Math 110, Fall 2008
The Mathematics of Voting

In advance of the 2008 presidential election, we will be taking a look at the mathematical theory of voting. Most of the reading is from the book Gaming The Vote: Why Elections Aren't Fair, by William Poundstone. This handout covers some preliminary information that you should be aware of before doing the reading from that book.

For some of the examples in this handout -- to avoid getting bogged down in actual politics -- let's imagine that our class is holding a vote to decide what color to paint our classroom. The problem is, how to hold the vote in such a way that the result best reflects the preference of the class as a whole.

Note that if there are only two possible options, then there is no problem, except in the unlikely event of a tie. In the absence of a tie, one of the options will have the support of the majority of the voters, and that is the option that should win. As for ties, in practice we would have to come up with some way to break ties (such as flipping a coin), but for the most part, we will simply ignore the possibility of a tie.

If there are three or more options, there is still no problem as long as one of the options receives majority support. The option that gets more than 50% of the vote wins. The problem arises when there are three or more options, and none of the options is the first choice of a majority of the voters. This means that no matter which option is selected, a majority of the voters do not get their first choice. The question is, how can we conduct the vote and decide on a winner in the way that best reflects the "will of the voters," even in cases where no option is the first choice of a majority?

Plurality Voting

The voting method that is used almost universally for elections in the United States is plurality voting. In a plurality vote, the option selected by the largest number of voters is declared to be the winner, even if it is not selected by a majority of the voters. For example, if there are three options that receive 31%, 42%, and 27% of the votes respectively, then the second option is declared to be the winner, even though only a minority of the voters selected it.

Now, there is a very strong argument that plurality voting is just about the worst possible voting method! Let's look at some of the things that can go wrong with it. Two of the main problems are spoilers and vote splitting.


Imagine that our class is preparing to vote on a new color for our classroom. There are two candidates: Peaceful Blue and Dynamic Yellow. Polls indicate that Dynamic Yellow is preferred by a small majority, 53% for yellow to 47% for blue. However, before the vote is held, a new candidate is nominated: Shocking Red. Not a lot people like red classrooms, but there are some yellow supporters who would like something even more stimulating than yellow, and those people actually support red. However, very few blue supporters like red. When the vote is held, Peaceful Blue gets 46%, Dynamic Yellow gets 44%, and Shocking Red gets the other 10%. Blue wins, under plurality voting.

Shocking Red is a spoiler. Without Red in the race, Yellow would win. Red has no real chance of winning, but it draws enough support away from Yellow so that Blue ends up with a plurality of the vote. Even though a majority of the voters prefer Yellow to Blue (in a two-way match-up), Yellow loses to Blue in a plurality vote among three options.

Maybe it wouldn't surprise you to learn that Shocking Red was actually nominated by a sneaky supporter of Peaceful Blue: He saw that Blue would lose in a two-way vote, and hoped that Red would draw enough support away from Yellow to throw the race to Blue. And indeed it did.

It's still a question whether anything like this happens in real life. In Gaming the Vote, William Poundstone surveys eight US Presidential elections for which it has been claimed that the presence of a spoiler changed the outcome of the election. Here's his view on these elections (page 91):

Let me recap. Five presidential elections were probably decided by spoilers (1844, 1848, 1884, 1912, 2000). At least two others (1892, 1992) are questionable cases. In still another race (1860), four-way vote splitting and the electoral college created such ambiguity that it was a factor in precipitating civil war.

In 1844, an abolitionist spoiler put a slave-owner in the White House.

In 1848, a former Democratic candidate sabotaged the Democratic Party's chances.

In 1884, a Prohibition Party candidate helped elect a supposed "ally of the saloon."

In 1912, a former Republican president prevented the reelection of a Republican president.

In 2000, a consumer and environmental advocate elected the favored candidate of corporate America.

There have been 45 presidential elections since 1828 [the first held under modern rules]. In at least five, the race went to the second most popular candidate because of a spoiler. That's over an 11 percent rate of catastrophic failure. Were the plurality vote a car or an airliner, it would be recognized for what it is -- a defective consumer product, unsafe at any speed.

