Cary’s Instant Runoff Elections: Fair?

This year, Cary followed state law (HR-1024, pdf here) by implementing a new method of casting and counting votes for mayor and several city council seats. (Ballot directions, and a Sample ballot)

The method was presented at a Cary Town Council meeting (view the presentation here) and it included several selling points, including one that piqued my interest. The new method “preserves majority rule”. That’s great, but it begs the question:

Does it really? In every case? Also, is majority rule the only fairness criterion that needs to be preserved? I’ve got the scoop.

We are all familiar with voting. And, after a recent presidential election we all know the truth of the Tom Stoppard quote, “It’s not the voting that’s democracy; it’s the counting.”

I’m going to use three simple counting methods to demonstrate that the same ballots can yield three different results. And I’m going to show three criteria for fairness to see if Cary’s method is truly fair.

Typically, we in the U.S. count votes using a simple majority system (jargon: a majority is a plurality of votes). All voters are asked to pick one candidate, and the winner is the one that gets a majority of the votes. If no one gets a majority, subsequent (runoff) elections are held between the top two vote-getters.

Runoffs are expensive, and are typically associated with lower voter turnouts (example). Also, since voters only express their first choice, some argue that we don’t get a clear picture of voter preferences.

To get a better picture, some elections allow you to specify more than your first choice. In some (like the election of the Heisman trophy winner), you rank all the candidates.

Here’s an example. Suppose a local land-grant university is electing their homecoming queen, and that students rank the four candidates (Heidi, Jackie, Lisa, and Kate) on each ballot. It’s obvious that there will be a lot of ballots that look identical, that is, that express the exact same ranking. Here’s how things shake out:

Who wins? That depends on how you count the votes.

This election consists of 740 preference ballots. I can summarize these results in a preference schedule:

Number of Voters ->	280	200	160	80	20
1st choice	Heidi	Kate	Lisa	Jackie	Kate
2nd choice	Jackie	Jackie	Kate	Lisa	Lisa
3rd choice	Kate	Lisa	Jackie	Kate	Jackie
4th choice	Lisa	Heidi	Heidi	Heidi	Heidi

Eliminating Candidates

One thing I assume about working with preference ballots is that individual preferences are transitive. That is, if someone prefers A over B, and prefers B over C, then we can assume that they prefer A over C.

So, in our example, suppose Jackie decides to move from this school to, say, a much more prestigious midwestern university. We don’t have to do a new election: just remove Jackie from the table above to get new counts. In the table, the second column (K, J ,L ,H) and the fifth column (K, L, J, H) become the same. Remove J from the picture and both columns are (K, J, H).

Simple Majority (Plurality) Method

This is the method that we all know: look only at the first place votes and the person with the most first place votes wins. In reality, preference ballot’s aren’t needed with this method, since first choice is the only choice.

Counted this way, we get the following:

Heidi: 280 first-place votes

Kate: 220 first-place votes

Lisa: 160 first-place votes

Jackie: 80 first-place votes

Hooray! Heidi is the winner!

This makes sense. We’ll call it the majority criterion.

Majority Criterion. If a candidate receives a majority of the first-place votes, then that choice should be the winner of the election.

I don’t want to trivialize this: it’s an important criterion for declaring a voting method as fair. If you change the way votes are counted (as Cary did), it’s important to preserve this. Any counting method that doesn’t allow the person with a majority of first place votes to win is said to violate the majority criterion. The plurality method satisfies the majority criterion.

So what’s wrong?

Well, it’s easy to come up with examples that illustrate the flaws inherent in simple plurality voting. Suppose SAS installs a new soda dispenser in the break room near my office, and they poll my entire floor to decide what to dispense. Here’s the preference schedule for the result:

Number of Voters ->	49	48	3
1st choice	Pepsi	Coke	Lemonade
2nd choice	Coke	Dr. Pepper	Coke
3rd choice	Lemonade	Orange	Dr. Pepper
4th choice	Orange	Lemonade	Orange
5th choice	Dr. Pepper	Pepsi	Pepsi

Using the plurality method, Pepsi barely wins with 49 first-place votes. However, note that Coke not only has 48 first-place votes, but 52 second-place votes. Coke is obviously a better choice for the dispenser. In fact, if we eliminate all choices other than Coke or Pepsi, Coke wins 51-49.

Compare Coke to Pepsi: it wins 51 to 49. Compare Coke to Lemonade: it wins 97 to 3. Compare Coke to Orange or Dr. Pepper and it wins all 100 votes.

