Answer to Question #139380 in Operations Research for Joseph Se

Question

A. Describe apportionment.
B. What are the different methods of apportionment? Describe each method.
C. Define Huntington-Hill Number.
D. What is Voting?
E. What are the different voting methods used to determine the winner among candidates or options? Describe each method.
F. What are the four basic criteria of fairness? Describe each criterion.

Answer 1

A. Apportionment is the act of dividing costs between different accounts in a fair way, or the amount that is put into each account.

B. Methods of apportionment:

Hamilton’s Method:

• Step 1. Calculate each state’s lower quota.
• Step 2. Round the standard quota’s down and give to each state its lower quota
• Step 3. Give the surplus seats (one at a time) to the states with the largest residues (fractional parts) until there are no more surplus seats.

Jefferson’s Method

• Step 1: Find a ‘suitable’ divisor d.
• Step 2: Using d as the divisor, compute each state’s modified quota (modified quote = state population/d).
• Step 3: Each state is apportioned its modified lower quota. 

Adam’s Method

• Step 1: Find a ‘suitable’ divisor d.
• Step 2: Using d as the divisor, compute each state’s modified quota (modified quota = state population/d).
• Step 3: Each state is apportioned its modified upper quota.

Webster’s Method

• Step 1: Find a ‘suitable’ divisor d.
• Step 2: Using d as the divisor, compute each state’s modified quota (modified quote = state population/d).
• Step 3: Find the apportionments by rounding each modified quota to the nearest integer.

Huntington-Hill Method

• Step 1: Find a ‘suitable’ divisor d. [Here a suitable divisor means a divisor that produces an apportionment of exactly M seats when the quotas (populations divided by d) are rounded using the Huntington-Hill rounding rule.]
• Step 2: Find the apportionment of each state by rounding its quota using the Huntington-Hill rounding rule.

C. Define Huntington-Hill Number:

D. Voting is a way for a group of people to select one from among several possibilities.

E. Voting methods

Plurality Method: The candidate with the most first-place votes wins the election.

The Borda Count Method (Point System): Each place on a preference ballot is assigned points. Last place receives one point, next to last place receives two points, and so on. Thus, if there are N candidates, then first-place receives N points. Now, multiply the point value for each place by the number of voters at the top of the column to find the points each candidate wins in a column. Lastly, total up all the points for each candidate. The candidate with the most points wins.

The Plurality with Elimination Method (Sequential Runoffs): Eliminate the candidate with the least amount of 1st place votes and re-distribute their votes amongst the other candidates. Repeat this process until you find a winner. Note: At any time during this process if a candidate has a majority of first-place votes, then that candidate is the winner.

The Method of Pairwise Comparisons: Compare each candidate to the other candidates in one-on-one match-ups. Give the winner of each pairwise comparison a point. The candidate with the most points wins

F. Four basic criteria of fairness

Majority Criterion: If candidate X has a majority of the first-place votes, then candidate X should be the winner of the election.

• The majority criterion is always satisfied by the Plurality Method, the Plurality with Elimination Method, and Pairwise Comparison Method.
• The Borda Count Method does not satisfy the majority criterion. This means that the Borda Count Method does not always select the candidate with the majority of first place rankings. 

Condorcet Criterion: If candidate X is preferred by the voters over each of the other candidates in a head-to-head comparison, then candidate X should be the winner of the election.

• The Condorcet criterion is always satisfied by the Method of Pairwise Comparison.
• The Borda Count Method, the Plurality with Elimination Method, and the Plurality Method might select a Condorcet candidate, but they can also fail to honor the criterion. 

Monotonicity Criterion: If candidate X is a winner of an election and, in a reelection, the only changes in the ballots are changes that favor X (and only X), then X should still be the winner.

• The Plurality Method, the Borda Count Method and the Pairwise Comparison Method always satisfy the monotonicity criterion.
• The Plurality with Elimination Method can violate the monotonicity criterion.

Independence of Irrelevant Alternatives Criterion (IIA): If candidate X is a winner of an election and in a recount one of the nonwinning candidates withdraws or is disqualified, then X should still be the winner.

• The Plurality Method, the Borda Count Method, the Pairwise Comparison Method, and the Plurality with Elimination Method can fail to satisfy the Irrelevant Alternative criterion.