A lot more damage than having the classroom painted the wrong color! By the way, when there is a potential spoiler in the race, voters often engage in what is called strategic voting. In strategic voting, voters essentially lie about their preferences, because they think that lying will increase their chances of being happy with the result. In our classroom color example, voters who sincerely support Red might decide to vote for Yellow because they worry that a vote for Red will increase the chance that Blue will win. These voters really like Red best, but they also greatly prefer Yellow to Blue. They are willing to cast a vote for their second choice in order to stop their least favorite choice from winning. Strategic voting is not just an issue with spoilers; you'll find more discussion of this in the reading.

Vote Splitting

A spoiler is an option that has no real chance of winning, but whose presence in the race can cause the most popular option to lose. Vote splitting is another way that plurality voting can go wrong, and in this case it is even possible for the least popular option to win!

Let's imagine a vote on classroom color in another class. In this election, there are three candidates: Peaceful Blue, Calming Green, and Morbid Black. Now, as it happens, there is a big Goth contingent in the class, so Morbid Black has a good deal of support, 40%. The other 60% is about evenly split in their support between Peaceful Blue and Calming Green, 32% to 28%. Under plurality voting, Black wins with 40%. Now, you have to understand that the supporters of Peaceful Blue and Calming Green really hate the idea of a black classroom. The candidate that won is the least favorite of a majority of the voters. On the other hand, Blue supporters would have been fairly happy with Green, while Green supporters would have been fairly happy with Blue. In a two-way race between Green and Black, Green would have won easily. In a two-way race between Blue and Black, Blue would have won easily. But in the three-way race, the Blue/Green voters (who really have very similar opinions) split their vote between two candidates, allowing the less popular Black to win.

In the real world, vote splitting can occur in a political election when there are, for example, two liberals and a conservative in the race. The liberal vote might be split between the two liberal candidates, allowing the conservative to win with less than a majority. So, a conservative wins, even though a majority of the voters support liberals. Things like this are actually pretty common in American elections, especially in primaries, where there can be many candidates who have a shot at winning.

Plurality Voting with Runoff

If the problem with plurality voting is that a candidate can be elected who has less than majority support, then maybe the solution is to have runoff elections. In plurality voting with runoff, a first round of voting is held. If any candidate has a majority in the first round, then that candidate is declared to be the winner. Otherwise, the two candidates who got the highest numbers of votes in the first round go into a runoff election. In the runoff, there are only two candidates, and (ignoring the possibility of a tie) one of them will get a majority of the votes. So, the ultimate winner of the election always has majority support? Or does he?

Runoff elections do more or less solve the problem of spoilers. By definition, a spoiler is someone who doesn't get many votes, so the spoiler won't make it into the runoff. In our first classroom color example, with Peaceful Blue, Dynamic Yellow, and Shocking Red getting 44%, 46%, and 10% of the vote in a first round, Red is eliminated, and Blue and Yellow go into the runoff election. In the runoff, most of the people who voted for Red will go for Yellow, and only a few of them go for Blue. As a result, Yellow wins the runoff, even though Blue had more votes than Yellow in the first round. This seems to be the right outcome, and the spoiler effect has been eliminated.

However, runoffs do not solve all the problems with plurality voting. In particular, they don't solve the vote splitting problem. In cases of vote splitting It is quite possible for the candidate with the most support to be eliminated in the first round, leaving a choice between two less popular candidates in the runoff. Consider a classroom color vote where the options are Basic Beige, Bright Orange, and Royal Purple. In the first round, the votes are 30% for Beige, 36% for Orange, and 34% for Purple. So Beige is eliminate, while Orange and Purple go into the runoff. Now, as it happens, pretty much everyone would have been reasonably happy with Beige, even though it was the first choice of only 30%. On the other hand, even though Orange got 36%, the other 64% don't like Orange at all. And although Purple got 34%, the other 66% can't stand purple. So between the two candidates in the runoff, neither one really has majority support. Beige would win in a two-way match between Beige and Orange. Beige would also win in a two-way match between Beige and Purple. But Beige was eliminated and is not available as a choice in the runoff -- even though it was arguably the option that should have won.

