ECONOMY

donald g. saari 403
Next, add “configurations of preferences” to alter the positional outcomes but
not the Borda and pairwise outcomes. According to the above, we must add NRR
reversal terms. To make C the plurality winner, add (see Figure 22.4b) y units of
(A ≻ B ≻ C, C ≻ B ≻ A) andz units of (C ≻ A ≻ B, B ≻ A ≻ C). According to
Figure 22.4b, the plurality (s = 0) tallies for A: B: C are 1 + y :2+z : y + z. Sothe
C ≻ B ≻ A plurality ranking occurs with y and z values satisfying:
y + z > 2+z > 1+y.
One solution has y = z = 3; along with the above x = 2, they create the Chair profile.
Only the Borda Count satisfies both NRR and NCR, so only this rule is immune
to these perturbations: the Borda ranking remains B ≻ A ≻ C. Notice that we
could have selected any ranking for the plurality method and created a supporting
profile.
Some consequences. Several consequences follow from this construction including
the historical debate whether the Condorcet or Borda winner better reflects the voters’
beliefs. As indicated above, all possible differences in the Borda and Condorcet winners
are caused by Condorcet terms! (This statement holds for any number of candidates.)
Consequently, this two-century debate reduces to determining the appropriate outcome
for the Condorcet triplet. Namely, if you support the Condorcet winner, then
you must explain why the Condorcet triplet outcome should not be a tie. As I
have never heard a convincing argument, I must support the Borda winner. This
comparison of Borda and Condorcet winners is illustrated with the construction of
the Chair election example where the bias introduced by Condorcet triplet changes
the Condorcet winner from B to A.
As another example, there are many papers that analyze whether a procedure must
elect the Condorcet winner. To understand the results, notice from NRR that any non-
Borda positional procedure, or rules using these positional procedures, cannot satisfy
this condition because their outcomes are changed by adding profile portions with
no effect on pairwise outcomes. (This comment is illustrated by the construction of
the Chair example.) Because the Condorcet winner can be altered by using Condorcet
triplets (or Condorcet n-tuples for n ≥ 3 candidates), only those procedures based on
pairwise outcomes can possibly satisfy this condition.
6 Strategic and Other Behavior
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The important Gibbard–Satterthwaite theorem (Gibbard 1973; Satterthwaite 1975)
shows that with three or more candidates, all “reasonable” election methods (e.g.
where each candidate can be elected and we exclude dictatorial rule) allow settings
where it is to the advantage of some voter to vote “strategically” rather than sincerely.
By doing so, the voter ensures a personally better election outcome. But we know
this; for instance, had more Nader voters voted strategically for Gore in Florida,