2.14: What’s Wrong with Approval Voting?

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Approval voting can very easily violate the Majority Criterion.

Example 13

Consider the voting schedule:

\(\begin{array}{|l|l|l|l|}
\hline & 80 & 15 & 5 \\
\hline 1^{\text {st }} \text { choice } & \text { A } & \text { B } & \text { C } \\
\hline 2^{\text {nd }} \text { choice } & \text { B } & \text { C } & \text { B } \\
\hline 3^{\text {rd }} \text { choice } & \text { C } & \text { A } & \text { A } \\
\hline
\end{array}\)

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.

\(\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline & \text { Bob } & \text { Ann } & \text { Marv } & \text { Alice } & \text { Eve } & \text { Omar } & \text { Lupe } & \text { Dave } & \text { Tish } & \text { Jim } \\
\hline \text { Titanic } & & \mathrm{X} & \mathrm{X} & & & \mathrm{X} & & \mathrm{X} & & \mathrm{X} \\
\hline \text { Scream } & \mathrm{X} & & \mathrm{X} & \mathrm{X} & & \mathrm{X} & \mathrm{X} & & \mathrm{X} & \\
\hline \text { The Matrix } & & \mathrm{X} & \mathrm{X} & & \mathrm{X} & & \mathrm{X} & & & \mathrm{X} \\
\hline
\end{array}\)

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.