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Study Guides > Math for Liberal Arts: Co-requisite Course

Approval Voting

Up until now, we’ve been considering voting methods that require ranking of candidates on a preference ballot. There is another method of voting that can be more appropriate in some decision making scenarios. In this lesson we will study the method of Approval Voting, its flaws, and how voting is done in the US. With Approval Voting, the ballot asks you to mark all choices that you find acceptable. The results are tallied, and the option with the most approval is the winner.


A group of friends is trying to decide upon a movie to watch. Three choices are provided, and each person is asked to mark with an “X” which movies they are willing to watch. The results are:
Bob Ann Marv Alice Eve Omar Lupe Dave Tish Jim
Titanic X X X X X
Scream X X X X X X
The Matrix X X X X X X X
Totaling the results, we find: Titanic received 5 approvals Scream received 6 approvals The Matrix received 7 approvals. In this vote, The Matrix would be the winner.
In the following video you will see the example from above. https://youtu.be/-8put6XKw20?list=PL1F887D3B8BF7C297

Try It Now

Our mathematicians deciding on a conference location from earlier decide to use Approval voting. Their votes are tallied below. Find the winner using Approval voting.
30 10 15 20 15 5 5
Seattle X X X X
Tacoma X X X X X
Puyallup X X X X
Olympia X X X

What’s Wrong with Approval Voting?

Approval voting can very easily violate the Majority Criterion.


Consider the voting schedule:
80 15 5
1st choice A B C
2nd choice B C B
3rd choice C A A
Clearly A is the majority winner. Now suppose that this election was held using Approval Voting, and every voter marked approval of their top two candidates. A would receive approval from 80 voters B would receive approval from 100 voters C would receive approval from 20 voters B would be the winner. Some argue that Approval Voting tends to vote the least disliked choice, rather than the most liked candidate.
Additionally, Approval Voting is susceptible to strategic insincere voting, in which a voter does not vote their true preference to try to increase the chances of their choice winning. For example, in the movie example above, suppose Bob and Alice would much rather watch Scream. They remove The Matrix from their approval list, resulting in a different result.
Bob Ann Marv Alice Eve Omar Lupe Dave Tish Jim
Titanic X X X X X
Scream X X X X X X
The Matrix X X X X X
Totaling the results, we find Titanic received 5 approvals, Scream received 6 approvals, and The Matrix received 5 approvals. By voting insincerely, Bob and Alice were able to sway the result in favor of their preference.

Voting in America

In American politics, there is a lot more to selecting our representatives than simply casting and counting ballots. The process of selecting the president is even more complicated, so we’ll save that for the next chapter. Instead, let’s look at the process by which state congressional representatives and local politicians get elected. For most offices, 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. Typically, the two candidates receiving the most votes in the primary will then move forward to the general election. While somewhat similar to instant runoff voting, this is actually an example of 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 competition shows like American Idol. Congressional, county, and city representatives are partisan offices, in which candidates usually declare themselves a member of a political party, like the Democrats, Republicans, the Green Party, or one of the many other smaller parties. As with non-partisan offices, a primary election is usually held to narrow down the field prior to the general election. Prior to the primary election, the candidate would have met with the political party leaders and gotten their approval to run under that party’s affiliation. In some states a closed primary is used, in which only voters who are members of the Democrat party can vote on the Democratic candidates, and similar for Republican voters. In other states, an open primary is used, in which any voter can pick the party whose primary they want to vote in. In other states, caucuses are used, which are basically 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 do have the disadvantage that they allow raiding, in which a voter will vote in their non-preferred party’s primary with the intent of selecting a weaker opponent for their preferred party’s candidate. Washington State currently uses a different method, called a top 2 primary, in which voters select from the candidates from all political parties on the primary, and the top two candidates, regardless of party affiliation, move on to the general election. While this method is liked by independent voters, it gives the political parties incentive to select a top candidate internally before the primary, so that two candidates will not split the party’s vote. Regardless of the primary type, the general election is the main election, open to all voters. Except in the case of the top 2 primary, the top candidate from each major political party would be included in the general 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.

How can someone win a US Presidential election, even if they don't win the popular vote?

In the 2016 US presidential election, Donald Trump won 306 electoral college votes compared to Hillary Clinton's 232, yet Hillary's tally for the popular vote was 65,794,399 while Trump's was 62,955,202.  How does this work?
According to the Pew Research Center[footnote]http://www.pewresearch.org/fact-tank/2016/12/20/why-electoral-college-landslides-are-easier-to-win-than-popular-vote-ones/[/footnote], this happened because Trump won large states such as Florida, Pennsylvania, and Wisconsin by very narrow margins. This allowed him to gain all the electoral college votes available for those states. How does the electoral college work? This ~5 minute video, provides a brief, and interesting introduction to how a citizen's vote is translated into an electoral college vote (or how it is supposed to be). https://youtu.be/OUS9mM8Xbbw?list=PL9936C719FF689E7D Scary, huh? In fact, according to the Pew Research Center,

In the vast majority of U.S. elections, in which the same candidate won both the popular and the electoral vote, the system usually makes the winner’s victory margin in the former a lot wider than in the latter. In 2012, for example, Barack Obama won 51% of the nationwide popular vote but nearly 62% of the electoral votes, or 332 out of 538.

Looking back at all presidential elections since 1828, the winner’s electoral vote share has, on average, been 1.36 times his popular vote share—what we’ll call the electoral vote (EV) inflation factor. Trump’s EV inflation factor, based on his winning 56.5% of the electoral votes (304 out of 538) is 1.22, similar to Obama’s in 2012 (1.21).

Since all but two states use a plurality system, recall that this means the winner of the popular vote—no matter how small the margin—is not necessarily the winner of the electoral college vote.

Licenses & Attributions

CC licensed content, Original

CC licensed content, Shared previously

  • Approval voting . Authored by: Lippman, David. License: CC BY: Attribution.
  • How the Electoral College Works. License: All Rights Reserved. License terms: Standard YouTube License.