MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Applications 91
(a)
Figure 3.19: Representation Triangle: (a) Ranking Regions. (b) Voting Paradox.
[261]
Application 19 (Representation Triangle (Voting Theory)). According
to D. G. Saari [261], each of three candidates may be assigned to the vertex
of an equilateral “representation triangle” which is then subdivided into six
“ranking regions” by its altitudes (Figure 3.19(a)) [261].
Each voter ranks the three candidates. The number of voters of each type
is then placed into the appropriate ranking region (Figure 3.19(b)). The representation
triangle is a useful device to illustrate voting theory paradoxes and
counterintuitive outcomes. The two regions adjacent to a vertex correspond to
first place votes, the two regions adjacent to these second place votes, and the
two most remote regions third place votes. Geometrically, the left-side of the
triangle is closer to A than to B and so represents voters preferring A to B;
ditto for the five regions likewise defined. For the displayed example, C wins a
plurality vote with 42 first place votes. In so-called Borda voting, where a first
place vote earns two points, a second place vote earns one point and a third
place vote earns no points, B wins with a Borda count of 128. In Condorcet
voting, where the victor wins all pairwise elections, A is the Condorcet winner.
So, who really won the election [261]?
Application 20 (Error-Correcting Code). The projective plane of order
2, which is modeled by an equilateral triangle together with its incircle and
altitudes, is shown in Figure 3.20(a).
There are 7 points (numbered) and 7 lines (one of which is curved); each
line contains three points and each point lies on three lines. This is also known
as the Steiner triple-system of order 7 since each pair of points lies on exactly
one line. The matrix representation is shown in Figure 3.20(b) where the rows
(b)