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therefore be selected under some tie-breaking rule. But {b, c} is not a stable AV
outcome: If either the 3 a-voters switch to ab, the 2 bc-voters switch to b, or the 2 dbc-
voters switch to db, candidate b will be selected, whom all three types of voters prefer to
{b, c}.
While Proposition 10 shows that strongly stable, stable, and unstable AV outcomes
may coexist, it is important to know the conditions under which each kind of outcome
can occur.
Proposition 11. A nontied AV outcome is strongly stable if and only if it is a
nontied Condorcet winner.
Proof. To prove the “if” part, suppose candidate i is a unique Condorcet winner at
P. We will show that i is a nontied AV outcome that is strongly stable at its critical
strategy profile, C
i
(P). Clearly, i is a nontied AV outcome at C
i
(P) by Proposition 2. To
show its strong stability, suppose there exists a coalition of voters K, comprising one or
more types, that prefers some other candidate j to candidate i and coordinates to induce
the selection of j. Because candidate i is a unique Condorcet winner, however, the
cardinality of K is strictly less than the cardinality of coalition L, whose members prefer i
to j. The members of L vote for i but not for j at C
i
(P). Hence, whatever sincere,
admissible strategy switch the members of K consider at candidate i’s critical strategy
profile to induce the election of candidate j, j will receive fewer votes than i, proving that
i is a strongly stable AV outcome.
To prove the “only if” part, suppose that candidate i is not a Condorcet winner.
Consequently, there exists a coalition of voters K, comprising one or more types, that
15
As we will show in Proposition 12, unstable AV outcomes do not necessarily include, as here, non-Pareto
Convention