Ranked voting

Concepts formulated by the Marquis de Condorcet in the 18th century continue to be important in the study of elections. One of these concepts is the Condorcet winner, a candidate who would win against any other candidate in a two-way race. A voting system that always elects this candidate is called a Condorcet method; however, it is possible for an election to have no Condorcet winner, a situation called a Condorcet cycle. Suppose an election with 3 candidates A, B, and C has 3 voters. One votes A > C > B, one votes B > A > C, and one votes C > B > A. In this case, no Condorcet winner exists: A cannot be a Condorcet winner as two-thirds of voters prefer B over A. Similarly, B cannot be the winner as two-thirds prefer C over B, and C cannot win as two-thirds prefer A over C. This forms a rock-paper-scissors style cycle with no Condorcet winner.

Voting systems can also be judged on their ability to deliver results that maximize the overall well-being of society, i.e. to choose the best candidate for society as a whole.^[9]

Spatial voting models, initially proposed by Duncan Black and further developed by Anthony Downs, provide a theoretical framework for understanding electoral behavior. In these models, each voter and candidate is positioned within an ideological space that can span multiple dimensions. It is assumed that voters tend to favor candidates who closely align with their ideological position over those more distant. A political spectrum is an example of a one-dimensional spatial model.

The accompanying diagram presents a simple one-dimensional spatial model, illustrating the voting methods discussed in subsequent sections of this article. It is assumed that supporters of candidate A cast their votes in the order of A > B > C, while candidate C's supporters vote in the sequence of C > B > A. Supporters of candidate B are equally divided between listing A or C as their second preference. From the data in the accompanying table, if there are 100 voters, the distribution of ballots will reflect the positioning of voters and candidates along the ideological spectrum.

Spatial models offer significant insights because they provide an intuitive visualization of voter preferences. These models give rise to an influential theorem—the median voter theorem—attributed to Duncan Black. This theorem stipulates that within a broad range of spatial models, including all one-dimensional models and all symmetric models across multiple dimensions, a Condorcet winner is guaranteed to exist. Moreover, this winner is the candidate closest to the median of the voter distribution. Empirical research has generally found that spatial voting models give a highly accurate explanation of most voting behavior.^[10]

Arrow's impossibility theorem is a generalization of Condorcet's result on the impossibility of majority rule. It demonstrates that every ranked voting algorithm is susceptible to the spoiler effect.^{[dubious – discuss]} Gibbard's theorem provides a closely related corollary, that no voting rule can have a single, always-best strategy that does not depend on other voters' ballots.

The Borda count is a weighted-rank system that assigns scores to each candidate based on their position in each ballot. If m is the total number of candidates, the candidate ranked first on a ballot receives m − 1 points, the second receives m − 2, and so on, until the last-ranked candidate who receives zero. In the given example, candidate B emerges as the winner with 130 out of a total 300 points. While the Borda count is simple to administer, it does not meet the Condorcet criterion. Also, it is heavily affected by the entry of candidates who have no real chance of winning.

Instant-runoff voting, often conflated with ranked-choice voting in general, is a contingent ranked-vote voting method that recursively eliminates the successive plurality loser during rounds of counting and transfers their votes, if possible, until one candidate has the majority of the votes. In the given example, candidate A is declared winner in the third round, having received a majority of votes through the accumulation of first-choice votes and redistributed votes from candidate B. This system embodies the voters' preferences between the final candidates, stopping when a candidate garners the preference of a majority of voters. Instant-runoff voting does not fulfill the Condorcet winner criterion.

Single transferable voting is a contingent ranked-vote election method that elects multiple members. It elects any candidates who achieve quota, and if necessary recursively eliminates the plurality loser at various stages of the vote count. It transfers the votes of eliminated candidates and surplus votes of winners until enough are elected by quota or by still being in the running when the field of candidates is thinned to the number of remaining open seats. Because elected members are elected with the same or about the same number of votes, each party popular enough for representation receives a number of seats appropriate to the vote tallies of its candidates. The transfers reduce waste to about one quota – which in a five-seat district is about 17 percent of valid votes; in districts with more members than five, the waste is smaller. All but one quota of votes approximately are used to actually elect someone in the district so the percentage of effective votes is dependably about 80 to 90 percent of valid votes in the district.^[11][12]

The defeat-dropping Condorcet methods all look for a Condorcet winner; i.e. a candidate who is not defeated by any other candidate in a one-on-one majority vote. If there is no Condorcet winner, they repeatedly drop (set the margin to zero) for the one-on-one matchups that are closest to being tied, until there is a Condorcet winner. How "closest to being tied" is defined depends on the specific rule. For the minimax Condorcet method, the elections with the smallest margin of victory are dropped, whereas in ranked pairs, only elections that create a cycle are eligible to be dropped (with defeats being dropped based on the margin of victory).