Mathematics

Thinking Mathematically

7th Edition

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Thinking Mathematically – eText (w/Access Code – MyMathLab ) – ISBN # 9780134705095

ANSWER

Voting and Apportionment

1. Preference table and how to use preference tables

A preference table is a table used to rank the voters choices in order of preference. It is used to show the most preferred candidate by showing the number of voters in the top row who voted for each option/candidate. Example is shown below for candidates A, B, C

From the preference table above the total votes are 11 + 9 + 5 + 3 = 28. In the first choice the most preferred candidate is candidate B who gets 11 + 9 = 20 votes, candidate A gets 5 + 3 = 8 votes, while candidate C gets 0 votes.

In the second choice candidate C gets 11 + 3 = 14 votes, B gets 5 votes, and A gets 9 votes.

In the third choice candidate C gets 14 votes, B gets 3 votes, and A gets 11 votes.

All the choices are used to determine the winner. Candidate B wins in first choice with 20 votes, second and third choices candidate C wins with 14 votes in both choices. Thus, Candidate B wins over candidates C and A.

1. Using plurality method to determine an election’s winner

In a plurality method, only the first choice is used to determine the winner. Thus, in the above table the winner is candidate B who has the most votes from the first choice. Votes from the first choice are candidate B = 11 + 9 = 20, candidate A = 5 + 3, and candidate C no votes. Thus, candidate B wins the election.

• Using Borda Count method to determine an election’s winner

Borda count method involves assigning candidates based on their ranking. This criterion is used to assign numbers from the last choice to the first choice. 1 point assigned for the last choice, 2 points assigned to the second-last choice. The highest point to the first choice. The total of the point values determines the winning candidate.

For example, in the table below;

Third choice is assigned 1 point, second choice is assigned 2 points, first choice is assigned 3 points. Therefore, the winning candidate is determine as follows

B: (11*3) + (9*3) + (5*2) = 70 points

C: (11*2) + (3*2) + (9*1) + (5*1) = 41 points

A: (11*1) + (9*2) + (5*3) + (3*3) = 53 points

The winner is candidate B, followed by candidate A then candidate C in third place.

1. Using plurality-with-elimination method to determine an election’s winner

Involves a series of rounds to determine the winner by counting first choice votes.

In round 1, count first choice votes and declare the winner if there is and if no majority winner then eliminate the candidate with the least votes.

In round two recount the first choice votes and check for majority winner. Repeat the process until a candidate has majority votes.

Example;

ROUND 1: there is a tie because candidate A and D have the majority votes, 22 votes each. Thus eliminate candidate with least votes, candidate B.

ROUND 2: Columns 1 and 2 represent the same candidates and thus can be combined to be as follows;

Candidate C has the least votes. Thus eliminate the candidate.

Both candidates A and E have the same majority votes, 39 votes each. Candidate D has the least votes. So eliminate candidate D.

From first choice; candidates A has 39 + 22 = 61 votes

While candidate E has 21 + 18 = 39 votes

Therefore candidate A has the majority votes and thus wins the election.