Said another way, even though Coke wins when compared head-to-head with all its competitors, the plurality method chooses another winner. This violates another criterion of fairness: The Condorcet criterion (from the French mathematician/politician the Marquis de Condorcet, right).

Condorcet Criterion. If any choice is preferred in a head-to-head comparison with all the other choices, then that choice should be the winner (the “Condorcet candidate”).

The Condorcet criterion simply says that if there is a Condorcet candidate, then that person should be the winner of the election. In many cases, the Condorcet winner is the plurality winner. The problem is that there are cases where they differ.

This, too, is not a trivial matter. Thinking back to the soda example, suppose you were one of the three voters who are represented in the last column. You may want Lemonade, but it’s an obvious loser. Your best strategy is to switch your votes and place Coke as your first choice. It’s really your second choice, but you decide to vote insincerely so as to not “waste” your vote.

Sound familiar? This is the reason that the American two-party system is so hard to crack for third-party candidates. People who want the third party candidate vote insincerely (like Ralph Nader voters who cast their votes for Al Gore). And it turns out of all the different vote counting methods, the plurality method is the one most affected by insincere voting.

Borda Counts

One way to address this problem is to assign points to each of the preferences. That is, give one point for last place, two points for next-to last, etc., until you get to first place. (This method is named after another Frenchman, Jean-Charles de Borda, right)

In our homecoming example,

Number of Voters ->	280	200	160	80	20
1st choice = 4 points	Heidi: 280 × 4 = 1120 points	Kate: 200 × 4 = 800	Lisa: 160 × 4 = 640	Jackie : 80 × 4 = 320	Kate: 20 × 4 = 80
2nd choice = 3 points	Jackie: 280 × 3 = 840	Jackie: 200 × 3 = 600	Kate: 160 × 3 = 480	Lisa: 80 × 3 = 240	Lisa: 20 × 3 = 60
3rd choice = 2 points	Kate: 280 × 2 = 560	Lisa: 200 × 2 = 400	Jackie: 160 × 2 = 320	Kate: 80 × 2 = 160	Jackie: 20 × 2 = 40
4th choice = 1 point	Lisa: 280 × 1 = 280	Heidi: 200 × 1 = 200	Heidi: 160 × 1 = 160	Heidi: 80 × 1 = 80	Heidi: 20 × 1 = 20

Now, we just add up the point totals:

Heidi gets 1120 + 200 + 160 + 80 + 20 = 1580 points.

Jackie gets 840 + 600 + 320 + 320 + 40 = 2120 points

Kate gets 560 + 800 + 480 + 160 + 80 = 2080 points

Lisa gets 280 + 400 + 640 + 60 = 1380 points.

Huzzah! Jackie wins!

So What’s Wrong?

Here’s another hypothetical example, where people express their choice among four candidates A, B, C, and D:

Number of Voters ->	51	5	23	21
1st choice	A	C	B	D
2nd choice	C	B	C	C
3rd choice	B	D	D	B
4th choice	D	A	A	A

(There are 100 votes here, so A has a majority with 51).

I won’t go through the calculations, but the Borda Count results are:

A: 153

B: 151

C: 205

D: 91

So under the Borda count method, C is the winner. However, A got a majority of first-place votes. That is, we have an example that shows that Borda counts violate the majority criterion.

And although I won’t prove it here, anything that violates the majority criterion also violates the Condorcet criterion. Despite both of these, the system is in use (as in the aforementioned Heisman trophy, other leagues MVPs, and in several music and movie awards).

Instant Runoff Voting (Plurality-With-Elimination)

In most elections, a candidate must have a majority of votes to get reelected. When there are three or more candidates running, some candidates are removed and another runoff election takes place.

Cary (and other places, like San Francisco) use an efficient way of conducting runoffs using preference ballots. The preferences in these ballots (and the transitivity discussed earlier) allow us to determine who would win in a runoff between any two candidates, obviating the need for an actual (time consuming, expensive) runoff.

The differences in various instant runoff methods involves how many candidates are removed after the first balloting. I’ll illustrate with a simple one, where we eliminate the last place voter, recount, eliminate last place, and so on. Cary actually eliminates all but the first two, but the mathematical properties of the two methods are the same.

Here again is the preference schedule for the homecoming queen example.

