Voting Paradox
Voting paradoxes can arise in any election involving three or more candidates; though they come in many different forms, they can all be summed up in a single statement: Even if every voter is individually rational, society as a whole is not.
The most commonly used voting method in the United States is the plurality vote: Each voter casts a ballot for one candidate, and the winner is the candidate who gets the most votes (even if this is less than 50 percent). Though this system is familiar, it is rife with inconsistencies. Consider, for example, a situation where 9 people are deciding which one of three restaurants to eat dinner at. Each person ranks the three restaurants, and the rankings are as follows:
- 4 people rank A best, B second, and C third.
- 3 people rank B best, C second, and A third.
- 2 people rank C best, B second, and A third.
In a plurality vote, A would be the restaurant chosen—even though more than half of the voters rank it last. To make the irrationality of the plurality vote even more evident, suppose that restaurant B happens to be closed. The 9 people recast their ballots, and now C, which used to be in last place, wins over A, 5 votes to 4. (The 3 who voted for B would now cast their ballots for C, since it is their second choice). If an individual voter reacted this way, we would find it utterly illogical. (Imagine going into an ice cream parlor that has three flavors--chocolate, vanilla, and strawberry. You choose chocolate, and then the vendor tells you, "Oops, we don't have strawberry today." Would you then change your choice to vanilla?)
The paradox here is called dependence on irrelevant alternatives. That is, the outcome of the election between A and C depends on where people rank B. Even though no one changed their minds about the relative merits of A and C, the simple act of dropping B to last place on everyone's list (because the restaurant was closed) caused the social choice to change.
In view of this paradox, one might say that a "reasonable" voting system should be independent of irrelevant alternatives. One might also argue that a "reasonable" voting system should also satisfy the "Pareto condition": If every voter ranks A above B, then A should be ranked above B in the final outcome. According to a seminal theorem proved by Kenneth Arrow in 1950 (who later won a Nobel Prize for this work), if there are three or more candidates, then there is only one "reasonable" voting system: a system where one voter is a dictator, and no one else's vote counts. In essence, the irrationality of society can only be eliminated by giving all the power to one rational voter. Because this is not considered an option in a democratic society, this result is often called Arrow's impossibility theorem--it is impossible to find a "reasonable" voting system.
Ironically, Arrow's impossibility theorem—the master paradox of voting theory--had the effect of stimulating research into alternative voting systems, rather than discouraging it. Granted that no system is perfectly rational and perfectly fair, mathematicians and political scientists are still interested in determining what systems might be most rational and most fair under realistic conditions. Some other systems in common use are:
- runoff elections,
- approval voting (in which voters may cast a vote for all the candidates they approve of), used in several scientific societies,
- single transferable vote (in which the bottom candidate is dropped after each round and the voters who voted for that candidate have their votes "transferred" to their next-favorite), used in Ireland and Australia;
- Borda count (in which a voter's top-ranked candidate gets n points, the second-ranked candidate gets n-1, and so on), used in the ranking of sports teams.
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