Devlin's Angle

November 2004

Election Math

A quick glance at the political map of the United States that appeared in most newspapers the morning after the US General Election decries the oft repeated claim that the people have decided. Both the map and the figures behind them show that the people, taken as a whole, are about as undecided as it is possible to be. Far from being united, our nation is now cleanly divided into two regions having very different political flavors, with the inland and southern states all Republican (colored red) and the west coast, north east, and north central regions all Democrat (colored blue).

I am hardly the first person to observe that (regardless of which candidate or party wins) it is quite ridiculous for a nation as large and powerful as the USA, whose founding ideals are built around the notion that the elected leaders represent the entire population, to have its most fundamental political choices come down to a few thousand voters in one state (Ohio this time round, Florida in 2000).

The culprit is not the Electoral College, as some have suggested, but the way we do the math of elections -- how we count the vote. If we really believed in the intentions of our Founding Fathers, we would use our intellectual talent to devise an electoral tally system that truly reflects the wishes of the electorate.

The electoral math used in the United States election process counts votes using a system known as plurality voting. In this system, also known as "first-past-the- post," the candidate with the most votes is declared the winner. This system has several major flaws. The most obvious one is the one I alluded to above, where the latest election has left 48% of the nation with a president they do not like, do not trust, and whose personal beliefs make it impossible for him to represent the views of many of the very citizens he is supposed to lead. To anyone who truly believes that the United States should in fact be united, that we should be "One Nation," that kind of outcome alone should be reason to seek a better system.

Another flaw with plurality voting, although this one did not cause any problems in the latest presidential election, is that the method can result in the election of a candidate whom almost two-thirds of voters detest.

For instance, in 1998, in a three-party race, plurality voting resulted in the election of former wrestler Jesse Ventura as Governor of Minnesota, despite the fact that only 37% of the electors voted for him. The almost two- thirds of electors who voted Democrat or Republican had to come to terms with a governor that none of them wanted -- or expected. Judging by the comments immediately after the election, the majority of Democrat and Republican voters were strongly opposed to Reform Party candidate Ventura moving into the Governor's mansion. In which case, he won not because the majority of voters chose him, but because plurality voting effectively thwarted the will of the people. Had the voters been able to vote in such a way that, if their preferred candidate were not going to win, their preference between the remaining two could be counted, the outcome could have been quite different.

Several countries, among them Australia, the Irish Republic, and Northern Ireland, use a system called single transferable vote. Introduced by Thomas Hare in England in the 1850s, this system takes account of the entire range of preferences each voter has for the candidates. All electors rank all the candidates in order of preference. When the votes are tallied, the candidates are first ranked based on the number of first-place votes each received. The candidate who comes out last is dropped from the list. This, of course, effectively "disenfranchises" all those voters who picked that candidate. So, their vote is automatically transferred to their second choice of candidate -- which means that their vote still counts. Then the process is repeated: the candidates are ranked a second time, according to the new distribution of votes. Again, the candidate who comes out last is dropped from the list. With just three candidates, this leaves one candidate, who is declared the winner. In a contest with more than three candidates, the process is repeated one or more additional times until only one candidate remains, with that individual winning the election. Since each voter ranks all the candidates in order, this method ensures that at every stage, every voter's preferences among the remaining candidates is taken into account.

An alternative system that avoids the kind of outcomes of the 1998 Minnesota Governor's race is the Borda count, named after Jean-Charles de Borda, who devised it in 1781. Again, the idea is to try to take account of each voter's overall preferences among all the candidates. As with the single transferable vote, in this system, when the poll takes place, each voter ranks all the candidates. If there are n candidates, then when the votes are tallied, the candidate receives n points for each first-place ranking, n-1 points for each second place ranking, n-2 points for each third place ranking, down to just 1 point for each last place ranking. The candidate with the greatest total number of points is then declared the winner.

