Short Answer

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.

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 $n = 1, 2, . . ., k$ .

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Step by Step Solution

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TABLE OF CONTENTS

Step 1: 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.

Step 2: Determination of the deferred acceptance algorithm

Let $P (n)$ 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, $P (1)$ holds true.

In the inductive step, let the series $P (1), P (2), . . . ., P (k)$ 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 as Q 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 $P (1), P (2), . . . ., P (k)$ 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 $n = 1, 2, . . ., k$ .

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Discrete Mathematics and its Applications

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Short Answer

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Step 1: Significance of the strong induction

Step 2: Determination of the deferred acceptance algorithm

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Discrete Mathematics and its Applications

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Step 1: Significance of the strong induction

Step 2: Determination of the deferred acceptance algorithm

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