The Pros and Cons of Various Systems

Proportional representation obviously does not work when you are voting for a single office like mayor or president, but there are still lots of ways to choose a winner. In fact, mathematicians have spent a fair amount of time on voting theory. When you talk to them about voting they usually start by referring to something called Arrow's theorem, which, they will tell you, proves that it is impossible to devise a perfect voting system.

Not exactly. This theorem, dreamed up by mathematician Kenneth Joseph Arrow (born August 23, 1921), actually just set up a number of criteria he deemed “necessary” and mathematically proved that it was impossible for any voting system to meet them all. He did not prove that his criteria were the only ones possible or necessary. For us non-mathematicians the point is simply that it is probably not possible to please everyone or even create a system that we can all agree is fair. What follows is a brief summary of the major voting systems and their pros and cons. For a more complete review, see Poundstone's Gaming the Vote.

Plurality Voting. This is the method used in most American elections. No matter how many candidates (issues, choices, etc.) on the ballot, each voter chooses only one. The choice with the most votes wins. This works fine if there are only two choices, but when there are more, it is prone to vote splitting. Minor candidates become spoilers, like Nader in 2000. They can distort the vote. It is quite possible that a majority of voters would have preferred Gore to Bush, but enough of those voters actually voted for Nader to cause Bush to win. Both conservatives and liberals have suffered from this sort of distortion. In this case, Republicans (who presumably would never actually vote for Nader) gave financial support to him hoping he would help defeat Gore. So voters get only two choices and have no way of express their degree of support for either.

The Borda Count. Named for French mathematician Jean-Charles de Borda (1733-99). Voters rank all of the choices. If there are x choices, voters rank them all, 1 through x. The rankings become points which are then tallied. The candidate with the most (or least, depending on how the ranking works) points wins. This eliminates the spoiler effect that plagues plurality voting. Unfortunately, it is highly subject to manipulation. It would work fine if all voters are honest, but like many voting systems, it can be distorted by strategic voters. If, say, there are two choices that are close in actual popularity and several others with little popularity, strategic voters could place the popular choice they do not like below the truly unpopular choice. If there were an election between candidates McCain, Obama, Stalin, Hitler, and Vlad the Impaler, Republicans might rank them:

  • McCain=1
  • Hitler=2
  • Stalin=3
  • Vlad=4
  • Obama=5

If enough Republicans (and no Democrats) did this, McCain might win even if Obama had a clear majority of first-place votes. But obviously, Democrats could use the same strategy. The worst (and quite possible) outcome is that one of the really awful choices wins. This is particularly likely if there are two popular (but contentious) choices and one dreadful one, say McCain, Obama, and Vlad. Vlad might actually win, even if no one ranks him first. Not a good outcome.

Approval Voting. Similar to plurality vote except you get to vote for more than one. This avoids vote splitting. In the Bush, Gore, Nader contest, voters could vote for both Gore and Nader if they so chose. Nader would probably have gotten more votes than he actually did, as people could vote for him and Gore and not feel that they were throwing away their vote. If you had a ballot with three Republicans and two Democrats, the Republicans could vote for all the Republicans, and the Democrats all the Democrats. No one is “throwing his vote away” (unless he votes for everyone on the ballot). The major problem is that it does not define exactly what “approval” means. In some sense you might not like anyone on the ballot. Or you might like them all. But if you vote either for all or none, you are wasting your vote. It is possible to vote “strategically” but strategic voting will not seriously distort the outcome. Strategic voting, in this case, means just voting for both for the choice that you like and the one you think has the best chance of winning.

The biggest thing against it seems to be that the voting theorist Donald Saari is adamantly, vociferously against it. These are the points that he and his supporters claim against it:

  1. Voters might mark their ballots foolishly or capriciously because they do not know what “approval” might mean. This would distort the election.
  2. No one in fact does know what “approval” means, thus no one knows what his vote might mean.
  3. Strategic voters will base their vote on what they think other voters will do. If enough people do this, the outcome is entirely unpredictable.
  4. The votes are not ranked, and this could distort the outcome. This is the hypothetical example they give: there are 10,000 voters and three candidates. 9,999 voters think “A” is excellent, “B” is mediocre, and “C” is disastrous. One nut-case thinks just the opposite. He loves “C” and hates “A” but agrees that “B” is mediocre. If mediocrity rates “approval,” then B is elected unanimously, A getting one less vote.

A number of objections can be raised to this analysis. Is the scenario in “4” ever really likely? And do not points 3 & 4 tend to cancel each other out? In this case it just takes two “strategic” voters to ensure that the “right” candidate wins. But perhaps there are better systems in any case.

Condorcet Voting. Named after the Marquis de Condorcet (1743-94). The idea is to choose the candidate who would beat all others in a two-way race. Ballots are marked with rankings (as in the Borda Count) but the tally is different. Condorcet voting was impractical before computers, but is simple now. Every possible pairing is calculated and the Condorcet winner is the one who is the majority winner with every other candidate. The problem is that there may be what is called a Condorcet cycle, which is a scissors-paper-rock situation: A beats B, B beats C, but C beats A. While this is theoretically possible, in real life situations, it is probably extremely rare—so rare as to be disregarded. But no one really knows. At all events, “Condorcet winner” is a concept often brought up in discussions about voting systems.

