Day 21 in Mat115H

Today:

• Announcements:
  • You have an exam coming up in a week. Again, it will feature some elements from Dodgson's life (from Fit the Fourth through Fit the Sixth), as well as some of the mathematics that we've discussed since our last exam. I'll give you a few more details next time, but an initial breakdown is as follows:
    • Modular arithmetic in the form of UPCs, ISBNs, checks, ...
    • Geometry
      • basic constructions (e.g. dropping perpendiculars, drawing equilateral triangles)
      • puzzles related to those that we did for our homework
      I'll be providing you with a compass and straight-edge for the exam.
    • Determinants -- simple example only! 2x2 case.
    • Election schemes, and how to game them
    • Dodgson's suggestions for Tournaments
    • Memoria Technica
  • Any other questions on "geometry homework"?
    • Last time:
      • Window problem (checked two strategies last time)
      • Fibonacci puzzling -- Check slopes!
      • Painting cubes
      • Cheating maps
      • How do you get the mirrored time, exactly?
      • The isosceles theorem
    • We didn't look at Graph traversals, and I think that we should!

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

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

  • Some results from elections from last we studied them:
    • 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 shows what happens when there's no transitivity in the system:

        	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.

    • 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.

• 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 ("instant run-off")-- p. 133
  • Method of Marks
  • Gaming the system: Dodgson writes a pamphlet on A Method of Taking Votes on More than Two Issues.
  • Method of Marks (revised) -- p. 135
  • Dodgson was not influential, according to Classics of Social Choice (which discusses Dodgson thoroughly)
  • 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" (also includes a section on how to write letters, and lots of other good stuff!).
  • 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"

  • Looking at his list of "desiderata" suggests that this is so (p. 147).

• 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.