Further Mathematics - Senior Secondary 3 - Game theory

Game theory

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TERM: 2ND TERM

WEEK: 3
Class: Senior Secondary School 3
Age: 17 years
Duration: 40 minutes of 4 periods
Subject: Further Mathematics
Topic: Game Theory
Focus: Introduction to game theory, types of games, solution of two-person, zero-sum games using pure and mixed strategies, matrix games.

SPECIFIC OBJECTIVES:
By the end of the lesson, students should be able to:

  1. Understand the concept of game theory and its applications.
  2. Identify and describe different types of games.
  3. Solve two-person, zero-sum games using pure strategies.
  4. Solve two-person, zero-sum games using mixed strategies.
  5. Represent games in matrix form.

INSTRUCTIONAL TECHNIQUES: • Question and answer
• Guided demonstration
• Discussion
• Practice exercises
• Use of analogies and real-life connections

INSTRUCTIONAL MATERIALS: • Whiteboard and markers
• Chart of various types of games
• Matrix game examples
• Flashcards for game types
• Worksheets for practice problems

 

PERIOD 1 & 2: Introduction to Game Theory and Types of Games

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of game theory as a mathematical framework to analyze competitive situations where the outcome depends on the decisions of others. Defines terms like players, strategies, payoffs, and outcomes.

Students listen and ask questions about game theory.

Step 2 - Explanation of Types of Games

Describes the types of games: Cooperative vs Non-cooperative, Zero-sum vs Non-zero sum, Simultaneous vs Sequential, and Perfect vs Imperfect information games. Provides real-life examples of each type.

Students take notes on the types of games and contribute examples from everyday life.

Step 3 - Discussion

Engages students in a discussion about games they have seen or played and categorizes these games according to the types described.

Students discuss and classify the games they know into the types mentioned.

Step 4 - Analogy

Uses the example of competitive sports (e.g., chess, football) to illustrate zero-sum games, where one player's gain is another's loss.

Students relate the examples to their own experiences in games and competitions.

NOTE ON BOARD:
Game Theory:

  • Players: Individuals or groups involved in the game.
  • Strategies: Plans of action available to players.
  • Payoffs: The outcomes or rewards a player receives based on strategies.
  • Types of Games:
    • Cooperative vs Non-cooperative
    • Zero-sum vs Non-zero-sum
    • Simultaneous vs Sequential
    • Perfect vs Imperfect Information

EVALUATION (5 exercises):

  1. What is game theory?
  2. Define zero-sum games and give an example.
  3. Explain the difference between cooperative and non-cooperative games.
  4. Give an example of a simultaneous game.
  5. What is meant by imperfect information in a game?

CLASSWORK (5 questions):

  1. What type of game is chess?
  2. Is a football match a zero-sum game? Why?
  3. Provide an example of a non-cooperative game.
  4. What are payoffs in a game?
  5. In what type of game does the outcome depend on both players' strategies?

ASSIGNMENT (5 tasks):

  1. Research a popular game and explain its type.
  2. List at least two examples of non-zero-sum games.
  3. Describe a game where perfect information is available to all players.
  4. Write a short paragraph on why game theory is important in economics.
  5. Research a real-life example where game theory is applied (e.g., auctions).

 

PERIOD 3 & 4: Solution of Two-Person, Zero-Sum Games Using Pure and Mixed Strategies

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of pure and mixed strategies. Explains that in a zero-sum game, one player's gain is exactly the other player's loss.

Students listen and ask questions for clarification.

Step 2 - Pure Strategy

Demonstrates how to solve a simple two-person, zero-sum game using pure strategy by using a matrix. Explains the concept of dominant strategies.

Students watch the demonstration and take notes.

Step 3 - Mixed Strategy

Introduces mixed strategy and explains how to find probabilities for players' strategies when no pure strategy is optimal.

Students ask questions and practice using probabilities in mixed strategies.

Step 4 - Matrix Representation

Demonstrates how to represent a game in matrix form. Explains how to find the optimal strategy for each player.

Students observe the steps and work on matrix examples.

NOTE ON BOARD:

  • Pure Strategy: A strategy where a player chooses one action with certainty.
  • Mixed Strategy: A strategy where a player chooses their actions based on probability.
  • Matrix Form: Represents the payoffs of each player depending on the chosen strategies.

NOTE (Workings for Examples):

  • Pure Strategy Example:
    • Player A's strategies: A1, A2
    • Player B's strategies: B1, B2
    • Matrix representation:
      • For each combination of strategies, the corresponding payoff is shown.
  • Mixed Strategy Example:
    • The probability for each strategy is calculated based on the expected payoff.

EVALUATION (5 exercises):

  1. Solve the two-person zero-sum game using pure strategies.
  2. Find the mixed strategy solution for a given game.
  3. What is the dominant strategy?
  4. Write the payoff matrix for a simple game with two players.
  5. If a player uses a mixed strategy, what does this mean for their decision-making process?

CLASSWORK (5 questions):

  1. Solve the game using pure strategy where Player A has two strategies and Player B has two.
  2. What are the steps to finding a mixed strategy?
  3. Provide a real-life example of a mixed strategy.
  4. How can a payoff matrix help in analyzing a game?
  5. What is the best response in a zero-sum game?

ASSIGNMENT (5 tasks):

  1. Solve a matrix game with more than two strategies for each player.
  2. Find the mixed strategy for a two-person, zero-sum game.
  3. Research a game where mixed strategies are commonly used.
  4. Provide an example of a game with a dominant strategy.
  5. Write an analysis of a competitive situation where game theory applies.