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The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. The method can also be used to create a sorted list of winners. The Schulze method is also known as Schwartz Sequential Dropping (SSD), Cloneproof Schwartz Sequential Dropping (CSSD), Beatpath Method, Beatpath Winner, Path Voting, and Path Winner. If there is a candidate who is preferred pairwise over the other candidates, when compared in turn with each of the others, the Schulze method guarantees that that candidate will win. Because of this property, the Schulze method is (by definition) a Condorcet method. Many different heuristics for the Schulze method have been proposed. The most important heuristics are the path heuristic and the Schwartz heuristic. The path heuristic Each ballot contains a complete list of all candidates. Each voter ranks these candidates in order of preference. Voters may give the same preference to more than one candidate and may keep candidates unranked. When a given voter does not rank all candidates, then it is presumed that this voter strictly prefers all ranked candidates to all not ranked candidates and that this voter is indifferent between all not ranked candidates. Procedure Suppose dV,W is the number of voters who strictly prefer candidate V to candidate W. A path from candidate X to candidate Y of strength z is an ordered set of candidates C(1),...,C(n) with the following four properties:
If there is a p such that there is a path from candidate A to candidate B of strength p and no path from candidate B to candidate A of strength p, then candidate A disqualifies candidate B. Candidate D is a potential winner if and only if there is no candidate E such that candidate E disqualifies candidate D. Examples A path from candidate X to candidate Y is an ordered set of candidates C(1),...,C(n) with the following three properties:
The strength of the path C(1),...,C(n) is min . In other words: The strength of a path is the strength of its weakest link. pA,B = max . pA,B = 0 if there is no path from candidate A to candidate B. In other words: pA,B is the strength of the strongest path from candidate A to candidate B. Then the Schulze method can be described as follows: Candidate A is a potential winner if and only if pA,B ≥ pB,A for every other candidate B. Example 1 Example (45 voters; 5 candidates): 5 ACBED 5 ADECB 8 BEDAC 3 CABED 7 CAEBD 2 CBADE 7 DCEBA 8 EBADC The critical defeats of the strongest paths are underlined. Candidate E is a potential winner, because pE,X ≥ pX,E for every other candidate X. Example 2 Example (30 voters; 4 candidates): 5 ACBD 2 ACDB 3 ADCB 4 BACD 3 CBDA 3 CDBA 1 DACB 5 DBAC 4 DCBA The critical defeats of the strongest paths are underlined. Candidate D is a potential winner, because pD,X ≥ pX,D for every other candidate X. Example 3 Example (30 voters; 5 candidates): 3 ABDEC 5 ADEBC 1 ADECB 2 BADEC 2 BDECA 4 CABDE 6 CBADE 2 DBECA 5 DECAB The critical defeats of the strongest paths are underlined. Candidate B is a potential winner, because pB,X ≥ pX,B for every other candidate X. Example 4 Example (9 voters; 4 candidates): 3 ABCD 2 DABC 2 DBCA 2 CBDA The critical defeats of the strongest paths are underlined. Candidate B and candidate D are potential winners, because pB,X ≥ pX,B for every other candidate X and pD,Y ≥ pY,D for every other candidate Y. The Schwartz set The definition of a Schwartz set, as used in the Schulze method, is as follows: Procedure The voters cast their ballots by ranking the candidates according to their preferences, just like for any other Condorcet election. The Schulze method uses Condorcet pairwise matchups between the candidates and a winner is chosen in each of the matchups. From there, the Schulze method operates as follows to select a winner (or create a ranked list): The situation The results would be tabulated as follows:
Pairwise winners First, list every pair, and determine the winner: Note that absolute counts of votes can be used, or percentages of the total number of votes; it makes no difference. Dropping Next we start with our list of cities and their matchup wins/defeats Technically, the Schwartz set is simply Nashville as it beat all others 3 to 0. Therefore, Nashville is the winner. Ambiguity resolution example Let's say there was an ambiguity. For a simple situation involving candidates A, B, and C. In this situation the Schwartz set is A, B, and C as they all beat someone. Schulze then says to drop the weakest defeat, so we drop C > A and are left with Therefore, A is the winner. (It may be more accessible to phrase that as "drop the weakest win", though purists may complain.) Summary In the (first) example election, the winner is Nashville. This would be true for any Condorcet method. Using the first-past-the-post system and some other systems, Memphis would have won the election by having the most people, even though Nashville won every simulated pairwise election outright. Nashville would also have been the winner in a Borda count. Using Instant-runoff voting in this example would result in Knoxville winning, even though more people preferred Nashville over Knoxville. Satisfied criteria The Schulze method satisfies the following criteria: If winning votes is used as the definition of defeat strength, it also satisfies: If margins as defeat strength is used, it also satisfies: Failed criteria The Schulze method violates the following criteria: Independence of irrelevant alternatives The Schulze method fails independence from irrelevant alternatives. However, the method adheres to a less strict property is sometimes called local independence from irrelevant alternatives. It says that if one candidate (X) wins an election, and a new alternative (Y) is added, X will win the election if Y is not in the Smith set. Local IIA implies the Condorcet criterion. Use of the Schulze method The Schulze method is not currently used in government elections. However, it is starting to receive support in some public organizations. Organizations which currently use the Schulze method are: External resources Note that these sources may refer to the Schulze method as CSSD, SSD, beatpath, path winner, etc. General Advocacy Research papers Books Software | ||||||||||||||||||||||||||||||||||||||||||||||||||||
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