One famous example of this took place in the election for Governor of Louisiana in 1991. Louisiana is one of the few places in the United States that uses plurality-with-runoff voting for its elections. In the first round, in addition to several other candidates, there were three candidates who had fairly high levels of support. One was the incumbent governor, Buddy Roemer. Another was Edwin Edwards, a former governor known for gambling, womanizing, and ethical issues. The third was David Duke, an avowed racist and former member of the Ku Klux Klan, in which he had been a "wizard." Although there was not a great deal of enthusiasm for Roemer, he would have defeated either Edwards or Duke in two-way match-ups. However, Roemer was eliminated in the first round, and it was Edwards and Duke who went on to the runoff. (The vote was 33.8% Edwards, 31.7% Duke, and 26.5% Roemer; no candidate was close to a majority.) As Poundstone says in Gaming the Vote (page 16), "The campaign then entered its pathological phase. Louisianans had to decide whether to vote for Edwards or for Duke. For some, this was like deciding whether to die slowly in a bear trap or to gnaw off a leg." Poundstone says (page 18) that "the bizarre tone of the election was captured in two bumper stickers: VOTE FOR THE CROOK -- IT'S IMPORTANT and VOTE FOR THE LIZARD, NOT THE WIZARD. In the end, the lizard (Edwards) won, and the country breathed a sigh of relief at avoiding the election of a Ku Klux Klansman as governor. This sort of "lizard versus wizard" dilemma can easily happen in plurality-with-runoff voting.

Instant Runoff Voting

Another type of runoff has become popular in recent years, with many people promoting it as a superior way to conduct our elections. Instant Runoff Voting (IRV) is similar to plurality-with-runoff, but the runoff is held "instantly." In IRV, instead of just voting for their top choice, each voter makes a ranked list of all the candidates in order of preference. Each voter has a first choice, a second choice, a third choice, and so on, and all this information is collected on the first (and only) ballot. If a candidate gets a majority of first choice votes, that candidate is declared to be the winner. Otherwise, the candidate with the least number of first choice votes is eliminated from consideration. Then the ballots are counted again, but ignoring the candidate who was eliminated. For example, for ballots that listed the eliminated candidate as first choice, those ballots' second choice is now counted as their first choice. So, the vote totals for all the other candidates can go up, and it is possible that one of them now has a majority of first place votes. If so, that candidate is declared to be the winner. If not, the process is repeated; the one with the fewest first place votes is eliminated, and the counting is done again. Eventually, some candidate will have a majority, even if it requires the elimination of all but two candidates before that happens.

Instant Runoff Voting is certainly superior to simple plurality voting. Whether is is an improvement on plurality-with-runoff is not so clear, although it is probably somewhat better. However, it does not eliminate the lizard-versus-wizard phenomenon. Even if IRV had been used in the 1991 Louisiana governor's race, Roemer would have been eliminated and it would have been the lizard (Edwards) or the wizard (Duke) in the end.

Approval Voting

This handout has already given you most of the background that you need before doing the reading from Gaming the Vote, but there is one other voting method that is mentioned but not explained in the assigned reading: approval voting.

In approval voting, a voter simply marks each candidate as "approve" or "disapprove." The winner is the candidate who is "approved" by the largest number of voters. (In some versions of approval voting, the winner also has to be approved by a majority of the voters. If no candidate gets approval from a majority, then a new election, with new candidates, must be held.)

Although approval voting seems to make sense, it has its own problems. For example, it can easily happen that there is a candidate who is the first choice of an actual majority of the voters, but still loses the election to another candidate.

Approval voting is similar to range voting, which is discussed in the reading, so I will not cover it further here.

The reading from Gaming the Vote consists of the following selections: Chapter 2, pages 46 through 51; Chapter 7, pages 133 through 148; and Chapter 14, pages 231 through 241.