Yet another system that avoids the Jesse Ventura phenomenon is approval voting. Here the philosophy is to try to ensure that the process does not lead to the election of someone whom the majority opposes. Each voter is allowed to vote for all those candidates of whom he or she approves, and the candidate who gets the most votes wins the election. This is the method used to elect the officers of both the American Mathematical Society and the Mathematical Association of America.

To see how these different systems can lead to very different results, let's consider a hypothetical scenario in which 15 million voters go to the polls in an election with three candidates, A, B, C. Their preferences between the three candidates are as follows:

6 million rank A first, then B, then C.

5 million rank C first, then B, then A.

4 million rank B first, then C, then A.

If the votes are tallied by the plurality vote -- the present system -- then A's 6 million (first-place) votes make him the clear winner. And yet, 9 million voters (60% of the total) rank him dead last! That hardly seems fair.

What happens if the votes are counted by the single transferable vote system -- the system used in Australia and Ireland? The first round of the tally process eliminates B, who is only ranked first by 4 million voters. Those 4 million voters all have C as their second choice, so in the second round of the tally process their votes are transferred to C. The result is that, in the second round, A gets 6 million first place votes while C gets 9 million. Thus, C wins by a whopping 9 million to 6 million margin.

But wait a minute. Looking at the original rankings, we see that 10 million voters prefer B to C -- that's 66% of the total vote. Can it really be fair for such a large majority of the electorate to have their preferences ignored so dramatically?

Thus, both the plurality vote and single transferable vote can lead to results that run counter to the overwhelming desires of the electorate. What happens if we use the Borda count? Well, with this method, A gets

6m x 3 + 5m x 1 + 4m x 1 = 27m points,
C gets
6m x 1 + 5m x 3 + 4m x 2 = 29m points,
and B gets
6m x 2 + 5m x 2 + 4m x 3 = 34m points.
The result is a decisive win for B, with C coming in second and A trailing in third place.

What happens with approval voting? Well, as I have set up the problem so far, we don't have enough information -- we don't know how many electors actively oppose each particular candidate. Let's assume that C's supporters and B's supporters could live with the others' candidate, but the voters in both groups really don't want to see A elected. In this case, B gets 15 million votes, C gets 9 million votes, and A gets a mere 6 million. All in all, it's beginning to look as though B is the one who should win.

Another approach is to choose the individual who would beat every other candidate in head-to-head, two-party contests. This method was suggested by the Marquis de Condorcet in 1785, and as a result is known today as the Condorcet system.

For the scenario in our example, B also wins according to the Condorcet system. He gets at least 10 million votes in a straight B-C contest and at least 9 million votes in a A-B match-up, in either case a majority of the 15 million voters. Unfortunately, although it works for this example, and despite the fact that it has considerable appeal, the Condorcet method suffers from a major disadvantage: it does not always produce a clear winner!

For example, suppose the voting profile were as follows:

5 million rank A first, then C, then B.

5 million rank C first, then B, then A.

5 million rank B first, then A, then C.

Then 10 million voters prefer A to C, so A would easily win an A-C battle. Also, 10 million voters prefer C to B, so C would romp home in a B-C contest. The remaining two-party match-up would pit A against B. But when we look at the preferences, we see that 10 million people prefer B to A, so B comes out on top in that contest. In other words, there is no clear winner. Each candidate wins one of the three possible two-party battles!

One worrying problem with the single transferable vote is that if some voters increase their evaluation of a particular candidate and raise him or her in their rankings, the result can be -- paradoxically -- that the candidate actually does worse! For example, consider an election in which there are four candidates, A, B, C, D, and 21 electors. Initially, the electors rank the candidates like this:

7 voters rank: A B C D

6 voters rank: B A C D

5 voters rank: C B A D

3 voters rank: D C B A

In the first round of the tally, the candidate with the fewest first-place votes is eliminated, namely D. After D's votes have been redistributed, the following ranking results:

7 voters rank: A B C

6 voters rank: B A C

5 + 3 = 8 voters rank: C B A

Then B is eliminated, leading to the new ranking:
7 + 6 = 13 voters rank: A C

8 voters rank: C A

Thus A wins the election.

