More than Three Candidates
When there are more than three candidates, we can still use many of the methods we have described.
Plurality: Only the top preference of each voter matters, as before. If a voter has candidate X as her top choice, then X gets one point.
Positional: We assign 1 point to top-ranked candidates and 0 points to bottom-ranked candidates. If there are n candidates, then we must also have n − 2 parameters to tell us how many points to assign to the intervening candidates. For example, with four candidates, candidates ranked second would receive s points, and candidates ranked third would receive t points, where s and t are values between 0 and 1 with s greater than or equal to t.
Pairwise: With n candidates, there are n(n − 1)/2 pairwise match-ups to check. The Condorcet winner (if there is one) is still defined as the candidate who wins all of her pairwise elections.
As with three candidates, all of these methods can yield ambiguous results. We typically focus on three candidates because this case is sufficient to illustrate the voting "paradoxes" that we have seen.
The triangle diagram does not easily generalize beyond three candidates. With four candidates, for example, it is natural to start with a tetrahedron and slice it up into 24 solid regions: one for each possible preference order.
However, this three-dimensional diagram would be very difficult to interpret, so it would be nice to find a two-dimensional representation. In , Saari explains how to "unfold" this tetrahedron, obtaining the diagram shown below.
In this diagram, we still use the rule of thumb that "closer is better": the regions that are closest to the point labeled A represent the voters who have A as their top choice. For example, the region marked with an "x" in the diagram above consists of points that are closest to A, second closest to B, next closest to D, and farthest from C. So this region represents the preference A > B > D > C.
To compute the plurality winner using this diagram, we simply need to add up numbers in the appropriate regions. In the diagram below, each region is marked with a dot indicating which candidate is top-ranked by the voters in that region.
Similarly we can locate the regions where each candidate is ranked second and third and use this to compute the positional winner, as described above.
There are many ways to determine a winner in an election with three candidates. These methods include the standard plurality method and the Borda and Condorcet methods developed in the late 18th Century. Representing the information in voter profiles graphically allows us to understand these methods in an intuitive way. In addition, we can use ideas from linear algebra and vectors to construct interesting examples that illustrate situations where these methods give conflicting results. For more information on the mathematics of voting and elections, please see the references.