Question

A stable assignment, defined in the preamble to Exercise 60 in Section 3.1, is called optimal for suitors if no stable assignment exists in which a suitor is paired with a suitee whom this suitor prefers to the person to whom this suitor is paired in this stable assignment. Use strong induction to show that the deferred acceptance algorithm produces a stable assignment that is optimal for suitors.

Short answer

The deferred acceptance algorithm produces a stable assignment that is optimal for suitors. The statement holds true for$n=1$ number of suitors and it also holds true for number of suitors. The statement also holds true$n=k+1$ for suitors of $n=1,2,...,k$.

Step-by-step solution

Significance of the strong induction

The strong induction is mainly described as a proof which is related to the simple induction. The strong induction is mainly used for proving a particular theorem.

Determination of the deferred acceptance algorithm

Let $P\left(n\right)$is described as a statement which states that “deferred acceptance algorithm produces a stable assignment that is optimal for suitors, when there are number of suitees and number of suitors”.

In the basic step where $n=1$, as there are only suitee and suitors, the suitor has been assigned by the stable assignment to the particular suitee and the particular assignment is also optimal for a particular suitor. Hence, $P\left(1\right)$holds true.

In the inductive step, let the series $P\left(1\right),P\left(2\right),....,P\left(k\right)$holds true, then it is to be considered that $P(k+1)$is also true. Let, the number of suitees and suitors are $k+1$respectively. If the $k+1$suitors are being divided into two different groups such asQ and p, then the algorithm of deferred acceptance mainly produces a stable assignment which is mainly optimal for the and also suitors which states that$P\left(1\right),P\left(2\right),....,P\left(k\right)$ holds true. Hence,$P(k+1)$ holds true.

Thus, the deferred acceptance algorithm produces a stable assignment that is optimal for suitors. The statement holds true for$n=1$ number of suitors and it also holds true for$n=k+1$ number of suitors. The statement also holds true for suitors of $n=1,2,...,k$.