Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

Answers without the blur.

Just sign up for free and you're in.

Short Answer

Suppose we have three men $m_{1}, m_{2}, and m_{3}$ and three women $w_{1}, w_{2}, and w_{3}$ . Furthermore, suppose that the preference rankings of the men for the three women, from highest to lowest, are $m_{1} : w_{3}, w_{1}, w_{2}; m_{2} : w_{1}, w_{2}, w_{3}; m_{3} : w_{2}, w_{3}, w_{1}$ and the preference rankings of the women for the three men, from highest to lowest, are $w_{1} : m_{1}, m_{1}, m_{3}; w_{2} : m_{2}, m_{1}, m_{3}; w_{3} : m_{3}, m_{2}, m_{1}$ . For each of the six possible matchings of men and women to form three couples, determine whether this matching is stable.

First matching: $m_{1} - w_{1}, m_{2} - w_{2}, m_{3} - w_{3}$ is stable.

Second matching: $m_{1} - w_{1}, m_{2} - w_{3}, m_{3} - w_{2}$ is not stable.

Third matching: $m_{1} - w_{2}, m_{2} - w_{1}, m_{3} - w_{3}$ is not stable.

Fourth matching: $m_{1} - w_{2}, m_{2} - w_{3}, m_{3} - w_{1}$ is not stable.

Fifth matching: $m_{1} - w_{3}, m_{2} - w_{1}, m_{3} - w_{2}$ is stable.

Sixth matching: $m_{1} - w_{3}, m_{2} - w_{2}, m_{3} - w_{1}$ is not stable.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1:

We have given three men: $m_{1}, m_{2}, m_{3}$ and three women $w_{1}, w_{2}, w_{3}$ .

For stable matching, we have:

$\begin{aligned} m_{1} : w_{3} > w_{2} > w_{1} \\ m_{2} : w_{1} > w_{2} > w_{3} \\ m_{3} : w_{2} > w_{3} > w_{1} \end{aligned}$

And

$\begin{aligned} w_{1} : m_{1} > m_{2} > m_{3} \\ w_{2} : m_{2} > m_{1} > m_{3} \\ w_{3} : m_{3} > m_{2} > m_{1} \end{aligned}$

Step 2:

Now, we will have six possible matching; we will check whether they are stable or unstable.

First matching: $m_{1} - w_{1}, m_{2} - w_{2}, m_{3} - w_{3}$ .

This is stable because all the women assigned their preferred partners.

Second matching: $m_{1} - w_{1}, m_{2} - w_{3}, m_{3} - w_{2}$ .

This is not stable because $m_{2}$ prefers $w_{2}$ over his current partner and $w_{2}$ prefers $m_{2}$ over her current partner.

Third matching: $m_{1} - w_{2}, m_{2} - w_{1}, m_{3} - w_{3}$ .

This is not stable because $m_{1}$ prefers $w_{1}$ over his current partner and $w_{1}$ prefers $m_{1}$ over her current partner.

Step 3:

Fourth matching: $m_{1} - w_{2}, m_{2} - w_{3}, m_{3} - w_{1}$ .

This is not stable because $m_{1}$ prefers $w_{1}$ over his current partner and $w_{1}$ prefers $m_{1}$ over her current partner.

Fifth matching: $m_{1} - w_{3}, m_{2} - w_{1}, m_{3} - w_{2}$ .

This is stable because all the men assigned their preferred partners.

Sixth matching: $m_{1} - w_{3}, m_{2} - w_{2}, m_{3} - w_{1}$ .

This is not stable because $m_{3}$ prefers $w_{3}$ over his current partner and $w_{3}$ prefers $m_{3}$ over her current partner.

Step 4:

Hence, we have

First matching: $m_{1} - w_{1}, m_{2} - w_{2}, m_{3} - w_{3}$ is stable.

Second matching: $m_{1} - w_{1}, m_{2} - w_{3}, m_{3} - w_{2}$ is not stable.

Third matching: $m_{1} - w_{2}, m_{2} - w_{1}, m_{3} - w_{3}$ is not stable.

Fourth matching: $m_{1} - w_{2}, m_{2} - w_{3}, m_{3} - w_{1}$ is not stable.

Fifth matching: $m_{1} - w_{3}, m_{2} - w_{1}, m_{3} - w_{2}$ is stable.

Sixth matching: $m_{1} - w_{3}, m_{2} - w_{2}, m_{3} - w_{1}$ is not stable.

Find Study Materials for

Create Study Materials

Select your language

Discrete Mathematics and its Applications

Answers without the blur.

Short Answer

Step by Step Solution

Step 1:

Step 2:

Step 3:

Step 4:

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Discrete Mathematics and its Applications

Answers without the blur.

Short Answer

Step by Step Solution

Step 1:

Step 2:

Step 3:

Step 4:

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths