ECONOMY

396 a tool kit for voting theory
For any number from {1, 2,...,7}, there exists a profile with precisely that
number of different positional outcomes.
Approval voting. To show how to compute outcomes for other rules, I illustrate
the approach with approval voting (AV). AV, analyzed and promoted by Brams and
Fishburn (1983), is where a voter votes “approval” for as many candidates as desired.
So a voter with A ≻ B ≻ C preferences could vote for Ann, or for Ann and Barbara.
While a vote for all three is admissible, Brams and Fishburn cogently argue that such
a vote is not rational because, by not distinguishing among the candidates, the voter’s
ballot has no effect.
Obviously, if voters vote in different ways, the AV outcome can change. Indeed,
Ann from the Chair election could receive thirteen different AV election tallies ranging
from 6 to 18 votes! A quick way to determine all values a candidate can receive is to
recognize that the range is defined by her plurality (vote for one) and antiplurality
(vote for two) tallies. As there are 13 different AV tallies for Ann, 10 for Barb, and 3 for
Connie, the Chair profile admits 13 × 10 × 3 = 390 different AV election tallies! (It
is easy to create scenarios that support each outcome.)
Plotting 390 points is not appealing, so to identify all AV outcomes plot the eight
extreme tallies constructed from the plurality (6, 8, 9) and antiplurality (18, 17, 11)
by interchanging values from each list. This selection represents where a candidate
receives only first-place votes (plurality tally) or all possible second-place votes (antiplurality
tally). As the eight extreme tallies are:
(6, 8, 9) (18, 8, 9) (6, 17, 9) (6, 8, 11)
(18, 17, 11) (6, 17, 11) (18, 8, 11) (18, 17, 9),
plot (Figure 22.3) the eight normalized tallies:
( 6
23 , 8
23 , )( 9 18
23 35 , 8
35 , )( 9 6
35 32 , 17
( 18
46 , 17
46 , )( 11 6
46 34 , 17
34 , 11
34
32 , 9
32
)( 18
37 , 8
37 , 11
37
(4)
)( 6
25 , 8 25 , )
11
25
)( 18
44 , 17
44 , ) (5)
9
44
Connect all points with straight lines: the enclosed region contains all 390 tallies
that are (essentially) equally spaced in this region. While “only” 390 points in this
Figure 22.3 AV hull are AV election tallies, because they are closely packed into a small
region, it is reasonable to view all points in this distorted rectangle as admissible AV
tallies.
The Figure 22.3 graph identifies several conclusions: a partial sample follows. (For
more AV properties, see Saari 1994, 2001a; Saari and van Newenhizen 1988.)
1. If the reader finds it disturbing that the Chair profile allows seven different
positional outcomes, the reader will be more disturbed by Figure 22.3 showing
that any ranking can be a sincere AV outcome for this profile! This is because
the AV hull meets all six short-line segments (representing a tie between candidates),
the center point (complete tie), and all six open regions (strict election
outcomes). This indeterminate phenomenon is not a peculiarity of the Chair
election example: multiple outcomes are to be expected. Indeed, even an unanimity
profile allows different election rankings.