Here in San Francisco, we are about to have a run-off election to choose our next mayor. The two contenders are Gavin Newsom and Matt Gonzalez.
We already had an election but among the field of candidates, nobody got more than 50% of the vote which is what would have been required to avoid a run-off. Still, Newsom actually pulled off about 40% which makes him the man to beat. In comparison, Gonzalez brought in about 20%.
Run-off elections are a good thing in that they more closely represent the choice of the voter than, say, plurality elections where the candidate with the most votes wins. The reason for this was starkly clear in the last presidential election where third-party candidates (e.g. Ralph Nader) can pull away votes from a favored candidate and tip the election towards a candidate that the majority of the population did not approve of.
No voting system is perfect but some are better than others. As early as the 18th century, the Marquis de Condorcet was one of the first to point out that in a simple plurality voting system, you can get counter-intuitive results. In a simple example, it is not hard to show that voter preferences can be set up among three candidates (A,B and C) such that if A was pitted against B, A would win; if B was pitted against C, B would win. But, counterintuitively, if A went up against C, C would win.
The Nobel Prize winning economist Kenneth Arrow expanded on this to show that it was in fact impossible to construct a perfect voting system. That is, given a basic set of requirements about how results should follow from voter choice, there is no system that guarantees these results!
None of this means, however, that some systems are not better than others. But plurality is one of the weakest. Much better are ranking systems. The best article I found on the web that gives a good laymans overview is this article from Science News. An excerpt from near the end of the article lays out the problem succinctly:
Consider 15 people deciding what beverage to serve at a party. Six prefer milk first, wine second, and beer third; five prefer beer first, wine second, and milk third; and four prefer wine first, beer second, and milk third.
In a plurality vote, milk is the clear winner. But if the group decides instead to hold a runoff election between the two top contenders, milk and beer, then beer wins, since nine people prefer it over milk. And if the group awards two points to a drink each time a voter ranks it first and one point each time a voter ranks it second, suddenly wine is the winner. Although this is a concocted example, it's not an anomaly
Though why anyone would choose milk for a party is beyond me.