January 24, 2006

From MathWiki

  • We talked a little about the solution to the number of 2^2 x 2^2 Sudoku puzzles. You should do this exercise but I wasn't planning to grade it and it isn't necessary for you to hand in. I would like to see your answer though.
  • We talked about some of the obsticals/issues with 'Arithmagons.' There were a number of people who were not in class last week so I gave people an extra week to hand the two problems that were due in.
  • I asked you to do three more problems due two weeks from tonight. 'Not Cricket,' 'Sums of squares,' and 'Gossip.' We will talk more about problem solving strategies next week and, in particular, these problems.
  • Then I started talking about elections. I had done a little bit of research on the mathematics of voting and elections and there was lots to talk about because of the election. This turned out to be a much bigger topic than I orginally suspected. We compared our predictions and expectations to the actual outcome of the elections.
  • We talked about polls and why national poll numbers might not correspond to election results or seats in parliment.
  • I asked the question: "Is it possible to have 3 parties in two ridings with the winner of the popular vote not getting either of the two ridings?" Danielle came up with an example:
Example poll
Party Quebec Ontario Total
C 49 49 98
B 51 0 51
L 0 51 51

The question is, does this happen 'in real life' and the answer is yes, because ridings/voting districts are sometimes artifical or have natural influences on the voting population (gerrymandering (http://dictionary.reference.com/search?q=gerrymandering)).

  • Sometimes there are weird election results where almost 2/3rds of the population can't stand a candidate but he/she wins anyway. Is this representative of 'the will of the people?'
  • We talked about some other voting schemes
  1. Preferential voting systems (http://www.australianpolitics.com/voting/systems/preferential.shtml)
  2. Approval Voting (http://garsia.math.yorku.ca/~zabrocki/math1590w06/articles/ApprovalVoting.pdf)
  3. First-past-the-post (the system that U.S. and Canada uses)
  4. weighted count voting (http://garsia.math.yorku.ca/~zabrocki/math1590w06/articles/BordaVoting.pdf)

There are at least four different fairness criteria (http://www.ctl.ua.edu/math103/Voting/whatdowe.htm#Fairness%20Criteria) by which we should evaulate if a voting system is 'good'.

  • Finally, given a parliment that was elected and now has members that are in various parties that vote in blocks (this seems to be what happens in Canada), how do we evaulate how much influence parties have? We talked about two different measures of power in a parlimentary system (http://garsia.math.yorku.ca/~zabrocki/math1590w06/articles/WeightedVotingSystems.pdf). We calculated the power index (both of them) for the current Canadian parliment and it turns out that the Bloc Quebecois has the same 'power' as the Liberals even though they have less than half the number of seats.

Previous class : January 17, 2006

Next class : January 31, 2006

Main class page : Mathematics 1590 Nature of Mathematics II