Examples for first day on voting for Math 36

In each of the first three examples there are three candidates and 100 voters

Example 1

36 voters like A best, then B, and last C
24 voters like B best, then A, and last C
40 voters like C best, then A, and last B

Example 2

36 prefer A to B to C
10 prefer A to C to B
20 prefer B to A to C
10 prefer B to C to A
24 prefer C to B to A

Example 3
Preferences indicated by the order of candidates in each column:

Group                  I    II   III   IV
Number of voters 25  28  32  15
A   B    C   A
B   C    B   C
C   A    A   B

Homework relative to Example 3:

a)  Determine the winner by Plurality, Plurality with runoff, Hare, Coombs, & Borda methods.

b)  For each of these methods of voting, determine whether some group of voters has an insincere strategy that will lead to a more preferred outcome (assuming the rest of the voters vote sincerely), and state what insincere ranking would accomplish such a change in outcome.  For each such group, how many voters would have to make the needed insincere ranking?

c)  Are any of these strategies in conflict with each other?  For each strategy in your answer to question b), is there some group that could use an insincere strategy to counter the one you propose, or are some of the strategies you propose unanswerable (in the sense that no insincere counterstrategy exists).

d)  Find the Condorcet winner if there is one; find the Condorcet loser if there is one.