19
side for all other candidates j under AV is less than or equal to the sum on the left side,
with equality if and only if candidate j is preferred to candidate i by all voters.
Consequently, if inequality (1) is satisfied under the scoring rule, it is satisfied under AV
at C
i
(P).
Thus, in both cases (i) and (ii), the satisfaction of inequality (1) under a scoring
rule implies its satisfaction under AV at candidate i’s critical strategy profile, C
i
(P).
Hence, a candidate chosen under any scoring rule is also an AV outcome.
To prove the second statement, consider the following 7-voter, 3-candidate
example (Fishburn and Brams, 1983, p. 211):
Example 4
1. 3 voters: a b c
2. 2 voters: b c a
3. 1 voter: b a c
4. 1 voter: c a b
Because candidate b receives at least as many first choices as a and c, and more first and
second choices than either, every scoring rule will select b as the winner. But a is the
Note that candidate b in Example 4 is not AV-dominant: The number of voters
who consider b as their best choice and a as their worst choice (2 voters), or b as their
best choice and c as their worst choice (1 voter), does not exceed the number of voters
who prefer a to b (4 voters) or c to b (1 voter) (see Lemma 4). Put another way, the
8
Example 4 provides an illustration in which BC, in particular, fails to elect the Condorcet winner.
Convention