The Math Anxiety Club Elections

The Math Anxiety Club, which has 37 members, needs to elect a president. Four club members agree to be candidates: Alice, Boris, Carmen, and Don. Assume that each member of the club orders the candidates by preference. The table below gives the number of members with each of these preference rankings.

Number of members
14
10
8
4
1
1st choice
A
C
D
B
C
2nd choice
B
B
C
D
D
3rd choice
C
D
B
C
B
4th choice
D
A
A
A
A

The question you have to decide is: who should win the election? Another way to put the question is this: what is the proper way of counting these votes? Some possible answers would be:

-look at the first place rankings only

-give points for the various ranks, and add up the points

-look at what happens when each candidate runs against one of the others

-do an "election with runoff"

What happens when you try these methods? How does one decide which method to use?

A different question about voting:

Suppose you have a committee composed of five people, each of which tends to make the right decision about 3/4 of the time. Does having the five people vote to decide increase or decrease the likelihood of making the right decision?

"Math Anxiety Club" data taken from Excursions in Modern Mathematics, by Peter Tannenbaum and Robert Arnold. (Prentice-Hall, 1992)