21-102: Exploring Modern Mathematics
Homework 1
• Your answers to the problems below should be formatted as a PDF file and uploaded
to Gradescope by Monday, January 26 at 11:59pm.
• Explain your answers clearly and coherently, and justify any claims you make.
• The points for each assigned problem are listed on the left-hand side and sum to 20.
1. (5 pts) The 2021 Minneapolis City Council election used ranked choice voting. In the
Ward 2 election there were 3 main candidates - Yusra Arab (A), Robin Wonsley (W),
and Cam Gordon (G). The tables below show the 8,915 ranked ballots, after two minor
candidates were eliminated in round 1.
(a) Determine the winner of the election with respect to each of the major four voting methods that we discussed (plurality, ranked choice, Borda count, pairwise
comparison).
(b) A scholarly article about this election also describes a “downward monotonicity
paradox” in this example:
“If 80 of the voters who cast the ballot Arab, Gordon, Wonsley had instead
moved Arab down on the ballot and voted Gordon, Arab, Wonsley, then Arab
would be the IRV winner.”
Confirm those claims; Rewrite the table of ballots with the proposed changes to
those 80 voters, and then determine the winner using IRV (ranked choice).
(c) Based on all of these results, who do you think “should” win? If possible, come
up with a convincing explanation for why Gordon could be the winner, then do
the same for Wonsley and Arab.
2. (5 pts) The Coombs Method. This method is just like ranked choice voting except
that in each round we eliminate the candidate with the largest number of last-place
votes (instead of the one with the fewest first-place votes).
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(a) Find the winner of the Math Club election using the Coombs method.
(b) Give an example (with fully ranked ballots) that illustrates why the Coombs
method violates the Condorcet criterion.
(c) Give an example (with fully ranked ballots) that illustrates why the Coombs
method violates the monotonicity criterion.
3. (5 pts) Bucklin Voting. (Used in the early 20th century in many places around the
US.) The method proceeds in rounds.
• Round 1: Count the first-place votes only. If a candidate has a majority of the
first-place votes, that candidate wins. Otherwise, go to the next round.
• Round 2: Count the first- and second-place votes only (both 1st and 2nd place
votes are weighted equally). If there are any candidates with a majority of votes,
the candidate with the most votes wins. Otherwise, go to the next round.
• Round 3: Count the first-, second-, and third-place votes only (all of which
are weighted equally). If there are any candidates with a majority of votes, the
candidate with the most votes wins. Otherwise, go to the next round.
• Repeat for as many rounds as necessary.
(a) Find the winner of the Math Club election using the Bucklin method.
(b) Give an example (with fully ranked ballots) that illustrates why the Bucklin
method violates the Condorcet criterion.
(c) Explain why the Bucklin method satisfies the monotonicity criterion.
4. (5 pts) Suppose the following was proposed as a fairness criterion:
If a majority of votes rank X above Y , then the results of the election should have X
ranked above Y .
Give a single example (with fully ranked ballots) in which all four voting methods
discussed in the chapter violate this criterion. (Hint: Consider an example with no
Condorcet candidate.)
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