9
strategy profile of the 7 voters is S = (a, a, a, bc, bc, dbc, dbc), whereby the 3 voters of
type (1) approve of only their top candidate, the 2 voters of type (2) approve of their top
two candidates, and the 2 voters of type (3) approve of all candidates except their lowest-
ranked. The number of votes of each candidate at S is 4 votes for b and c, 3 votes for a,
and 2 votes for d. Hence, AV selects candidates {b, c} as the (tied) winners at S.
3. Election Outcomes under AV and Other Voting Systems
Given a preference profile P, we consider the set of all candidates that can be
chosen by AV when voters use sincere, admissible strategies. We call this set AV
outcomes. Clearly, a candidate ranked last by all voters cannot be in this set, because it is
inadmissible for any voter to vote for this candidate.
Define an AV critical strategy profile for candidate i at preference profile P as
follows: Every voter who ranks i as his or her worst candidate votes only for the
candidate that he or she ranks top. The remaining voters vote for i and all candidates
they prefer to i.
Let C
i
(P) be the AV critical strategy profile of candidate i. In Example 1, the
critical strategy profile for candidate a is C
a
(P) = (a, a, a, bca, bca, d, d), giving a 5 votes
compared to 2 votes each for b, c, and d. It can easily be seen that C
i
(P) is admissible and
sincere.
We next give four lemmata that provide a theoretical foundation for several of our
subsequent propositions. They (i) show that under AV candidate i cannot do better than
at C
i
(P); (ii) characterize AV outcomes; (iii) characterize outcomes that can never be
chosen under AV; and (iv) characterize outcomes that must be chosen under AV.
Convention