We're all familiar with voting: You punch your chad or fill in the bubble or pull the lever next to the name of your candidate, and the candidate with the most votes wins. Simple enough. However, this is not the only way to do an election.
To see why, suppose we have 3 candidates, and candidate A gets 40%, candidate B gets 45%, and candidate C gets 15%. Candidate B wins in our normal voting method in the US. However, many people are bothered by the fact that candidate B lacks majority support.
One answer would be to hold a runoff election between A and B, to figure out who has majority support. B can still win, if at least a third of C's supporters (5% or more) support B and push him over the edge. However, A could also win, depending on how C's supporters vote.
Now, holding a second election is cumbersome. What if on election day every voter had to indicate his or her first choice, second choice, 3rd choice, etc.? We could look and see if anybody is the favorite of a majority. If so, great, that candidate wins. If not, we could eliminate the candidate with the fewest first place votes from consideration, look at ballots that listed that candidate first, and see who those voters' second choice is. With their first choice gone, their vote goes to their second choice, and we again see if anybody is the favorite of a majority. We keep doing this until somebody has a majority.
There is a name for this approach: Instant Runoff Voting (IRV). It's used in a few US locales, and there are organizations that want to use it in more places. It has some problems (which I will talk about in another post) but it's an intuitive illustration of a ranked voting method. Everybody ranks candidates (1st choice, 2nd choice, 3rd choice, etc.) and that ranking information is used to determine the winner.
It isn't the only ranked voting method, however. Consider a few more possibilities:
1) We again cast ballots where we indicate 1st choice, 2nd choice, 3rd choice, etc., and then those choices get points. In a simple example, the 1st choice gets 2 points, the 2nd choice gets 1 point, and the last choice gets 0 points. This would be an example of the Borda Count, which has been extensively studied by Donald Saari, a mathematician at UC Irvine. It isn't the only possible point system, of course. We could have a point system where 1st choice gets 5 points, 2nd choice gets a token 1 point, and the others get 0 points. Or point systems with any number of other possible scoring rules. The key thing to understand is that, again, each voter indicates a ranking of candidates, and based on the number of people giving each ranking we figure out the winner.
2) Condorcet: Again, we all rank the candidates. Then, the people counting the votes look at our ballots and see if a majority prefers A to B or B to A. They look at our ballots and figure out if a majority prefers A to C or C to A. They do the same with B and C. And if there are more candidates, they keep doing this until every pair of candidates has been compared.
If somebody wins every contest, that person is the winner. But what if A beats B, B beats C, and C beats A? It can happen. In technical terms, it can happen with rational voters if issue space is more than 2D (more on that in another post). But the idea is paper, rock, scissors. When it happens, you need a backup plan. And if you Google for information on Condorcet Voting you'll discover that every fan of Condorcet has a preferred way of resolving that dilemma.
The key thing for my work is that Condorcet, much like IRV, is a multi-stage method. In IRV, you first check to see if anybody has a majority. If not, you do eliminations and transfers. In Condorcet, you check to see if anybody beats all of the other candidates one-on-one. If not, you do something else.
And, as you can imagine, there are countless more examples of ranked voting methods. For my work, I just take it as a given that voters rank candidates, and based on those rankings a decision is made. I don't care about the specifics of how the decision is made, as long as it satisfies a few criteria. The key thing is that the election concept has been expanded from just voting for a single candidate, to a situation where voters indicate much more detailed information, and that information is processed according to (possibly quite complicated) rules.
Future topics (in no particular order):
*Arrow's Theorem
*Issue space, squeezed centrists, and Condorcet paradoxes
*Gibbard-Satterthwaite Theorem
*Elections and geometry.
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