ECONOMY

392 a tool kit for voting theory
Figure 22.1a represent:
Label Ranking Label Ranking Label Ranking
1 A ≻ B ≻ C 2 A ≻ C ≻ B 3 C ≻ A ≻ B
4 C ≻ B ≻ A 5 B ≻ C ≻ A 6 B ≻ A ≻ C
To represent a profile for {Ann, Barb, Connie}:
Number Ranking Number Ranking Number Ranking
6 A ≻ B ≻ C 0 A ≻ C ≻ B 6 C ≻ A ≻ B
3 C ≻ B ≻ A 2 B ≻ C ≻ A 6 B ≻ A ≻ C
to elect a departmental chair, place the number of voters with each ranking in the
appropriate ranking region: this equation 2 “Chair”profileisinFigure22.1b.
As described in Saari (2001), the geometric representation sorts profile entries in
a manner that simplifies tallying elections; e.g. as all voters preferring A ≻ B are to
theleftoftheverticalline,theirsum6+0+6=12isA’s tally in the {A, B} election.
This number, with B’s 3 + 2 + 6 = 11 tally from the right of the line, is listed under the
edge. All pairwise outcomes are similarly computed with the outcomes listed near the
appropriate triangle edge. According to these votes, Ann is the Condorcet winner—she
beats all others in pairwise majority votes.
Our standard voting system, where a voter votes for his favorite candidate and the
candidate with the most votes wins, is called the plurality vote. But rather than voting
just for one candidate, we might give 9, 7, 0 points, respectively, to a voter’s top-,
second-, and third-ranked candidate. This is called a “positional election:” it is where
the weights w 1 ≥ w 2 ≥ w 3 = 0 are assigned, respectively, to a voter’s top, second, and
bottom “positioned” (ranked) candidates and the societal ranking is determined by
the sum of points each candidate receives. Without loss of generality, scale all weights
by dividing by w 1 to obtain (1, s, 0). So with the earlier 9, 7, 0 points assignment,
the rescaling assigns 1, s = 7 , 0 points to a voter’s top-, second-, and third-ranked
9
candidate. To use weights that play a major role in what follows, the Borda Count is
where 2, 1, 0 points are assigned to the candidates, so its scaled weights are (1, 1 2 , 0).
The antiplurality vote, 1, 1, 0 (called this because by voting for two, you are effectively
voting against someone), already is in scaled form.
To compute tallies, notice that Ann’s outcome is [number of voters with A
top-ranked]+ s [number of voters with A second ranked]. In Figure 22.1b, thefirst
bracket’s value is A’s plurality outcome, or the sum of numbers in regions with A as a
vertex: 6 + 0 = 6. The second bracket—the number of voters with A second ranked—
involves the two triangles one step removed from where A is top ranked—the adjacent
regions that both contain 6. ThusA’s positional tally is [6 + 0] + s [6 + 6] = 6 + 12s .
All positional tallies are similarly computed and listed by the appropriate Figure 22.1b
vertex. Readers unfamiliar with this process should experiment by computing election
outcomes for other profiles.
According to Figure 22.1b the plurality (s =0)rankingisC ≻ B ≻ A with a 9 : 8 : 6
tally, so Connie, rather than Condorcet winner Ann, wins. But as the antiplurality system
(s =1)hastheoppositeoutcomeofA ≻ B ≻ C with a 18 : 17 : 11 tally, Condorcet
(1)
(2)