Approval Voting

Section 1.7 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 1.7.1

Figure 1.7.2 Video solution by David Lippman

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.

Try it Now 1.7.3

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

Subsection 1.7.1 What’s Wrong with Approval Voting?

Approval voting can very easily violate the Majority Criterion.

Example 1.7.4

Figure 1.7.5 Video solution by David Lippman

Consider the voting schedule:

	80	15	5
1 st choice	A	B	C
2 nd choice	B	C	B
3 rd 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.