Condorcet Voting ExplainedIn the Condorcet election method, voters rank the candidates in order of preference. The vote counting procedure then takes into account each preference of each voter for one candidate over another. It does so by conceptually breaking the election down into a series of separate races between each possible pairing of candidates, hence it is sometimes referred to as a ``pairwise'' method. If one of the candidates beats each of the other candidates in their one-on-one race, then that candidate wins. Otherwise, the result is ambiguous and an optimal procedure is used to resolve the ambiguity. Unlike our current plurality election method, the Condorcet system gives voters little incentive to falsify their true preferences.Pairwise Matrix ConstructionThe basics of Condorcet voting are best illustrated by example. Suppose an election has four candidates designated A, B, C, and D. Each voter ranks the candidates in order of preference. For example, the vote (B,D,C) ranks B first, D second, and C third. The last choice is implied. Voters are not required to fully rank the entire list. For example, the vote (D,B) indicates that the voter has no preference between A and C. Each vote translates to a pairwise matrix showing the equivalent vote for each possible pairing of candidates, as illustrated in the tables below.
Table 1 shows the pairwise matrix for the vote (B,D,C). Since B is ranked first, the B row gets a 1 in each of the columns for the other three candidates. Since D is ranked above C and A, the D row gets a 1 in the C and A columns. Finally, since C is ranked above A, the C row gets a 1 in the A column. Since A is not ranked, the A row is filled with zeros. All cells that do not get a 1 get a 0 (except that the diagonal cells are left blank, of course). As another example, Table 2 shows the pairwise matrix for the vote (D,B). The pairwise matrix for each vote are summed to get the pairwise matrix sum. Table 3 shows the sum for the two votes discussed here. Corresponding elements of the individual pairwise matrices are simply added together to get the pairwise matrix sum. The final pairwise matrix sum is used to determine the winner. If one of the candidates wins his or her one-on-one race with each of the other candidates, that candidate wins. For example, consider the pairwise matrix sum shown in Table 4 below. In this example, B beats A by 87-63, B beats C by 78-72, and B beats D by 73-51. Because B beats each of the other candidates, B wins.
Cyclic Ambiguity ResolutionSometimes no candidate beats each of the other candidates. The result is then ambiguous, and the ambiguity must be resolved. Such cyclic ambiguities are true ambiguities in the preferences of the electorate, and the fact that Condorcet accurately reflects them is not a problem with the Condorcet method itself, as is often erroneously assumed. Several excellent methods exist for resolving ambiguities, each with minor advantages and disadvantages compared to the other methods. The determination of the preferable method is an ongoing area of research and discussion. Two of those methods will be discussed here.The ambiguous pairwise matrix of Table 5 will be used as an example to show how various methods resolve cyclical ambiguities. In this example, none of the candidates is unbeaten: A is beaten by D and C; B is beaten by A and D; C is beaten by B; and D is beaten by C. The notation A/B (``A over B'') will be used to denote the defeat of B by A. A cyclic ambiguity is a set of defeats that ``wraps around'' on itself. For example, the list of defeats A/B, B/C, C/A forms a cycle. A cycle always involves at least three and possibly more candidates. A cycle also involves at least three and possibly more defeats. The magnitude of a defeat is the number of votes cast against the defeated candidate. For example, in Table 5, D defeated B and got 60 votes against B in doing so, hence the magnitude of the defeat is 60 votes. Contrary to intuition, the margin of victory or defeat is irrelevant because dropping or ignoring a defeat, when necessary, is tantamount to overruling those votes for the defeat--and fairness requires that as few votes be overruled as possible (the 50 votes against the defeat were ``overruled'' by the voters themselves). The notation D/B (60) will be used to denote the defeat of B by D with a magnitude of 60 votes. Here are the defeats in Table 5, ordered by magnitude:
SSD is the preferred method. It is relatively easy to program on a computer, and for all practical purposes can be computed instantly. If the number of candidates is three or less, PC and SSD give the same result, but for more than three candidates they could give different results. Plain Condorcet (PC)The Plain Condorcet method of ambiguity resolution is the simplest method: drop the weakest (smallest magnitude) defeat, repeating if necessary until one of the candidates is unbeaten.The Plain Condorcet method proceeds as follows for the example: D/A is the weakest defeat, so it is dropped. But A has another defeat, so there's still no unbeaten candidate. Next, C/D is the weakest defeat, so it is dropped. Now D is unbeaten, so D wins. The Plain Condorcet method is very simple and usually works well. It is superior to plurality and Instant Runoff Voting, but it suffers from some technical deficiencies compared to SSD and is not recommended for use in actual elections unless the public demands an extremely simple method. Schwartz Sequential Dropping (SSD)The Schwartz Sequential Dropping (SSD) method is as follows: drop the weakest defeat among an innermost unbeaten set, repeating if necessary until one of the candidates is unbeaten. An unbeaten set is a set of which none are beaten by anyone outside the set. An innermost unbeaten set is an unbeaten set that doesn't contain a smaller unbeaten set. In the absence of ties and defeats of equal magnitude, there can be only one innermost unbeaten set, known as the Schwartz set.The ordered list of defeats for the example is repeated here for convenience:
Note that PC and SSD produced the same winner in this case. |