= Solution
For each history of the first two votes, voter 3 selects its favourite achievable winner. Writing rows for voter 1's vote and columns for voter 2's vote, the resulting continuation winners are
$$
\begin{array}{c|ccc}
&A&B&C\\\hline
A&A&B&C\\
B&B&B&B\\
C&C&B&C
\end{array}.
$$
For example, after votes $B,C$, voter 3 can make either $B$ or $C$ win, and prefers $B$; a vote for $A$ would instead produce the tie-break winner $A$.
Voter 2 prefers $C$ to $B$ to $A$. Optimizing each row therefore gives continuation winners $C,B,C$ after voter 1's respective choices $A,B,C$. Voter 1 prefers $B$ to $C$ and therefore votes $B$, giving \b[winner $B$ in every <subgame perfect equilibrium>].
Some histories admit several equally good votes, but they produce the same winning candidate. Since each voter strictly ranks the three possible winners, <unique outcome under strict outcome preferences> makes this winner independent of those action ties. This is <backward induction> over outcomes, not a claim that the entire equilibrium strategy profile is unique.
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