Now suppose that the 3 voters who originally ranked the candidates D C B A change their mind about A, moving him from their last place choice to their first place: A D C B. These voters do not change their evaluation of the other three candidates, nor do any of the other voters change their rankings of any of the candidates. But when the votes are tallied this time, the end result is that B wins. (If you don't believe this, just work through the tally process one round at a time. The first round eliminates D, the second round eliminates C, and the final result is that 10 voters prefer A to B and 11 voters prefer B to A.)

For all the advantages offered by the single transferable vote system, the fact that a candidate can actually harm her chances by increasing her voter appeal -- to the point of losing an election that she would otherwise have won -- leads some mathematicians to conclude that the method should not be used.

The Borda count has at least two weaknesses. First, it is easy for blocks of voters to manipulate the outcome. For example, suppose there are 3 candidates A, B, C and 5 electors, who initially rank the candidates:

3 voters rank: A B C

2 voters rank: B C A

The Borda count for this ranking is as follows:

A: 3x3 + 2x1 = 11

B: 3x2 + 2x3 = 12

C: 3x1 + 2x2 = 7

Thus, B wins. Suppose now that A's supporters realize what is likely to happen and deliberately change their ranking from A B C to A C B. The Borda count then changes to:

A: 11; B: 9; C: 10.

This time, A wins. By putting B lower on their lists, A's supporters are able to deprive him of the victory he would otherwise have had.

Of course, almost any method is subject to strategic voting by a sophisticated electorate, and Borda himself acknowledged that his system was particularly vulnerable, commenting: "My scheme is intended only for honest men." Somewhat more worrying to the student of electoral math is the fact that the entry of an additional candidate into the race can dramatically alter the final rankings, even if that additional candidate has no chance of winning, and even if none of the voters changes their rankings of the original candidates. For example, suppose that there are 3 candidates, A, B, C, in an election with 7 voters. The voters rank the candidates as follows:

3 voters rank: C B A

2 voters rank: A C B

2 voters rank: B A C

The Borda count for this ranking is:

A: 13; B: 14; C: 15.

Thus, the candidates final ranking is C B A. Now candidate X enters the race, and the voters' ranking becomes:
3 voters rank: C B A X

2 voters rank: A X C B

2 voters rank: B A X C

The new Borda count is:

A: 20; B: 19; C: 18; X: 13.

Thus, the entry of the losing candidate X into the race has completely reversed the ranking of A, B, and C, giving the result A B C X.

With even seemingly "sophisticated" vote-tallying methods having such drawbacks, how are we to decide which is the best method? Of course, the democratic way to settle the matter would be to vote on the available systems. But then, how do we tally the votes of that election? When it comes to elections, it seems that even the math used to count the votes is subject to debate!

But that is no reason not to look for a better way. Many Americans, and I am one of them, would like to see our country return to being a world leader in democratic government, a country that spreads democratic ideals by example, not by military force. What better place to start than with the system by which governments are elected? Let's use our world-leading scientific and technological know-how to improve the democratic system itself.

The examples I gave above show that there is no easy solution. But two things are clear. First, as Winston Churchill said, democracy is a terrible form of government, but all the rest are much worse. Second, all voting systems have drawbacks, but plurality voting, our present system, is the worst, and any of the other systems described here would surely do a better job of representing the preferences of the electorate.

For the record, I regularly vote in elections, but I do so "scientifically," based on information about and past performance of the candidates; I am not affiliated with any political party. Part of this column is adapted from my column of November 2000.


Alan D. Taylor, Mathematics and Politics: Strategy, Voting, Power and Proof, Springer-Verlag (1995).

Donald Saari, Chaotic Elections! A Mathematician Looks at Voting, American Mathematical Society (2001).

Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin ( [email protected]) is the Executive Director of the Center for the Study of Language and Information at Stanford University and The Math Guy on NPR's Weekend Edition. Devlin's newest book, THE MATH INSTINCT: Why You're a Mathematical Genius (along with Lobsters, Birds, Cats, and Dogs), will be published next spring by Thunder's Mouth Press.