Instant-Runoff Voting (IRV). We discussed this earlier, so here we will just discuss the technical pros and cons. Proponents argue that candidates are encouraged to appeal broadly to many voters, thus bringing politicians to the center and good sense. The strategic voter cannot use the same stunt as in the Borda count because lesser rankings do not matter unless a candidate is eliminated. Your second-place ranking will not matter unless you vote for someone likely to be eliminated.

Critics say that moderates are actually punished. IRV can produce something called “center-squeeze.” That means that a very popular moderate can be “squeezed” by candidates to the left or the right who inspire limited but passionate support. If the moderate gets nearly everyone's second-place vote, but very few first place votes, the moderate will be eliminated in the first “round.” This could be true even if the moderate would be a Condorcet winner (would win in two-person match-up with every other candidate.)

There is also the peculiar “winner-turns-loser paradox” This is a strange situation in which voting for a candidate would cause him to lose. It is best to explain by example: Suppose you have three candidates, each fairly popular. There is a liberal, a conservative, and a moderate. Thirty-six percent are for the liberal, thirty-three percent are for the conservative and thirty-one percent for the moderate. If everyone votes honestly, the moderate will be eliminated. If roughly half of moderate voters ranked the liberal in second place, the liberal wins. But suppose some sneaky conservatives who had read the pre-election poll numbers and knew about this strange paradox decided to vote for the liberal instead. Enough of them to constitute four percent of the electorate do this. Now the liberal has forty percent, the conservative is down to twenty-nine percent and the moderate still has thirty-one. But it is now the conservative who is eliminated. Presumably all the conservative second-rank votes go to the moderate rather than the liberal, so the moderate collects all those votes for a grand total of sixty percent! Knowing their candidate was going to lose, the tricky conservatives made sure that the liberal lost as well.

This may not be as weird—or as bad—as it seems. Consider the original situation. For the sake of simplicity let us imagine only 100 voters. How would a Condorcet Vote turn out? (The Condorcet system looks to see how everyone does in two-way races). If, as originally supposed, all the liberal voters and all the conservatives rank the moderate in second place, and suppose that the moderate voters were more or less evenly split between liberals and conservatives for second place: say, 15 for the liberal and 14 for conservative, then these are the two-way pairings:

  • Liberal (52) beats Conservative (48)
  • Moderate (67) beats Conservative (33)
  • Moderate (64) beats Liberal (36)

So the Liberal does beat the Conservative by a small margin, but the Moderate beats them both by a landslide in Condorcet pairings! The Moderate is the clear Condorcet winner!

Here, as with the objections to approval voting, one of the supposed failings of the system cancels out another—in this case the “winner-turns-loser paradox” merely corrects the “center squeeze.” In fact, the “winner-turns-loser paradox” can only work when it is a corrective to the “center squeeze.” The center-squeeze is the real problem here.

And the center-squeeze is only going to arise when the electorate is deeply polarized. This is not usually the case, particularly in American politics. (Stark polarization usually occurs along ethnic or religious lines.) If political sentiment is thought of as a linear distribution from left to right (this is an oversimplification, but it works for this illustration) then it generally follows a classic bell curve, with most people in the center and the curve sloping as you approach either extreme. For the center squeeze to actually be a problem, the curve must plateau in the center, or even dip so that it looks like the profile of a two-humped Bactrian camel. In real politics (at least in America) most people occupy the center. And that means that most people are likely to rate the Moderate as number one. In American politics, even the extremists try to present themselves as occupying the center. And they all try to portray their opponents as off on the edges. That is why people like Rush Limbaugh (who really is on the edge) try to claim that Barack Obama (a left-leaning moderate) is somewhere on the lunatic fringe.

There are real problems with IRV, mostly logistical. If there are lots of choices on the ballot, it can be a pain to vote (who wants to rank 30 candidates, most of whom you know little about). And tallying the votes can be very complex, again, particularly when there are lots of candidates. If there are lots of fringe candidates with close votes between them, the votes must be carefully counted, then counted again making sure that every vote is counted. You might have to wait days to know whether it is the Prohibition or the Flat-Earth candidate who gets eliminated. This can be tiresome (and expensive). But is it really likely to be a problem? Who knows?

Range Voting. Mathematical theorists like this one. Every voter ranks each candidate on a given scale (1-10, 1-5, etc.). The choice with the most total points wins. It is used for beauty contests, internet polls, and the like, but almost never in public elections. It is possible to vote strategically, but with minimal effect on the outcome. If, say, there are two candidates who are close to each other but way ahead of the others in the race and one of them is your favorite, you could give your favorite the maximum score (as you would voting honestly) and the other fellow the minimum. If many people did that it would be unlikely to greatly effect the outcome (unless all of the voters for one candidate did it, but none of the other candidate's voters did) but would make the overall race closer. Computer studies suggest that range voting produces a more accurate outcome than any of the other popular systems (which is why the mathematicians like it). Another plus is that present voting machines would not have to be modified or abandoned for new ones with range voting (also true of approval voting, but an important criticism of IRV which would require different and more complex voting machines).

The main complaint about range voting is that it is too complicated. That is a questionable assessment. The people who design internet polls use it because they think it simple. It is easy to tally. If there are a lot of choices, it is more trouble for voters than a plurality vote (where you mark the one you like and ignore the rest), but not excessively so. It is easy to go through the no-hope candidates and give them all the lowest score.

More Information. Here are some websites with lots more information:

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