Most Republicans with whom I have discussed the Republican primaries say they would like anybody but former Massachusetts Gov. Mitt Romney to be the Republican nominee. But with so many other candidates splitting the vote, Romney narrowly won the Iowa caucuses and seems poised to walk away with the nomination. Assuming most Republicans have this “anybody but Romney” attitude, it seems unfair that he should represent the Republican Party in the general election. This raises the question of whether different voting systems exist that would better reflect Republican — and more generally society’s — preferences.

In a ranked voting system, voters rank candidates in order of preference. The voting system then tells us society’s preferences. Note that the plurality rules voting system by which Michigan elects its officials is a ranked voting system: Voters rank candidates (in their heads) and then vote for their first choice. Plurality rules tells us that the candidate with the most first choice votes is society’s preference.

For example, suppose Charlie, Snoopy and Woodstock run for governor of Michigan. In the election, Charlie receives 40 percent of the vote, Snoopy receives 35 percent of the vote, and Woodstock receives 25 percent of the vote. Using a plurality rules voting system, Charlie would win the election.

But what if every person who voted for Woodstock preferred Snoopy over Charlie? That is 60 percent of voters who would rather have Snoopy as governor than Charlie. But Charlie won the election. There would seem to be a problem with this voting system.

A runoff voting system, in which more than one round of voting can take place to find a winner, seeks to rectify this problem by choosing society’s preference in a different way. For example, in Louisiana’s gubernatorial elections, if no candidate receives more than 50 percent of the vote in the first round of voting, a second round of voting is held between the two candidates who received the most votes in the first round. The candidate who wins the second round is elected governor.

Another example of a runoff system follows: In each round, the candidate with the fewest number of votes is eliminated. The next round of voting has only candidates who were not previously eliminated. Rounds of voting continue until a candidate receives more than half of the vote, and this candidate wins the election. This runoff system is used for city elections in Minnesota’s Twin Cities and Duluth. Ann Arbor also used it for its mayoral election in 1975. Using either of these runoff systems, Snoopy would win the above election.

Readers may now ask: What is the best voting system, and why isn’t it used in all elections? Before answering the question, one must first define what best means. Mathematicians say a perfect voting system must have the following three properties: First, there is no dictator. That is, there is no voter whose vote completely determines the outcome of the election. Second, if a majority of voters prefer candidate X over candidate Y, then the voting system should tell us society prefers candidate X over candidate Y. Third, the voting system is independent of irrelevant alternatives. This means that if the majority of voters prefer candidate X over candidate Y and voters change their preferences for other candidates A, B, C without changing their preferences for X over Y, then the voting system still tells us society prefers X over Y.

We’ve seen above that the plurality rules system fails the second property, so it is not a perfect voting system. But what about the two runoff systems? They also fail the second property, so these runoff systems aren’t perfect either. In fact, the mathematical economist Kenneth Arrow proved in his Ph.D. thesis the amazing result that, as long as more than two candidates run in an election, one can find a problem with whatever voting system is used. No perfect voting system exists. Both a plurality rules system and the runoff systems have their merits and problems.

I hope readers walk away with two thoughts after reading this column. First, many questions — especially in public policy — often have no perfect solution. Competing solutions can be suggested and it may be that neither is wrong. This is a theme I hope to explore in my remaining columns — disagreement and debate on such issues should be expected and encouraged. It might sound powerful when a politician labels his or her opponent’s position as “simply wrong,” but without further information and a competing position, intelligent citizens should ignore such empty rhetoric.

Second, mathematicians often work on and find answers to very interesting questions that, one would think, have nothing to do with mathematics. Question: Does a perfect voting system exist? Math’s answer: No. You’ll never invent one. Stop looking.

Isn’t that neat?

Matthew Zabka can be reached at mzbka@umich.edu.

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