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Mathematics LibreTexts

2.7: What’s Wrong with IRV?

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    34182
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    Example 6

    Let’s return to our City Council Election

    \(\begin{array}{|l|l|l|l|}
    \hline & 342 & 214 & 298 \\
    \hline 1^{\text {st }} \text { choice } & \text { Elle } & \text { Don } & \text { Key } \\
    \hline 2^{\text {nd }} \text { choice } & \text { Don } & \text { Key } & \text { Don } \\
    \hline 3^{\text {rd }} \text { choice } & \text { Key } & \text { Elle } & \text { Elle } \\
    \hline
    \end{array}\)

    Solution

    In this election, Don has the smallest number of first place votes, so Don is eliminated in the first round. The 214 people who voted for Don have their votes transferred to their second choice, Key.

    \(\begin{array}{|l|l|l|}
    \hline & 342 & 512 \\
    \hline 1^{\text {st }} \text { choice } & \text { Elle } & \text { Key } \\
    \hline 2^{\text {nd }} \text { choice } & \text { Key } & \text { Elle } \\
    \hline
    \end{array}\)

    So Key is the winner under the IRV method.

    We can immediately notice that in this election, IRV violates the Condorcet Criterion, since we determined earlier that Don was the Condorcet winner. On the other hand, the temptation has been removed for Don’s supporters to vote for Key; they now know their vote will be transferred to Key, not simply discarded.

    Example 7

    Consider the voting system below.

    \(\begin{array}{|l|l|l|l|l|}
    \hline & 37 & 22 & 12 & 29 \\
    \hline 1^{\text {st }} \text { choice } & \text { Adams } & \text { Brown } & \text { Brown } & \text { Carter } \\
    \hline 2^{\text {nd }} \text { choice } & \text { Brown } & \text { Carter } & \text { Adams } & \text { Adams } \\
    \hline 3^{\text {rd }} \text { choice } & \text { Carter } & \text { Adams } & \text { Carter } & \text { Brown } \\
    \hline
    \end{array}\)

    Solution

    In this election, Carter would be eliminated in the first round, and Adams would be the winner with 66 votes to 34 for Brown.

    Now suppose that the results were announced, but election officials accidentally destroyed the ballots before they could be certified, and the votes had to be recast. Wanting to “jump on the bandwagon”, 10 of the voters who had originally voted in the order Brown, Adams, Carter change their vote to favor the presumed winner, changing those votes to Adams, Brown, Carter.

    \(\begin{array}{|l|l|l|l|l|}
    \hline & 47 & 22 & 2 & 29 \\
    \hline 1^{\text {st }} \text { choice } & \text { Adams } & \text { Brown } & \text { Brown } & \text { Carter } \\
    \hline 2^{\text {nd }} \text { choice } & \text { Brown } & \text { Carter } & \text { Adams } & \text { Adams } \\
    \hline 3^{\text {rd }} \text { choice } & \text { Carter } & \text { Adams } & \text { Carter } & \text { Brown } \\
    \hline
    \end{array}\)

    In this re-vote, Brown will be eliminated in the first round, having the fewest first-place votes. After transferring votes, we find that Carter will win this election with 51 votes to Adams’ 49 votes! Even though the only vote changes made favored Adams, the change ended up costing Adams the election. This doesn’t seem right, and introduces our second fairness criterion:

    Monotonicity Criterion

    If voters change their votes to increase the preference for a candidate, it should not harm that candidate’s chances of winning.

    This criterion is violated by this election. Note that even though the criterion is violated in this particular election, it does not mean that IRV always violates the criterion; just that IRV has the potential to violate the criterion in certain elections.


    2.7: What’s Wrong with IRV? is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by David Lippman via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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