Number of Voters ->	280	200	160	80	20
1st choice	Heidi	Kate	Lisa	Jackie	Kate
2nd choice	Jackie	Jackie	Kate	Lisa	Lisa
3rd choice	Kate	Lisa	Jackie	Kate	Jackie
4th choice	Lisa	Heidi	Heidi	Heidi	Heidi

Round 1:

Candidate	Heidi	Jackie	Kate	Lisa
Number of first-place votes	280	80	220	160

Since Jackie has the fewest number of first-place votes, she’s eliminated.

Round 2:

Removing Jackie from the list means that her 80 votes go to Lisa.

Candidate	Heidi	Kate	Lisa
Number of first-place votes	280	220	240

We now eliminate Kate. You can check the original preference ballot (where Heidi was listed last all the columns but one) to see that all of Kate’s votes go to Lisa.

Round 3:

We’re now down to two candidates, Heidi with 280 votes and Lisa with 440.

Lisa is the winner!

So what’s wrong?

There are two issues to illustrate with instant runoff voting. First, as in the Cary elections, you do not have to express a preference for all the candidates. You have the choice of only marking your first-place vote, or only your first two, and so on.

So here’s very divisive election with three candidates: Lee, Ross, and Ben. No one likes all three. Most only write down preferences for two, and Lee has eight people that only write his name.

Number of Voters ->	84	78	20	10	8
1st choice	Ben	Ross	Lee	Lee	Lee
2nd choice	Lee	Ben	Ross	Ben

There are 200 voters, so 101 are needed for a majority.

Round 1:

Here are the counts for first-place votes:

Ben: 84 (42%)

Ross: 78 (39%)

Lee: 38 (19%)

So Lee’s out on the first ballot. Eight ballots no longer are used, since there are no preferences for anyone else.

Round 2:

Of Lee’s remaining votes, 20 go to Ross, and 10 go to Ben.

Ben: 94

Ross: 98

So Ross is the winner, but with only 98 votes. That is, although he has a majority of the remaining ballots after round 2, only 49% of the original voters wrote his name down.

Therefore, instant runoff voting does not satisfy the majority criterion if some voters express no preference among the voters in the last round.

But there’s a more subtle problem — one that deals with human nature. Suppose that there are three candidates A, B, and C running for mayor, and a poll before the election gives the following results.

Number of Voters ->	10	8	7	4
1st choice	C	B	A	A
2nd choice	A	C	B	C
3rd choice	B	A	C	B

A has 11 first-place votes, B has 8, and C has 10. So B is eliminated, its votes and the win go to C, who handily wins with 18 votes.

If this poll becomes public (and what polls don’t!), some people may decide to change their votes. Let’s say the people in the last column decide that since C is going to win,so they’ll switch to now vote C. It won’t matter, right?

Number of Voters ->	14	8	7
1st choice	C	B	A
2nd choice	A	C	B
3rd choice	B	A	C

In this case, the votes shake our quite differently. A gets eliminated in the first round, so its votes go to B. Now, B has 15 votes to C’s 14 votes, so B is the winner.

Think closely about what happened here. C was the original winner, by a lot of votes.The only thing that happened was that people changed their votes in favor of C, and now C is the loser. That is, in this voting system, it’s possible to lose by getting more first place votes.

Mathematicians call this the monotonicity criterion.

The Monotonicity Criterion. If a choice is the winner of an election, and changes in the ballot only favor this candidate, then this candidate should still be the winner.

So instant runoffs violate the monotonicity criterion. It’s not difficult to show that this method also violates the Condorcet criterion (again: not always, not even typically, but in some cases. And that’s all it takes).

The One True Fair Method

Here’s the upshot. There is no method of voting that doesn’t violate at least one of these criteria. This was proven by Kenneth Arrow, a Stanford economist, in the 1940s. His Impossibility Theorem says that something’s going to be violated, and we just have to pick which criteria we can live with.

There is of course much more to this subject than I’ve detailed here. The proof involves game theory (which isn’t technically difficult mathematics, but you do have to be really clever to figure it all out). Cary’s decision to use instant runoffs is a much better choice than any of the other methods I’ve presented, including the majority method.

But it’s not perfect. Things seldom are.

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Sources not linked in the text: Wake County’s Instant Runoff Voting site, Wikipedia entries on Borda counts, plurality voting, and the Condorcet criterion. Also, Excursions in Modern Mathematics by Peter Tannenbaum, and various bits from the Consortium for Mathematics and its Applications.