Voting Theory: Learn It 4

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Voting Theory: Learn It 4

Copeland’s Method

So far none of our voting methods have satisfied the Condorcet Criterion. The Copeland Method specifically attempts to satisfy the Condorcet Criterion by looking at pairwise (one-to-one) comparisons.

Copeland’s method

In this method, each pair of candidates is compared, using all preferences to determine which of the two is more preferred. The more preferred candidate is awarded [latex]1[/latex] point. If there is a tie, each candidate is awarded [latex]\frac{1}{2}[/latex] point. After all pairwise comparisons are made, the candidate with the most points, and hence the most pairwise wins, is declared the winner.

Variations of Copeland’s Method are used in many professional organizations, including election of the Board of Trustees for the Wikimedia Foundation that runs Wikipedia.

Consider our vacation group example from the beginning of the module. A vacation club is trying to decide which destination to visit this year: Hawaii (H), Orlando (O), or Anaheim (A). Their votes are shown below. Determine the winner using Copeland’s Method.

	[latex]1[/latex]	[latex]3[/latex]	[latex]3[/latex]	[latex]3[/latex]
1st choice	A	A	O	H
2nd choice	O	H	H	A
3rd choice	H	O	A	O

We need to look at each pair of choices, and see which choice would win in a one-to-one comparison. You may recall we did this earlier when determining the Condorcet Winner.

For example, comparing Hawaii vs Orlando, we see that [latex]6[/latex] voters, those shaded below in the first table below, would prefer Hawaii to Orlando. Note that Hawaii doesn’t have to be the voter’s first choice — we’re imagining that Anaheim wasn’t an option. If it helps, you can imagine removing Anaheim, as in the second table below.

	[latex]1[/latex]	[latex]3[/latex]	[latex]3[/latex]	[latex]3[/latex]
1st choice	A	A	O	H
2nd choice	O	H	H	A
3rd choice	H	O	A	O

	[latex]1[/latex]	[latex]3[/latex]	[latex]3[/latex]	[latex]3[/latex]
1st choice			O	H
2nd choice	O	H	H
3rd choice	H	O		O

Based on this, in the comparison of Hawaii vs Orlando, Hawaii wins, and receives [latex]1[/latex] point.

Comparing Anaheim to Orlando, the [latex]1[/latex] voter in the first column clearly prefers Anaheim, as do the [latex]3[/latex] voters in the second column. The [latex]3[/latex] voters in the third column clearly prefer Orlando. The [latex]3[/latex] voters in the last column prefer Hawaii as their first choice, but if they had to choose between Anaheim and Orlando, they’d choose Anaheim, their second choice overall. So, altogether [latex]1+3+3=7[/latex] voters prefer Anaheim over Orlando, and [latex]3[/latex] prefer Orlando over Anaheim. So, comparing Anaheim vs Orlando: [latex]7[/latex] votes to [latex]3[/latex] votes: Anaheim gets [latex]1[/latex] point.

All together,

Hawaii vs Orlando: [latex]6[/latex] votes to [latex]4[/latex] votes: Hawaii gets [latex]1[/latex] point

Anaheim vs Orlando: [latex]7[/latex] votes to [latex]3[/latex] votes: Anaheim gets [latex]1[/latex] point

Hawaii vs Anaheim: [latex]6[/latex] votes to [latex]4[/latex] votes: Hawaii gets [latex]1[/latex] point

Hawaii is the winner under Copeland’s Method, having earned the most points. Notice this process is consistent with our determination of a Condorcet Winner.

Here is the same example presented in a video.

You can view the transcript for “Copeland’s method / Pairwise comparison 1” here (opens in new window).

What’s Wrong with Copeland’s Method?

As already noted, Copeland’s Method does satisfy the Condorcet Criterion. It also satisfies the Majority Criterion and the Monotonicity Criterion. So is this the perfect method? Well, in a word, no. Let’s look at an example to explore this further.

A committee is trying to award a scholarship to one of four students, Anna (A), Brian (B), Carlos (C), and Dimitry (D). The votes are shown below:

	[latex]5[/latex]	[latex]5[/latex]	[latex]6[/latex]	[latex]4[/latex]
1st choice	D	A	C	B
2nd choice	A	C	B	D
3rd choice	C	B	D	A
4th choice	B	D	A	C

Making the comparisons:

A vs B: [latex]10[/latex] votes to [latex]10[/latex] votes	A gets [latex]\frac{1}{2}[/latex] point, B gets [latex]\frac{1}{2}[/latex] point
A vs C: [latex]14[/latex] votes to [latex]6[/latex] votes:	A gets [latex]1[/latex] point
A vs D: [latex]5[/latex] votes to [latex]15[/latex] votes:	D gets [latex]1[/latex] point
B vs C: [latex]4[/latex] votes to [latex]16[/latex] votes:	C gets [latex]1[/latex] point
B vs D: [latex]15[/latex] votes to [latex]5[/latex] votes:	B gets [latex]1[/latex] point
C vs D: [latex]11[/latex] votes to [latex]9[/latex] votes:	C gets [latex]1[/latex] point

Totaling:

A has [latex]1 \frac{1}{2}[/latex] points	B has [latex]1 \frac{1}{2}[/latex] points
C has [latex]2[/latex] points	D has [latex]1[/latex] point

So Carlos is awarded the scholarship. However, the committee then discovers that Dimitry was not eligible for the scholarship (he is a sophomore and the scholarship is only for freshman). Even though this seems like it shouldn’t affect the outcome, the committee decides to recount the vote, removing Dimitry from consideration. This reduces the preference schedule to:

	[latex]5[/latex]	[latex]5[/latex]	[latex]6[/latex]	[latex]4[/latex]
1st choice	A	A	C	B
2nd choice	C	C	B	A
3rd choice	B	B	A	C

Making the comparisons:

A vs B: [latex]10[/latex] votes to [latex]10[/latex] votes	A gets [latex]\frac{1}{2}[/latex] point, B gets [latex]\frac{1}{2}[/latex] point
A vs C: [latex]14[/latex] votes to [latex]6[/latex] votes:	A gets [latex]1[/latex] point
B vs C: [latex]4[/latex] votes to [latex]16[/latex] votes:	C gets [latex]1[/latex] point

Totaling:

A has [latex]1 \frac{1}{2}[/latex] points	B has [latex]\frac{1}{2}[/latex] point
C has [latex]1[/latex] point

Suddenly Anna is the winner! This leads us to another fairness criterion.

Watch this video to see this example worked out again.

You can view the transcript for “Copelands and the IIA criterion” here (opens in new window).

the independence of irrelevant alternatives (IIA) criterion

If a non-winning choice is removed from the ballot, it should not change the winner of the election.

Equivalently, if choice A is preferred over choice B, introducing or removing a choice C should not cause B to be preferred over A.

In the election from the last example, the IIA Criterion was violated.

Another disadvantage of Copeland’s method is that it is fairly easy for the election to end in a tie. For this reason, Copeland’s method is usually the first part of a more advanced method that uses more sophisticated methods for breaking ties and determining the winner when there is not a condorcet candidate.