Three-Candidate Elections Using Saari Triangles
A Saari triangle provides a way to visualize the possible outcomes of a three-candidate election under different voting procedures. Each candidate is represented by a vertex of the triangle; the outcome is represented as a point inside the triangle. One week of data from the AP college football poll is included as a source of real-world data.
Contributed by: Matthew Romney, Yukun Tan, and Mengzhou Tang (February 2016)
Based on: An undergraduate research project at the Illinois Geometry Lab by Daoyu Duan, Zelin Li, Yukun Tan, and Mengzhou Tang, directed by Matthew Romney and A. J. Hildebrand.
Open content licensed under CC BY-NC-SA
A Saari triangle provides a way to visualize the possible outcomes of a three-candidate election under different voting procedures. Each candidate is represented by a vertex of a triangle labeled , , . Taking convex combinations of these vertices according to the number of votes received gives a point inside the triangle. The vertex closest to this point is the winner of the election.
In an election with complete information, voters not only choose their favorite candidate but rank all candidates according to their preferences. In a three-candidate election, there are six possible sets of preferences: , , , , , . The notation , for example, refers to the number of voters who prefer over and over .
A first-place vote is worth 1 point, a last-place vote is worth 0 points, and a second-place vote is worth points, where is the voting method parameter (). For example, the weight for the vertex for candidate is equal to . As the value of the parameter varies, the winner of the election may change.
The procedure line spans the range of possible outcomes as the voting method parameter varies. The background shading shows the range of all possible election outcomes for that value of the voting method parameter.
College football polls such as the AP poll are perhaps the most familiar real-world example of a nonplurality voting system. Approximately 60 members of the media vote for their top 25 teams. The results are tallied using a Borda count: a first-place vote is worth 25 points, a second-place vote is worth 24 points, and so on. However, this point system is fairly arbitrary, and the rankings might be different under another method. We have included data from one week of the AP poll to illustrate this, considering only the voters' relative preferences for the top three teams. While Ohio State is the indisputable first-ranked team, it is not unambiguously determined whether Utah or Baylor should be second.
 D. G. Saari, Basic Geometry of Voting, Berlin: Springer-Verlag, 1995.
 D. G. Saari and F. Valognes, "Geometry, Voting, and Paradoxes," Mathematics Magazine, 71(4), 1998 pp. 243–259. www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1999/0025570x.di021206.02p0091u.pdf.