Mathematics 209: Finite Mathematics

Study Guide

Unit 8: Games and Decisions

We finish this course with one more application of matrices, linear systems, and linear programming. Game theory is a relatively new field in mathematics, with applications in many disciplines, among them economics, psychology and political science.

Objectives

When you have completed this unit, you should be able to

  1. find saddle values and optimal strategies.
  2. distinguish between strictly and nonstrictly determined games.
  3. solve a 2 × 2 nonstrictly determined matrix game.
  4. find optimal strategies for each player of a matrix game.
  5. calculate the value of a matrix game.
  6. use linear programming to analyse matrix games.

Strictly Determined Games

Indications

  1. Read Section 10-1, Strictly Determined Games, on pages 396–400 of the textbook.
  2. Do odd-numbered exercises 1–29 on page 401. If you have difficulty, consult your tutor to discuss the problem.
  3. Solve problems 41 and 43 on pages 401 and 402. If you have difficulty, consult your tutor to discuss the problem.
  4. Answer the questions listed below, and then compare your answers with those given in the Answers to Study Guide Questions.

Questions

  1. What is a strictly determined matrix game?
  2. What is a nonstrictly determined matrix game?
  3. Why are there sixteen 2 × 2 matrices with entries that are either 1 or - 1?
  4. When is a game fair?

Mixed Strategy Games

Indications

  1. Read Section 10-2, Mixed Strategy Games, on pages 402–411 of the textbook.
  2. Do odd-numbered exercises 1–39 on pages 411–412. If you have difficulty, consult your tutor to discuss the problem.
  3. Solve problems 41 and 43 on pages 412–413. If you have difficulty, consult your tutor to discuss the problem.

Linear Programming and 2 × 2 Games: Geometric Approach

Indications

  1. Read Section 10-3, Linear Programming and 2 × 2 Games: Geometric Approach, on pages 413–419 of the textbook.

    2. Note: The maximization problems on pages 417 and 418 of the textbook contain typos. See the Textbook Corrections document for the correct wording.

  2. Do odd-numbered exercises 1–21 on page 419. If you have difficulty, consult your tutor to discuss the problem.
  3. Solve problems 25 and 27 on page 419. If you have difficulty, consult your tutor to discuss the problem.

Linear Programming and m × n Games: Simplex Method and the Dual Problem

Indications

  1. Read Section 10-4, Linear Programming and m × n Games: Simplex Method and the Dual Problem, on pages 420–424 of the textbook.
  2. Do odd-numbered exercises 1–9 on page 424. If you have difficulty, consult your tutor to discuss the problem.
  3. Solve problem 11 on page 425. If you have difficulty, consult your tutor to discuss the problem.

Finishing This Unit

  1. Review the objectives of this unit and make sure you are able to meet each of them.
  2. Study the section of the Chapter 10 Review titled Important Terms, Symbols and Concepts, on pages 425–426 of the textbook.
  3. If there is s concept, definition, example or exercise that is not yet clear to you, go back and re-read it. Contact your tutor if you need help.
  4. You might want to do odd-numbered exercises 1–31 from the Review Exercise section on pages 426–428. The questions on the practice examination are taken from this exercise. If you have difficulty, consult your tutor to discuss the problem.
  5. Complete the practice examination provided for this unit. Evaluate yourself, first checking your answers against those provided in the Answers section at the end of the textbook, and then comparing your solutions with those provided on pages S-442 to S-457 of the Student Solutions Manual portion of the textbook. The number of points in a question may indicate the number of steps in the solution. Give yourself full credit if your answer is correct and you give a complete solution, even if your solution differs from that shown in the Student Solutions Manual.
  6. Once you have written and evaluated the practice examination, complete and submit the fourth assignment. You will find instructions in the Assignment Drop Box on the course home page. Also, please leave enough time to enable your tutor to return your marked assignment, with feedback, before you write your examination.

Practice Examination

Time: 2 hours
Total points: 30
Passing grade: 50%

Do the following exercises from the Chapter 10 Review Exercise on pages 426–428 of the textbook.

To obtain full credit you must justify all your answers and show your work.

  1. Exercise 4  (Marks: 5 pts.)
  2. Exercise 12  (Marks: 5 pts.)
  3. Exercise 20  (Marks: 6 pts.)
  4. Exercise 26  (Marks: 6 pts.)
  5. Exercise 30  (Marks: 8 pts.)

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