= Solution
\b[The assignment is <Pareto optimal>.] Write $a_i$ for the room selected by the $i$th student. Suppose another assignment $a'$ makes every student at least as well off. The first student already selected their uniquely most valuable room, so $a'_1=a_1$. Once the rooms of students $1,\ldots,i-1$ are fixed, the rooms available to student $i$ in the original <serial dictatorship> are exactly the rooms that remain after those fixed assignments. Student $i$ chose the unique maximum among them, so weak improvement requires $a'_i=a_i$.
<Mathematical induction> fixes every student's room. Thus $a'=a$ and nobody can be strictly better off, contradicting the definition of a <Pareto improvement>. Strict room rankings and the absence of <allocation externalities> are the essential assumptions: a student's <utility> is determined solely by their own room.<One-sided matchings: house allocation problems in the Cornell networks and markets text>[§13.2, Definition 13.1 and Theorem 13.2: Pareto efficiency for strict preferences]{ref}
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