Gerrymandering and the Political Process week 6

Mattie Di Giovanni

 

Investigation 6: Exploring Different Voting Methods

 

In the lecture, we discussed several different voting methods and talked about what properties make a voting method “fair.”  In this investigation, you will explore some of these methods and investigate which methods satisfy which fairness properties.

 

  1. The method currently used in the U.S. for most elections is plurality.  Under plurality, the voters only vote for their first-place candidate. The candidate with the most votes wins.

 

Suppose there are five candidates running (A, B, C, D, and E) and 20 voters in total.  The preferences of each of the voters are summarized in the following table, called a voter profile.

5 voters 4 voters 4 voters 4  voters 3 voters
First A B C D E
Second B C B B D
Third C E D E B
Fourth E D E C C
Fifth D A A A A
  1. a)  Who would win under plurality?  Do you see anything troubling about this result?

A, This is troubling because 15 voters have A as their last pick, and A doesn’t have that large of a majority either so it just doesn’t seem like a good way to pick

  1. b) In the lecture, (and your readings) we talked about an alternative called Ranked-Choice Voting (RVC), in which voters rank all the candidates and the result is determined in a series of steps.  

Step 1: The first place votes are counted and if a candidate receives more than 50% of the vote, they are named the winner.  If no one receives more than 50% of the vote, the candidate with the fewest first-place votes is eliminated and for those voters who supported the eliminated candidate, their second-place votes are moved up to the first-place spot.

Step 2: The first place votes are recounted, using the results from step 1. If a candidate receives more than 50% of the vote, they are named the winner.  If no one receives more than 50% of the vote, the next candidate with the fewest votes is eliminated and the process repeats.

 

Who would win the election under the RCV method?  

  1. c) Do you think this is a “better” result? Why?

D is a better result because over all more people would rather have D than any other candidate.

  1. The Marquis de Condorcet (1743-1791) was a French mathematician and philosopher who proposed that if a candidate beats every other candidate in a head-to-head election, then that candidate ought to be the winner. Such a candidate is now called a “Condorcet winner.”

 

Suppose there are five candidates running (A, B, C, D, and E) and 5 voters in total.

 

1 1 1 1 1
A E A D B
C C B E D
B A D C C
D B C A E
E D E B A
  1. a) If only A and B were running, who would win?  (Assume if a voter’ first choice is someone else, they vote for whichever of A or B they prefer.) A wins
  2. b) If only A and C were running, who would win? C wins
  3. c)  Continue, looking at enough possible pairs of candidates to determine if there is a Condorcet winner.  
  4. d)  Is there a Condorcet winner for this voter profile? No There is no condorcet winner
  5. e) Is there a Condorcet winner for the voter profile in problem 1? (Don’t forget there is more than 1 voter in each column.) B is the condorcet winner in profile one

 

  1.    The Borda method is named after the 18th-century French mathematician and political scientist Jean-Charles de Borda who devised the system in 1770.   Variations of the method are currently used to elect officials in Iceland and Slovenia. In the Borda method, each voter ranks their candidates from first to last and assigns point values based on their rankings: 0 points for last place, 1 point for second to last, 2 points for third to last, etc.  The winning candidate is the one with the highest total point value.

 

Who would win the election under the Borda method in problem 1?

For Profile one in Problem one the winner is: B with 61 votes

  1. The Borda method is attractive because, like RCV,  it uses information about the voters’ preferences for all the candidates (not just their top candidate).  However, it has some disadvantages.
  2. a)  Which of A and B has the larger Borda score in the voter profile below?

B

5 4 4 4 3
B C A D E
C A B A A
E B E B B
D E D E D
A D C C C
  1. b) Suppose Candidates C, D and E dropped out, and voters had to choose between A and B.  Who would win, based on the preferences listed above? Why might this be a problem?

A would win 15-5 and this is a problem because it becomes a 2 party system essentially, because the second candidate gets zero votes.  

 

  1. In the Condorcet method and several other methods, candidates are paired up in head-to-head matches.  More commonly in sports, there is an agenda which determines who plays who.  For an election, an agenda is often a list that determines which candidates run against each other.  For example, suppose the agenda is B, A, C, D, E. This means that the first two candidates (B and A) have a head-to-head election.  The winner of that moves on to a new head-to-head election with the next person on the list; the winner of that election moves to the next candidate on the list, and so on.  The last person standing is declared a winner. This type of agenda is called a sequential agenda.

