Voting has got a lot to do with math, whether we like it or not. Every vote matters they say, and it really does, because if everyone thought going to vote was not important, the result would be truly skewed and not reflective of what society wants. Well especially with the US presidential election just behind our backs, elections are everywhere on the news. We think that the voting system is fair, and that it mathematically gives the presidency, or whatever else to the best candidate in the public’s viewpoint. Yet the system used for voting is far from perfect, and in reality creating a voting system that produces a winner according to basic intuitive rules is impossible.
One example of the complexity of voting involves that of the presidential race with Bush, Al Gore, and Nader where Bush won by a mere 517 votes. Let us assume that the supporters of Nader, who was highly liberal, would have for the most part voted for Gore, a less liberal Democrat, had Nader not been running. Well that means that had Nader not been running, Al Gore would have won, because the extra votes from Ralph Nader supporters would have brought him more votes than Bush. That brings to table a major dilemma: The majority of people in the US would have preferred Gore to Bush, and yet Bush still won. The concept of this idea was first thought of by a man named Condorcet, who conjectured that if for any candidate A, if a majority of people prefer a candidate B to A, then A cannot be the winner. And this makes perfect sense.
The mathematics of voting systems is a quirky area which has had results that are discomforting to the everyday citizen who believes that the president should be chosen by real social choice, not a flaw in the system.
Given a set of society’s preferences, there are many ways to ascertain a winner. One is by plurality, which is the normal method of voting used in the US where the person with the most first place votes wins. Another is by plurality with elimination, which eliminates a person who has the least first place votes and then compares again and so on. Another is to see which candidate is preferred by the majority to any other candidate (which is not always possible). Here is a case, and the winners in each case.
499 people: A > B > C
3 people: B > C > A
498 people: C > B > A
Winner under plurality: B
Winner under plurality with elimination: C
Condorcet Winner: A
See the problems? The unintuitive and complex nature of voting systems makes it all the more fascinating. Who ever knew you had to think so much before voting?