“Democracy Versus Game Theory" by Professor H. L. Bray, coming out soon
Preferential ballot elections allow voters to specify a 1st choice, a 2nd choice, a 3rd choice, and so on. Since they collect more information from voters, preferential ballot elections often produce winners who better represent voters than single vote ballot elections, in ways that can be made mathematically precise.
However, since more information is collected from voters, the question of how to determine the winner of a preferential ballot election is more complex, both mathematically and philosophically. This is the topic of Professor Bray's book, "Democracy Versus Game Theory: The Case for Preferential Ballot Elections."
One of the subtleties is that there are many ways to determine the winner of a preferential ballot election, and often these different methods lead to different results. More importantly, each vote counting method has its own game theory, which incentivizes candidates and voters to behave in certain ways.
For example, single vote ballot elections incentivize candidates to join one of two major political parties to avoid "splitting the vote" with other candidates who have similar political positions. Furthermore, voters are incentivized to vote for one of the two major polical parties to avoid "throwing their vote away." This mathematically precise effect, known as Duverger's law, is very evident in American politics, which has favored two major political parties for most of its history.
The algorithm this web page uses to determine the winner (and ranking of all of the candidates) is the Ranked Pairs algorithm. This algorithm is used because it is Condorcet, clone invariant, monotone, and last place loser independent. These properties incentivize voters to vote more honestly and candidates to seek more centrist positions. They also decrease the influence of political parties and fringe candidates.
Condorcet means that a candidate who wins every head-to-head contest against every other candidate, if such a candidate exists, should win the election, thereby rewarding centrist candidates. Clone invariant means that similar candidates do not penalize (or reward) each other (for example, by splitting the vote), thereby making political parties less necessary. Monotone means that if a voter ranks a candidate higher, this should not ever hurt the candidate, allowing voters to vote honestly. Last place loser independent means that if the candidate who came in last place drops out of the election, the outcome does not change, thereby minimizing the influence of fringe candidates.
The only major preferential ballot vote counting method with all four of these properties is Ranked Pairs. Philosophically, Ranked Pairs chooses the ordering of the candidates which gives the highest priority to the greatest margins of victory in all of the head-to-head contests (where only two candidates compete at a time). Professor Bray's book, among other things, compares and contrasts this method to the other major methods, ultimately arguing the the Ranked Pairs method is the best preferential ballot vote counting method to use in most situations, for the reasons just described.
A brief review of other algorithms: Instant Runoff Voting is neither Condorcet (and hence not good at picking centrist candidates) nor monotone (so that, for example, voting for a candidate can actually decrease their odds of winning, in some circumstances). The Borda Count is monotone, but is still not Condorcet, and incentivizes voters to vote very dishonestly in some cases, including highly ranking candidates that they don't like, as a way of hurting candidates they view as competition for the candidate that they do like. Neither of these methods should be used to select a single winner,* if the above considerations are a concern. On the other hand, the Schulze method, which is Condorcet, clone invariant, and monotone, is a reasonable method, even though it is not last place loser independent. However, while it always determines a winner, it does not always provide a ranking for all of the candidates, even when all ties are broken, which could be a problem for some applications.
(* Instant Runoff Voting is okay when there are many winners, in certain circumstances where a broad diversity of winners is desired. However, for single winner elections, it is only slightly better than a single vote ballot election at selecting centrist candidates and, in general, has a very complicated game theory, thereby making it more vulnerable to manipulation. There is also a chaotic element to Instant Runoff Voting, where relatively small changes in the votes can lead to the single winner being drastically different.)