Day 19 in Mat115H

Today:

• Announcement:
  • Thanks to Dylan for an emailed couplet
  • If interested, I found a sermon written by Charles's father at Google Books, from 1838. I haven't read it yet!

• Fit the Sixth
  • Memoria Technica Couplets
    • and their use in mathematics: the logarithm.

  • 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!
    • The Colas versus the UnCola
    • Simple Majority -- 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

        4. Here's my favorite system: Instant run-off, until we have a majority. Who wins?

        Look, it seems like this should be easy. 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.

      • "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.

      • I want to share a story from my days in the Peace Corps in Togo.

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

      Carroll basically proposes double-elimination tournaments.

  • Parlimentary Representations
    • Principles of Parliamentary Representation
    • His work is considered a brilliant contribution.

  • 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.