ECONOMY

402 a tool kit for voting theory
C
y + z
C
2x
x 0
x
z
y
3 + x
0
x
3 + 2x
0 0
1 + x 2
1 + y 2 + z
A
B
A
1 + 2x 2 + x
1 + y + (2 + 2z)s
B
2 + z + (1+ 2y)s
(a) NCR and pairwise votes
(b) NRR and positional outcomes
Fig. 22.4 Constructing examples
for this bias is that by myopically concentrating on how a pair fares in each ranking,
the pairwise vote misses the broader symmetry supporting a complete tie. Surprisingly
(Saari 1999, 2001a), all possible differences between the Borda Count and pairwise
outcomes, and all possible differences among methods based on pairwise outcomes, are
caused by profile components given by the Condorcet triplets.
Constructing examples. Surprisingly, the two-century-old topic of comparing voting
procedures reduces to the simple observation whether a profile has components
in the NRR and NCR directions. Readers interested in exploring this approach
for three candidates should check Saari (1999, 2001a); see Saari (2000) for n ≥ 3
candidates. To illustrate, I show how to construct examples that do anything that
can arise.
Start with the profile where one voter prefers A ≻ B ≻ C and two prefer
B ≻ A ≻ C. This profile has the positional outcomes A, B, C of, respectively,
1+2s, 2+s, 0. Thus for s < 1, the outcome is B ≻ A ≻ C. The pairwise votes are
compatible with B ≻ A, B ≻ C, A ≻ C outcomes that crown Barb as the Condorcet
winner. With this profile, most standard methods agree giving the same B ≻ A ≻ C
outcome.
To distort this profile to create the Chair profile, first change the Condorcet winner
from B to A without affecting any positional ranking. According to NCR, this requires
adding x units of the Condorcet triplet, A ≻ B ≻ C, B ≻ C ≻ A, C ≻ A ≻ B,
that favors A over B. Adding this configuration, depicted by the x values in Figure
22.4a, hasnoeffect on the positional rankings (each candidate receives the same
x + xs bonus points). Figure 22.4a shows that the pairwise tallies are:
A : B with 1 + 2x :2+x, A : C with 3 + x :2x, B : C with 3 + 2x : x.
Thus A is the Condorcet winner with an x value satisfying
1+2x > 2+x, 3+x > 2x,
or x = 2. These new terms affect the pairwise outcomes but not the positional or
Borda outcomes.