2.4 What’s Wrong with Plurality?

The election from Example 2 may seem totally clean, but there is a problem lurking that arises whenever there are three or more choices. Looking back at our preference table, how would our members vote if they only had two choices?

Anaheim vs Orlando: 7 out of the 10 would prefer Anaheim over Orlando

Preference schedule for the vacation vote: column headings 1, 3, 3 and 3 voters; first choices A, A, O and H; second choices O, H, H and A; third choices H, O, A and O. The first, second and fourth columns are shaded, marking the 1 plus 3 plus 3 equals 7 voters who rank Anaheim above Orlando.

Anaheim vs Hawaii: 6 out of 10 would prefer Hawaii over Anaheim

The same vacation preference schedule, columns of 1, 3, 3 and 3 voters with first choices A, A, O and H, second choices O, H, H and A, third choices H, O, A and O, now with the third and fourth columns shaded to mark the 3 plus 3 equals 6 voters who rank Hawaii above Anaheim.

This doesn’t seem right, does it? Anaheim just won the election, yet 6 out of 10 voters, 60% of them, would have preferred Hawaii! That hardly seems fair. Marquis de Condorcet, a French philosopher, mathematician, and political scientist wrote about how this could happen in 1785, and for him we name our first fairness criterion.

Since Hawaii is preferred in a one-to-one comparison to both other choices, Hawaii is the Condorcet Winner.

Example 2

Consider a city council election in a district that is historically 60% Democratic voters and 40% Republican voters. Even though city council is technically a nonpartisan office, people generally know the affiliations of the candidates. In this election there are three candidates: Don and Key, both Democrats, and Elle, a Republican. A preference schedule for the votes looks as follows:

	342	214	298
$1 to the power st choice$	Elle	Don	Key
$2 to the power nd choice$	Don	Key	Don
$3 to the power rd choice$	Key	Elle	Elle

Show solution

We can see a total of $342 + 214 + 298 = 854$ voters participated in this election. Computing percentage of first place votes:

Don: $214 / 854 equals 25.1 %$

Key: $298 / 854 equals 34.9 %$

Elle: $342 / 854 equals 40.0 %$

So in this election, the Democratic voters split their vote over the two Democratic candidates, allowing the Republican candidate Elle to win under the plurality method with 40% of the vote.

Analyzing this election closer, we see that it violates the Condorcet Criterion. Analyzing the one-to-one comparisons:

Elle vs Don: 342 prefer Elle; 512 prefer Don: Don is preferred

Elle vs Key: 342 prefer Elle; 512 prefer Key: Key is preferred

Don vs Key: 556 prefer Don; 298 prefer Key: Don is preferred

So even though Don had the smallest number of first-place votes in the election, he is the Condorcet winner, being preferred in every one-to-one comparison with the other candidates.

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.