Cardinal Methods

Common Core Standards and Mathematical Practices

7th Grade: Number Systems

7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract integers and other rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

7.NS.A.3 Solve real-world and mathematical problems involving the four operations with integers and other rational numbers.

Mathematical Practices

MP.2 Reason abstractly and quantitatively.

MP.3 Construct viable arguments and critique the reasoning of others.

MP.4 Modeling with Mathematics

Learning Objectives

Understand the process of cardinal voting.
Identify the differences between cardinal and ordinal voting methods.
Explain the pros and cons of cardinal voting.
Explore real life examples where cardinal voting methods are used.
Think further about the effects of cardinal voting on the larger political environment using the discussion questions and further resources.

Vocabulary Used

Cardinal voting
Expressive
Independent rating
Approval voting
Range voting
Cumulative voting

Terms & Definitions

Comparisons

Examples

Practice & Discussion

What Is Cardinal Voting?

Cardinal voting is any system that allows for voters to give each candidate an independent rating.

The key word here is independent which is what makes cardinal methods different from the previously learned methods such as ranked choice voting. With cardinal voting, candidates aren't ranked against each other which means that Arrow's Theorem doesn't apply!

Cardinal voting can be expressive. When a ballot is expressive, voters can demonstrate on a scale how much they approve of a candidate.

Are there different forms of cardinal voting? YES!

It's not this!

But rather more like this...

Some Forms of Cardinal Voting Methods

Approval Voting

A voter has the choice of approving any number of candidates they like by somehow marking their names.
The candidate with the most approvals wins.
Connect with previously learned methods: If each voter is only allowed to approve of one candidate, this method gets reduced to the usual plurality vote.

Range Voting

A voter can score each candidate with some score in some range, like from 0-9.
Candidate with the highest score or the biggest average score wins.
If voters are only allowed to score with 0’s and 9’s, this method reduces to approval voting.

Cumulative Voting

A voter is asked to split one vote into fractional ballots, and distribute them among the candidates any way they like.
Candidate with the largest sum wins.
This method could also be used for elections electing multiple winners, where voters can be given more than one vote for them to split among candidates. More information here.

How Do These Methods Work?

Approval Voting

With this approval profile, we see that (5+3+2+2)=12 voters approve candidate A as the winner, (2+2+1+1)=6 voters approve B, and (3+2+1)=6 voters approve C.
So A is the winner of this election.

Range Voting

With this scored profile, we see that candidates are scored independently. Candidate A got (0+3+0+4)=7 points in total, B got (8+6+7+3)=24, and C got (5+6+5+4)=20 points.
So B got the most points and thus the winner of this election.

Cumulative Voting

With this fractional profile, we see that candidate A got (0+7/8+0+1/2+1/7+9/10)=4.17 fractional ballots in total, B got (2/3+1/8+0+0+3/7+1/10)=4.24 ballots, and C got (1/3+0+1+1/2+3/7+0)=4.60 ballots.
Candidate C has the largest sum and thus the winner of this election.

Comparisons of Voting Methods

How Is Cardinal Voting Different From Ordinal Voting?

As you've learned from the ranked choice voting module, ordinal voting methods ask voters to rank all candidates in an order of preference. So all candidates are considered against one another, and the same number of rank in an order can't be given to more than one candidate.

While for all the cardinal voting methods introduced above, all voters get to evaluate all candidates independently. i.e. their preference for one candidate need not affect their expression of preference for another candidate.

Pros & Cons of Various Types of Cardinal Voting

Real Life Examples Where Cardinal Methods Are Used

In governmental elections, the only place approval voting has been used is in Greece during the 19th and early 20th centuries. For each candidate, there were two “ballot boxes”, and each voter would get to put a single lead counter in one of them. You’d put your hand through a hole as you chose which, to hide your choice from poll watchers.

Approval voting is used by the UN in the multi-step process to elect the UN Secretary General. While three vote categories are used (“encourage”, “discourage”, and “no opinion”), only “encourage” votes are actually counted in the process, which matches how approval voting works.

Approval voting is often used in choosing officers for scientific organizations and for college student governments.

Dartmouth Alumni Association elections used approval voting till 2009; and Dartmouth students started to use approval voting to elect their student body president in 2011.

Approval voting is often used in choosing officers for scientific organizations.

The Mathematical Association of America, American Mathematical Society, American Statistical Association, Society for Industrial and Applied Mathematics...

Just recently in June, 2020, Fargo, a city of North Dakota, just voted to switch to an approval voting system. Becoming the first U.S. city to try approval voting.

Range voting is being used in this partisan office representative election.

Practice & Discussion

Try it out for yourself!

Check out the solutions once finished!

Approval voting is used for this election. Candidate A got 8+7=15 approvals, B got 15+8+5=28 approvals, and C got 15+7=22 approvals. Thus candidate B is the winner.
Voters are asked to score each candidate according to a given range, so range voting is used here. From the profile we can see that for each voter, they can give the same score to multiple candidates. This shows that each candidate is evaluated independently. How voters vote for one candidate isn’t influenced by how the voters felt toward other candidates. These are all properties of a cardinal voting method. For this election, candidate A got 7 points, B got 2+5+1+3=11 points, C got 5+5+5+4=19 points, and D got 4+3+4+2=13 points, so C is the winner.
Cumulative voting is used here, generating a fractional ballot. Candidate A got 1+⅓+¼ = 1.583 ballots, B got ¼+⅔ = 0.917, C got ⅓+¼ = 0.583, and D got ⅓+¼+⅓ = 0.917. So candidate A is the winner of this election.

Discussion Questions

Suppose that voters submit a mixed preference/approval ballot. A preference ballot is one that represents the preference order one voter from highest to lowest ranking. In a mixed ballot, voters give a full preference ballot and a full approval ballot for the slate. How that a candidate with the majority of first-place votes can fail to win the approval voting election.
Which of these methods are subject to strategic voting? For more information on strategic voting, check this out.
Taking into account the pros and cons of cardinal voting, do you think that these methods are worth implementing? Should we be using cardinal voting more instead of the more commonly used ordinal voting?
How does the "expressiveness" of a ballot allow for more voter subjectivity and voter choice?

Further Resources

The Mathematics of Politics by E. Arthur Robinson & Daniel H. Ullman
STAR Voting
Expressiveness in Approval vs. Ranked Ballots
What is an "equal vote"? Are there organizations working towards implementing more equal voting systems? Many of these organizations promote methods of cardinal voting!
- Equal Vote Coalition (US)
- FairVote (US)
- Electoral Reform Society (UK)

Angel Xu - Louisa Park - Lulu Alsaud

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