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Section 11.7 Approval Voting

Subsection 11.7.1 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. 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.

Example 11.7.1.

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:

Table 11.7.2.
Bob Ann Marv Alice Eve Omar Lupe Dave Tish Jim
Titantic X X X X X
Scream X X X X X X
The Matrix X X X X X X X
Solution.

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.

Problem 11.7.3. 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.

Table 11.7.4.
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
Answer.

Using Approval voting, Tacoma is the winner:

Seattle has \(30+10+15+5=60\) votes

Tacoma has \(30+15+20+15+5=85\) votes

Puyallup has \(10+20+25+5=50\) votes

Olympia has \(15+15+5=35\) votes

Subsection 11.7.2 What's Wrong wtih Approval Voting?

Approval voting can very easily violate the Majority Criterion.

Example 11.7.5.

Consider the voting schedule:

Table 11.7.6.
80 15 5
1st Choice A B C
2nd Choice B C B
3rd Choice C A A
Solution.

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.

Table 11.7.7.
Bob Ann Marv Alice Eve Omar Lupe Dave Tish Jim
Titantic 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.