Math in Society, 2nd Edition: Tools for decision making

Now that we have studied how to apportion the number of representatives for each state, along with a brief history of the U.S. government, we want to look at how representatives and other public officials are elected. There are also many other situations where a group consensus is needed, and some type of voting occurs. You may be most familiar with systems where you get one vote and the candidate with the most votes wins. However, there are many other ways of determining a winner. We will look at several voting methods in this section.

For an election using any of these methods to be fair, each person must have an equal right to vote and an equal ability to exercise that right. The Voting Rights Act of 1965 prohibited any election practice that denies the right to vote based on race, but key portions of the law have been invalidated since then. Voter suppression continues today in different forms such as purging voter rolls, eliminating polling places and voter identification laws, which can also affect transgender people.

The John Lewis Voting Rights Advancement Act was passed in the U.S. House of Representatives in 2019 to restore elements of the Voting Rights Act and make it possible to enforce the law. It has not been taken up in the Senate, though. Follow this link if you would like to read a more detailed timeline of voting rights

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aclu.org/voting-rights-act-major-dates-history

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To use some of the methods we are going to study, we need to know more than just each person’s first choice. We also need to know their 2nd and 3rd choices, and so on. A ballot where each person ranks all of the choices in order of their preference is called a ranked choice ballot.

To simplify this data, we create a preference schedule which combines all the people who voted with the exact same ranking. For example, Tasha, Lupe, and Helena, all voted in the same way, HAO. So, we will list that combination once, with the number 3 at the top of that column. Three students also voted for AHO, and OHA. One student voted for the ranking AOH. Below is the preference schedule.

The voting method you may be most familiar with in the United States is the plurality method. In the plurality method, the candidate with the most first-choice votes is declared the winner.

There is a difference between a majority and a plurality. For a majority, a candidate must have over 50% of the votes. There are some states that require a majority in order to win an election. In more cases, though, only a plurality is required, which means having more votes than any other candidate. If a tie occurs without using ranked choice voting, then a new run-off election would be required.

If we know how many voters there are, we can calculate the minimum number of votes needed for a majority win.

Continuing Example 5.2.3 from above, there are 10 voters, so we divide that by 2 or multiply by 0.5 to find 50%. Then we use the next whole number to get just over 50%.

\(10 \div 2=5\text{.}\) Then we add one and get 6 votes.

Exactly half of the votes would be a tie, so we go to the next whole number. A candidate would need 6 out of 10 votes to win a majority. If there was an odd number of voters we would divide by 2 and get a decimal. Let’s say there were 13 voters:

\(13 \div 2=6.5\text{.}\) Then we go to the next whole number and get 7 votes. Votes have to be a whole number and rounding a decimal up also gives us just over 50%.

Whether we end up with a decimal or not, we always use the next whole number after dividing by 2.

A candidate can win a plurality with fewer votes than the majority, as long as they have more than any other candidate. It depends on the rule for a particular election.

In a system with only two major political parties like the United States, it can be nearly impossible for third-party candidates to break through. You may feel like you would be wasting your vote if you voted for a third-party candidate. Consider this example.

Situations like this can lead to insincere voting. Insincere voting is when a person doesn’t vote according to their actual preference for strategic purposes. In the case above, a person who wants to vote for a third-party candidate realizes it would take a vote away from their major party candidate. Not wanting to see their party lose the election, they vote for the major party candidate.

The Instant Runoff Method is a modification of the plurality method that attempts to address the issue of insincere voting.

The voting is done with ranked choice ballots and a preference schedule is generated. Then we tally all the first-choice votes. The candidate with the least first-place votes is eliminated, but any votes for that candidate are transferred to the voters’ next choice. The process continues until one candidate has a majority. We will show you how it works with an example.

The Instant Runoff Method is similar to the idea of holding runoff elections, but since every voter’s order of preference is recorded on the ballot, the runoff can be computed quickly without requiring a second costly election. It also makes it safe to vote for a third-party candidate knowing that your vote will go to your second choice if your first choice can’t win.

This voting method is used in several places around the world, including the election of members of the Australian House of Representatives, statewide elections in Maine, and local elections in Benton County, Oregon (Ranked Choice Voting Resource Center, 2020).

Borda Count is another voting method, named for Jean-Charles de Borda, who developed the system in 1770. In this method, points are assigned to each candidate based on their ranking; 1 point for last choice, 2 points for second-to-last choice, and so on. The point values are totaled, and the candidate with the largest point total is the winner. We’ll show you how to do it in the next example.

Note that Sylvania would have won with the plurality method. Borda count is sometimes described as a consensus-based voting system, since it can be used to choose a more broadly acceptable option over the one with the most first-choice support. This is a different approach than plurality and instant runoff voting that focus on first-choice votes; Borda Count considers every voter’s entire ranking to determine the outcome.

Because of this consensus behavior, Borda Count, or some variation of it, is commonly used in awarding sports awards. Variations are used to determine the Most Valuable Player in baseball, to rank teams in NCAA sports, and to award the Heisman trophy.

Another method, called pairwise comparison, the Condorcet method or Copeland’s method, attempts to be fair by looking at each pair of candidates separately.

In this method, we look at each pair as if they were the only two candidates running and determine which of the two is more preferred. In Copeland’s method, the more preferred candidate is awarded 1 point. If there is a tie, each candidate is awarded \(\frac{1}{2}\) point. After all the pairwise comparisons are made, the candidate with the most points, and hence the most pairwise wins, is declared the winner.

Variations of Copeland’s Method are used in many professional organizations, including election of the Board of Trustees for the Wikimedia Foundation that runs Wikipedia.

Now that we’ve seen an example of each voting method, let’s run through each method for the same scenario.

To see a very simple example of how difficult voting can be, consider the election below:

Notice that in this election: 10 people prefer A to B, 10 people prefer B to C, and 10 people prefer C to A.

No matter whom we choose as the winner, \(\frac{2}{3}\) of voters would prefer someone else! This scenario is called Condorcet’s Voting Paradox. In this situation, there is no fair resolution. There are many additional paradoxes and fairness criteria that are explained in David Lippman’s original version of Math in Society

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Math in Society: Voting Theory, by David Lippman

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Since it is not possible to have one method that is fair in every situation, it is important to decide which method is the most fair for any given situation.

For many public offices in the U.S., a sequence of two public votes is held: a primary election and the general election. For non-partisan offices like sheriff and judge, in which political party affiliation is not declared, the primary election is usually used to narrow the field of candidates.

For partisan positions, typically, either the top two candidates receiving the most votes move forward, or the top candidate from each party will move forward. This is called sequential voting - a process in which voters cast totally new ballots after each round of eliminations. Sequential voting has become quite common in television, where it is used in reality shows like The Voice.

In some states a closed primary is used, in which only voters of each party can vote for that party’s candidates. In other states, an open primary is used, where voters can pick the party whose primary they want to vote in. In other states, caucuses are used, which are meetings of the political parties, only open to party members. Closed primaries are often disliked by independent voters, who like the flexibility to change which party they are voting in. Open primaries have the disadvantage of raiding, though, where a voter could vote in their non-preferred party’s primary with the intent of selecting a weaker opponent for their preferred party’s candidate.

Regardless of the primary type, the general election is the main election. While rules vary state-to-state, for an independent or minor party candidate to get listed on the ballot, they typically have to gather a certain number of signatures to petition for inclusion.

In the next section we will look at a different type of election for the President of the United States.