 

  1. a) Using the voter profile in problem 2, would win using the sequential agenda A, B, C, D, E? (Assume that in each head-to-head election, each voter votes for the candidate they prefer among the two running.)  D would win
  2. b) Would your answer in (a) change if the sequential agenda was B D A C, E? Yes, C would be the winner this way
  3. c) Who would win in the voter profile in problem 1 using the sequential agenda A, B, C, D, E?  
  4. d) Would your answer change in (c) if another agenda was used? No Because B is the condorcet winner so B will always win.

 

Fairness Criteria for Voting Methods

 

Rather than argue whether the results of an individual election are fair, mathematicians look at the properties of each voting method to determine if the method is fair (or not).  There are several well-known criteria that people use to assess voting methods. We will look at a number of them below.

 

  1. A voting procedure satisfies monotonicity (MONO) if when a voter changes their preferences in favor of a winner, the candidate should remain a winner.  

 

First, we will think about the plurality method. Consider the voter profile in problem 1.

  1. a) Who won under plurality?   A
  2. b) Now suppose the 3 voters in the last column switched their preferences so that the winner from (a) moved up the list.  Would the winner still be the winner? Explain in a sentence why the plurality method always satisfies monotonicity. Yes because they had most votes originally but now they have even more because the 3 voters changed to candidate A. . So when people change their preferences in favor of the winner they remain the winner, as per the example I just completed.
  3. Now we will think about RCV and the monotonicity property  
  4. a) Find the winner under RCV for the following voter profile:
5 4 6 2
C A B A
B C A B
A B C C

 

Candidate B would win

  1. b) Suppose the two voters in the last column switch their preferences for A and B.  Find the winner under RCV for this new voter profile.
  2. c) Explain why this example shows that the RVC method violates monotonicity.

Because if A and B are switched for the two voters than candidate C wins and this violates MONO because C was no the original winner.

  1.  Next, we will think about the Borda method and the monotonicity property.  Consider the voter profile in problem 3.
  2. a) Who was the winner under the Borda method? B
  3. b) Suppose the last 3 voters in the last column switched their preferences so that the Borda winner moved up the list.  Would the Borda winner still be the Borda winner? Explain. Yes, because the winner had already won but moving them up to a higher rank only increases their points having them remain as the winner.
  4. c) Explain why, using the Borda method, if when a voter changes their preferences in favor of a winner, the candidate always remains a winner. Because the candidate only gains more points and if they were already the winner then they only gain more points nothing is subtracted to they remain as the winner. They just have more points than before.

 

  1.  A voting procedure satisfies Independence of irrelevant alternatives (IIA) if it is impossible for a “loser” to become a “winner” unless at least one voter reverses their order of the “loser” and the “winner”.

 

An example is below:

 

In the left voter profile, the plurality winner is A.

In the right voter profile, the voters in the middle column switched their preferences for A and C. And now the plurality winner is B.

Notice that no one changed their minds about their A/B preference-  yet A lost to B because of the “irrelevant alternative” of C.

2 2 3 2 2 3
A A B A C B
B C A B A A
C B C C B C

This shows that the plurality method does not satisfy the Independence of irrelevant alternatives criterion.

 

  1. a) Now you will look at the Borda method and the IIA criterion.  Determine the Borda winner in the voter profile below left. C would win with 13 points.
  2. b) Suppose the voters in the last 2 columns switch their preferences of B and D as shown below right.

Determine the new Borda winner. The winner would change to B.

  1. c) Explain why this example shows that the Borda method violates Independence of irrelevant alternatives The Borda method violates IIA because an Irrelevant alternative- a candidate with lower votes then most can be switched in the Borda method to help another candidate win without really affecting the irrelevant alternative. For example B and C were very close in points whereas D had less than 3 other candidates. But when it was switched B now became the winner and D only dropped down a few more points. So even though D had really no chance at winning, by switching them B was able to win which is why its violated.
3 1 1 3 1 1
A C E A C E
B D C B B C
C E D C E B
D B B D D D
E A A E A A

 

  1. A voting procedure satisfies the Pareto condition if when everyone prefers one candidate to another,  then the second candidate cannot win. (Note that this doesn’t mean that first must win, because there might be a third candidate that people like even better.)  The Pareto condition is named after the Italian economist Vilfredo Pareto (1848-1923).

