Comments for Homework

Chapter 1

I graded problems 6, 20, and 42. The most common mistake made was on problem 20, parts b and c (why the Borda Count election violated the Majority and Condorcet Critera). Carefully read the clarifications below.

Also, I took off a half-point for calculation errors, such as adding up the points wrong for the Borda Count. Always double-check your calculations.


Clarification of the Majority Criterion

In an election, a majority candidate is a candidate who receives more than 50% of the first-place votes. Not every election will have a majority candidate. For example, in an election where candidate A gets 4 first-place votes and B and C each get 3 first-place votes, no one is a majority candidate - there are 10 votes total, but no one has the requisite 6 or more first-place votes. Do not got "majority candidate" mixed up with "winner by plurality." Even though a majority candidate always wins by plurality, there will be a winner by plurality even when there is no majority candidate (read that sentence again until it make sense).

The Majority Criterion can be summarized "The majority candidate, if there is one, should win." Thus an election can only violate the Majority Criterion if there is a majority candidate. If there is no majority candidate, there can be no violation. To show that an election violates the Majority Criterion (e.g. Exercise 20b), you must demonstrate:

  1. One of the candidates has over half of the first-place votes, and so is a majority candidate
  2. That majority candidate did not win according to the method used

Clarification of the Condorcet Criterion

This will read a lot like the Clarification of the Majority Criterion above, because a lot of the underlying logic is the same.

In an election, a Condorcet candidate is a candidate who wins every head-to-head comparison against the other candidates. Not every election will have a Condorcet candidate. Checking for a Condorcet candidate looks a lot like doing the Method of Pairwise Comparisons, but do not get "Condorcet candidate" mixed up with "winner by pairwise comparisons." A Condorcet candidate always wins by the method of pairwise comparisons, but there will be a winner (or winners) by pairwise comparisons even when there is no Condorcet candidate. The winner by pairwise comparisons is the candidate who wins the most match-ups, but to be a Condorcet candidate you must win all match-ups. For a sports analogy, winning by pairwise comparisons is like having the best record in a tournament, but being a Condorcet candidate means going undefeated. Notice that a majority candidate (if there is one) will also be a Condorcet candidate, but there can be a Condorcet candidate when there is not a majority candidate.

The Condorcet Criterion can be summarized "The Condorcet candidate, if there is one, should win." Thus an election can only violate the Condorcet Criterion if there is a Condorcet candidate. If there is no Condorcet candidate, there can be no violation. To show that an election violates the Condorcet Criterion (e.g. Exercise 20c), you must demonstrate:

  1. One of the candidates wins in any head-to-head comparison with the other candidates, and thus is a Condorcet candidate
  2. The Condorcet candidate did not win according to the method used