Day 20 in Mat115H

Today:

• Announcement:
  • Please read Fit the Seventh for next time. Your next assignment comes out of the reading from the Heart of Mathematics.
  • Return of your "geometry homework", and questions from it.
    • Window problem:
      • Dylan's approach to the window problem
      • Those of you who tilted, and made a diamond -- how do you know the amount of light let in? (Kayla!)
    • Fibonacci puzzling -- Check slopes.
    • Graph traversals
    • Painting cubes
    • Cheating maps
    • How do you get the mirrored time, exactly? (Hannah)
    • The isosceles theorem
    • Finishing p. 28

• Fit the Sixth
  • Last time:
    • Memoria Technica Couplets
      • and their use in mathematics: the logarithm.
    • Some results from elections:
      • Problems:
        • Winning by swamping the opposition: "Colas versus the UnCola"
        • "Transitivity" means that, if we prefer A to B, and B to C, then we'll prefer A to C. Condorcet's Paradox says there's no promise of transitivity!

          Example:

          	5 voters' ranking	5 voters' ranking	5 voters' ranking
          First Place	A	B	C
          Second Place	B	C	A
          Third Place	C	A	B

          Look at the beautiful symmetry in this table.... It's responsible for the problem, in many ways.
        • Gaming the system

      • We want a system that has the following properties, given each voter's list of ordered rankings for the candidates:
        1. Better is better: If a voter changes his or her vote in favor of Candidate X, then that shouldn't hurt Candidate X.
        2. Ignore the irrelevant: throwing out a loser will not change the outcome.
        3. Go along with the consensus: if everyone agrees, then we have a winner.

        Here's the main, astonishing result:

        The only voting scheme that satisfies these three very reasonable requirements is a dictatorship!

        That is, one person makes the decision.

        It was not a "fluke" that different methods gave different results. As mentioned in the text, mathematicians can "rig" a seemingly benign system that will hand the election over to a preferred candidate.

      • an example from my days in the Peace Corps in Togo.

  • Slide rules -- their basis in logarithms
    • an animation of a slide rule

  • More Memoria Technica Couplets
    • How can I remember my office phone number: X5377?

  • Elections -- Dodgson becomes an administrator, and hence has to deal with decision-making at the university. You'll find yourself remembering his plight some day!
    • Simple "Majority" (i.e. plurality) -- p. 133
    • Method of Elimination -- p. 133
    • Method of Marks
    • Method of Marks (revised) -- p. 135
    • A Method of taking votes on more than two issues (preview)
    • Section 8.4 of the Heart of Mathematics: Peril at the Polls
      • "Sixth grade vice-presidential vote"

        	6 boys prefer	4 boys prefer	8 girls prefer	4 girls prefer
        First Choice	Chad	Chad	Gwyn	Courtney
        Second Choice	Courtney	Gwyn	Courtney	Gwyn
        Third Choice	Gwyn	Courtney	Chad	Chad

        Three voting schemes, three different winners.

        1. Plurality voting -- Chad wins
        2. Vote for Two -- Courtney wins
        3. Borda Count Method -- Gwyn wins

      • My favorite system is instant run-off, until we have a majority. Who wins?

        This one was proposed for situations like the "Nader voting" that helped defeat Al Gore and so elect George Bush in 2000: if Nader voters could have used instant run-off, they could have voted for their favorite (Nader), but then fallen back to Gore when Nader didn't take the vote.

  • Lawn Tennis Tournaments
    • This Contribution of Carroll in "social choice"
    • See the (bad) scheme on page 143.
    • We consider the scheme in which 8 players play each other so that the top three players are not knocked out prematurely, then how can we proceed?

      Carroll basically proposes double-elimination tournaments.

    • Lawn Tennis Tournaments: The True Method of Assigning Prizes with a Proof of the Fallacy of the Present Method (p. 143)

  • Parlimentary Representations
    • Principles of Parliamentary Representation
    • Francine Abeles: Dodgson "made important contributions both to political science with his proposals for proportional representation and to mathematics with ideas that half a century later became known as game theory."

      Abeles points out, "Dodgson's main concerns were selecting the most preferred candidates to hold office, conducting elections properly, promoting minority representation, and achieving fair representation"

  • What new things do we learn about Dodgson in this Fit?
    • At work
    • At "play"
      • Ariel's Song from The Tempest

        Full fathom five thy father lies;
                      Of his bones are coral made;
            Those are pearls that were his eyes:
                      Nothing of him that doth fade,
            But doth suffer a sea-change
            Into something rich and strange.
            Sea-nymphs hourly ring his knell:
                                      Ding-dong.
            Hark! now I hear them -- Ding-dong, bell.