 

  1. a) Suppose an election was held with several candidates using the plurality method.  Explain why if everyone prefers one candidate to another then the second candidate cannot win. Because it follows the Majority. So in plurality everyone only votes for the person they want. So whoever wins the majority is the winner.
  2. b) Repeat (a) for the Borda method. This rule also applies to the Borda method because if everyone prefers the top candidate then the top candidate will have the most votes in the rank.
  3. c) Repeat (a) for RCV. For RCV if everyone votes for the same top candidate then they will win in the first round because they will have more than 50% of the votes. This rule works for all 3 voting methods.

 

  1.  In this question, you will begin to fill in the chart below, answering yes or no depending on whether the election method listed satisfies the corresponding fairness property. The CWC refers to the Condorcet winner criterion which says that if a voter profile has a Condorcet winner, than the voting method should always choose that candidate.
  2. a) There are several entries in the chart that you should be able to fill in based on the previous problems. Enter those answers in the chart.
  3. b) Construct an example to show that RCV does not satisfy the Condorcet Winner Criterion.

The RVC Method does not satisfy the CVC because it is not possible to always have one, an example is in Voter Profile number 2, it basically shows that it’s split too evenly among the different candidates and 1 candidate does not dominate.

 

CWC IIA Pareto Condition Monotonicity
Plurality Yes Yes Yes Yes
Borda Count Yes No Yes Yes
RVC No No Yes No
Sequential Agenda No yes yes No

 

  1. In 1951, economist Kenneth Arrow proved a famous theory saying that the only voting method satisfying both the Pareto condition and the IIA criteria was a dictatorship.  (Meaning one voter’s choice always wins.) For many years, this was considered proof that there can never be a “fair” voting method. Recently, however, people have begun rethinking about that the IIA property may not be that important.  

 

Most of the methods listed in the chart satisfy the Pareto condition and monotonicity, however, RVC violates monotonicity.  This makes it a seriously flawed method, in many mathematicians eyes. Others argue that failures of monotonicity in real life aren’t very likely, and so it’s not a serious concern.

 

Mathematician Donald Saari uses some geometric arguments to suggest that the Borda count is the most appropriate method. Others argue for other voting methods. There are many!  Here is a partial list:

 

(i) Under Range voting (sometimes called Score voting), everyone ranks each candidate on a scale of 1 to 10 (or some other number).  The candidate with the highest total score wins.

(ii)  Under Approval Voting (championed by political scientist Steven Brams), everyone simply votes for all candidates that they “approve” of.  (This could be one candidate or more than one candidate). The candidate with the most votes wins.

(iii) Under Cumulative voting, each voter is given a fixed number of points, (such as 10), and they may distribute them among the candidates however they wish.  (They can give them all to one candidate, or divide them over several.) The candidate with the most points wins.

 

  1. a) Discuss what you see as the advantages and disadvantages of each of the three methods above.  

Under range Voting: Pro- every candidate gets a vote of some kind con: what if you don’t know/ like some of the candidates- creates inaccurate voting.

Under Approval Voting: pro: makes voting easier for the voter because they can pick several candidates they would like. Con: Too many options to pick from, you would need to be very educated on each person

Under Cumulative voting: Pro: Can distribute points among multiple candidates: con: is basically like having 10 votes so why not just vote for one candidate? It doesn’t make the most sense to me, I also see that could make it hard for voters to determine how to split up their points.

  1. b) There are many reasons to like one voting method over another. In addition to the fairness criteria we have discussed, voting methods can be critiqued based on the likely impact on the candidates’ platforms or campaigns. For instance, advocates of approval voting claim that it would encourage candidates to appeal to a wide spectrum of voters (rather than focus on the left or right wing of the political spectrum).

Why is such a claim made for approval voting and do you agree with it? Do you think it is a good thing to adopt a voting method which encourages candidates to “move to the center” of the political spectrum?

I think that in the US it appeals to most people to have a candidate who they support on many issues then one central person who they agree with on some issues. I think that it makes a lot more sense to have a spectrum because a centralized political cant really satisfy both ends. So no I don’t agree with it.

 

  1. c) Which method (among all the ones we have discussed) do you think should be used for the next presidential election?  Why?

I think the RVC method should be used because it allows for voters to have a chance at having their second pick go into office. I think that the US should move away from a two party system and expand into multiple parties or there should be multiple candidates for each party. I think RVC allows for more people to be happy if one of there top picks gets picked rather than either the person they vote for wins or loses. It creates more sense of win for more people. And creates a less Us versus Them approach to votings.

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