Some voting methods

This sheet describes some voting methods which can be used when there are
three or more candidates.  All of these methods presuppose that a preference
schedule is given.  We will assume that ties do not occur.  (This is a 
reasonable assumption when there are millions of voters, but not when (e.g.)
the U.S. senate is voting.)

1.  Plurality.  The winner is the candidate who receives the most
    first place votes.

2.  Plurality with one runoff.  There are two rounds of voting.  In the first
    round, the top two candidates are selected.  A second round of voting is
    used to determine which of these is the winner.

3.  Plurality with repeated runoffs (eliminating one candidate each time).

4.  Condorcet.  If there is one candidate who would beat each of the other
    candidates in a head-to-head contest, that candidate is called the
    Condorcet winner.  This is not really a method, because there does not
    have to be a Condorcet winner.

5.  Borda count.  Assign some number of points (called a weight) to a 
    first-place vote, some number to a second place vote, and so forth.  Then 
    sum the points to arrive at a score for each candidate.  The candidate with 
    the highest score is declared the winner.  The outcome depends on the 
    particular assignment of weights.  For three candidates, you could use
    the weights 3,2,1.

6.  The Hare system.  There is a single round of voting, in which voters
    write their individual preference schedule on the ballot.  Once the ballots 
    have been tabulated, yielding a preference schedule for the entire
    electorate, the winner is determined according to the following scheme.
    First the candidate with the least number of first place votes is 
    eliminated and removed from the preference schedule.  This procedure is
    repeated until only one candidate remains.  That candidate is declared
    the winner.  (How is this method different from #3?)

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                  Assignment 1 for Chapter 11

I. Read pages 411-424, skipping the subsections on
"Elections with two alternatives", "Independence of Irrelevant
Alternatives" (near top of 418 -- middle of 419), and "Sequential Pairwise
Voting and the Pareto Condition" (about one page, starting near top of 420).

II. For each of the six voting methods, determine who wins, if the following
preference schedule is given:

              Number of Voters

Rank          3   7   5   4
-------------------------------
First         C   D   C   A
Second        A   A   D   D
Third         D   B   A   B
Fourth        B   C   B   C

Use the weights 4,3,2,1 for the Borda count.

III. Do problems 9, 10, 11, 12abc on the photocopied pages from the 3rd
edition.