The Baldwin method satisfy the Condorcet criterion: since Borda always gives any existing Condorcet winner more than the average Borda points, the Condorcet winner will never be eliminated. It does not satisfy the independence of irrelevant alternatives criterion, the monotonicity criterion, the participation criterion, the consistency criterion and the independence of clones criterion, while it does satisfy the majority criterion, the mutual majority criterion, the Condorcet loser criterion, and the Smith criterion. Also, the Baldwin method violates reversal symmetry.
Copeland’s method or Copeland’s pairwise aggregation method is a Condorcet method in which candidates are ordered by the number of pairwise victories, minus the number of pairwise defeats.
Proponents argue that this method is easily understood by the general populace, which is generally familiar with the sporting equivalent. In many round-robin tournaments, the winner is the competitor with the most victories.
Copeland requires a Smith set containing at least five candidates to give a clear winner unless two or more candidates tie in pairwise comparisons.
Dodgson’s Method is a voting system proposed by Charles Dodgson.
In Dodgson’s method, each voter submits an ordered list of all candidates according to their own preference (from best to worst). The winner is defined to be the candidate for whom we need to perform the minimum number of pairwise swaps (added over all candidates) before they become a Condorcet winner. In particular, if there is already a Condorcet winner, they win the election.
In short, we must find the voting profile with minimum Kendall tau distance from the input, such that it has a Condorcet winner; they are declared the victor. Computing the winner is an NP-hard problem.
The Kemeny-Young method is a voting system that uses preferential ballots and pairwise comparison counts to identify the most popular choices in an election. It is a Condorcet method because if there is a Condorcet winner, it will always be ranked as the most popular choice.
This method assigns a score for each possible sequence, where each sequence considers which choice might be most popular, which choice might be second-most popular, which choice might be third-most popular, and so on down to which choice might be least-popular. The sequence that has the highest score is the winning sequence, and the first choice in the winning sequence is the most popular choice. (As explained below, ties can occur at any ranking level.)
The Kemeny-Young method is also known as the Kemeny rule, VoteFair popularity ranking, the maximum likelihood method, and the median relation.
The Kemeny-Young method uses preferential ballots on which voters rank choices according to their order of preference. A voter is allowed to rank more than one choice at the same preference level. Unranked choices are usually interpreted as least-preferred.
Another way to view the ordering is that it is the one which minimizes the sum of the Kendall tau distances (bubble sort distance) to the voters’ lists.
Kemeny-Young calculations are usually done in two steps. The first step is to create a matrix or table that counts pairwise voter preferences. The second step is to test all possible rankings, calculate a score for each such ranking, and compare the scores. Each ranking score equals the sum of the pairwise counts that apply to that ranking.
The ranking that has the largest score is identified as the overall ranking. (If more than one ranking has the same largest score, all these possible rankings are tied, and typically the overall ranking involves one or more ties.)
The Nanson method is based on the original work of the mathematician Edward J. Nanson.
Nanson’s method eliminates those choices from a Borda count tally that are at or below the average Borda count score, then the ballots are retallied as if the remaining candidates were exclusively on the ballot. This process is repeated if necessary until a single winner remains.
The Nanson method and the Baldwin method satisfy the Condorcet criterion: since Borda always gives any existing Condorcet winner more than the average Borda points, the Condorcet winner will never be eliminated. They do not satisfy the independence of irrelevant alternatives criterion, the monotonicity criterion, the participation criterion, the consistency criterion and the independence of clones criterion, while they do satisfy the majority criterion, the mutual majority criterion, the Condorcet loser criterion, and the Smith criterion. The Nanson method satisfies reversal symmetry, while the Baldwin method violates reversal symmetry.
Nanson’s method was used in city elections in the U.S. town of Marquette, Michigan in the 1920s. It was formally used by the Anglican Diocese of Melbourne and in the election of members of the University Council of the University of Adelaide. It was used by the University of Melbourne until 1983.
Ranked pairs (RP) or the Tideman method is a voting system developed in 1987 by Nicolaus Tideman that selects a single winner using votes that express preferences. RP can also be used to create a sorted list of winners.
If there is a candidate who is preferred over the other candidates, when compared in turn with each of the others, RP guarantees that candidate will win. Because of this property, RP is (by definition) a Condorcet method.
The RP procedure is as follows:
- Tally the vote count comparing each pair of candidates, and determine the winner of each pair (provided there is not a tie).
- Sort (rank) each pair, by the largest margin of victory first to smallest last. “Lock in” each pair, starting with the one with the largest number of winning votes, and add one in turn to a graph as long as they do not create a cycle (which would create an ambiguity). The completed graph shows the winner.
- RP can also be used to create a sorted list of preferred candidates. To create a sorted list, repeatedly use RP to select a winner, remove that winner from the list of candidates, and repeat (to find the next runner up, and so forth).
Tally:
To tally the votes, consider each voter’s preferences. For example, if a voter states “A > B > C” (A is better than B, and B is better than C), the tally should add one for A in A vs. B, one for A in A vs. C, and one for B in B vs. C. Voters may also express indifference (e.g., A = B), and unstated candidates are assumed to be equally worse than the stated candidates.
Once tallied the majorities can be determined. If “Vxy” is the number of Votes that rank x over y, then “x” wins if Vxy > Vyx, and “y” wins if Vyx > Vxy.
Sort:
The pairs of winners, called the “majorities”, are then sorted from the largest majority to the smallest majority. A majority for x over y precedes a majority for z over w if and only if one of the following conditions holds:
- Vxy > Vzw. In other words, the majority having more support for its alternative is ranked first.
- Vxy = Vzw and Vwz > Vyx. Where the majorities are equal, the majority with the smaller minority opposition is ranked first.
Lock:
The next step is to examine each pair in turn to determine which pairs to “lock in”. This can be visualized by drawing an arrow from the pair’s winner to the pair’s loser in a directed graph. Using the sorted list above, lock in each pair in turn unless the pair will create a circularity in the graph (e.g., where A is more than B, B is more than C, but C is more than A).
Winner:
In the resulting graph, the source corresponds to the winner. A source is bound to exist because the graph is a directed acyclic graph by construction, and such graphs always have sources. In the absence of pairwise ties, the source is also unique (because whenever two nodes appear as sources, there would be no valid reason not to connect them, leaving only one of them as a source).
Of the formal voting system criteria, the ranked pairs method passes the majority criterion, the monotonicity criterion, the Condorcet criterion, the Condorcet loser criterion, and the independence of clones criterion. Ranked pairs fails the consistency criterion and the participation criterion. While ranked pairs is not fully independent of irrelevant alternatives, it does satisfy local independence of irrelevant alternatives.