properties.
- Kenneth Arrow's theorem says that a voting system that satisfies certain seemingly innocuous conditions must be a dictatorship.
- There is one more system that's popular, but rarely discussed with the others because it doesn't involve preference lists.
- In approval voting, each voter lists all the candidates that they would be satisfied with, in no particular order.
- Could be just one candidate, could be all but one candidate, could be every candidate or no candidate (in which case your vote wouldn't count).
- Whoever gets the highest number approving is the winner.
What about ties?
- Even in very large elections, ties are possible. (See Florida, 2000.)
- In small elections, such as a student club voting for its executives, they are even more likely.
- Any voting system we propose has to have a method of dealing with all possible scenarios; in particular it has to tell you what happens
if there is a tie.
- The convention used in the textbook is the following:
- Voting systems do not have to give just one social choice.
- The "social choice" could be two or more candidates.
- In this case it is called a "social choice set."
- The voting system tells you that any of the candidates in the social choice set is equally qualified.
- If you end up with a tie, you can pick the winner randomly (e.g. toss a coin) or use a dictatorship (e.g. the Supreme Court) or do something else.
- But we don't allow ties in the preference list. No individual voter can rank two candidates equally.
- How individual systems handle ties:
- Plurality: If several candidates all get the same highest number of first-place votes, then they are all winners.
- Instant runoff: If at any step,
several candidates all get the same lowest number of first-place votes, then they all get deleted from the list (unless they are all that's left).
If they are all that's left, then they are all winners.
- Sequential pairwise: f at any pairwise competition there is a tie, then both candidates remain on the list and both compete with the next candidate on the list, in order.
Example: if the agenda is Bush, Gore, Nader, Buchanan, and Bush and Gore tie in a pairwise contest, then the next pairwise contest is Bush-Nader.
- If Bush loses to Nader: Then the list is Gore, Nader, Buchanan. The next competition is Gore-Nader, and the sequential pairwise voting proceeds in the usual way.
- If Nader loses to Bush: Then the list is Bush, Gore, Buchanan. The next competition is Bush-Buchanan.
- If Bush loses to Buchanan, then Gore competes with Buchanan.
- If Buchanan loses to Bush, then Bush and Gore are both declared the winners.
- Borda count: If several candidates get the same highest Borda count total, then they are all declared winners.
- Dictatorship: Ties are impossible in a dictatorship.
An explicit example
- Suppose there are four candidates: A, B, C, and D.
- Suppose there are 8 voters, and that their preference lists are:
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A
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C
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A
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B
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B
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D
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A
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C
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B
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D
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D
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A
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D
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B
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D
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D
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B
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C
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D
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C
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D
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B
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C
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A
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B
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A
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A
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- Determine the winner(s) if each of the five systems is used. Assume: In sequential pairwise voting, the agenda is ABCD. Assume: voter #8 serves as the dictator.
- (on board)
- Answers:
- Plurality: A
- Instant runoff: A
- Sequential pairwise: Tie--B and D
- Borda count: D
- Dictatorship: C