L18 game theory

Microeconomics Lecture Overview

• Course: ECA002

• Topic: Game Theory

• Lecturer: Luke Garrod

Aims of the Lecture

• Understand Game Theory which:

  • Highlights strategies for maximizing gains and minimizing losses within constraints.

  • Helps predict behavior of economic agents in various situations.

  • Has applications in politics and psychology, making it a transferable skill for everyday life.

  • Focus on representing strategic interactions as games and generating predictions about behavior.

Lecture Outline

1. Introduction to Game Theory

2. Simultaneous-Move Games and Normal Form (Strategic Form)

3. Dominant Strategy Equilibria

4. Nash Equilibria

5. Sequential-Move Games and Extensive Form

6. Subgame Perfect Nash Equilibria

• Reading Materials:

  • Core: Lipsey & Chrystal, ch. 8, 185-190

  • Extra: Perloff, Ch. 14

Introduction to Game Theory

• Game Theory models strategic interaction:

  • GAME: All interactions considered as 'games'.

  • PLAYERS: Individuals making decisions.

  • STRATEGIES: Choices players can make.

  • INFORMATION: What players know when making decisions.

  • TIMING: When players can make their decisions.

  • PAYOFFS: Outcomes determined by players' strategies.

• Examples:

  • Game: Rock-Paper-Scissors

    • Players: Children

    • Strategies: Paper, Scissors, Rock

    • Payoffs: Bragging rights

  • Game: Oligopoly

    • Players: Sellers

    • Strategies: Pricing decisions

    • Payoffs: Profits

Representation of Games

• Need a convenient representation including:

  • Players, Strategies, Information, Timing, Payoffs

• Approaches:

  • Extensive Form

  • Normal Form

• Example Game 1:

  • Players: A (Up, Down), B (Left, Right)

  • Payoff matrix:

    • (U, L): 10, 10

    • (U, R): 8, 0

    • (D, L): 5, 6

    • (D, R): 5, 5

Simultaneous-Move Games and Normal Form

• Represented via a Matrix.

• Example Payoffs:

  • (U, L): 10, 10

  • (U, R): 8, 0

  • (D, L): 5, 6

  • (D, R): 5, 5

• Matrix Representation:

  • Player A's payoff written first, followed by Player B's:

  	LEFT	RIGHT
  UP	10,10	8,0
  DOWN	5,6	5,5

Dominant Strategy Equilibria

• Dominant strategy: provides highest payoff regardless of opponent's strategy.

• Example Analysis:

  • Player A:

    • Compare payoffs against Player B's strategies.

    • Best response determination:

      • If Player B plays LEFT, Player A's best is UP.

      • If Player B plays RIGHT, Player A's best is DOWN.

  • Resulting Dominant Strategy: LEFT for Player A.

Dominant Strategy Equilibria for Player B

• Assessing Player B's strategy:

  • If Player A plays UP, Player B’s best response.

  • If Player A plays DOWN, Player B’s best response.

  • Dominant Strategy Equilibrium: Signify equilibrium with a star.

Nash Equilibria

• Nash Equilibrium Concept (introduced by John Nash in 1950):

  • Occurs when no player can benefit from changing strategy given other players' strategies.

• Requirements for Nash equilibrium:

  1. Each player must play a best response to others.

  2. Beliefs about other players' strategies must align.

• Multiple or No Nash equilibria possible.

Example of Nash Equilibrium

• Game 2 Example:

  • Payoff Matrix:

    • (U, L): 3, 10

    • (U, R): 4, 0

    • (D, L): 4, 4

    • (D, R): 5, 5

Sequential-Move Games and Extensive Form

• Example of sequential moves: Player A moves first, then Player B.

• Payoffs:

  • (U,L): 10, 10

  • (U,R): 16, 12

  • (D,L): 12, 16

  • (D,R): 14, 14

• Predicting outcomes with consideration of sequential nature.

The Extensive Form Representation

• Represent games using tree diagrams to reflect decision-making.

• Player's moves and corresponding payoffs illustrated clearly.

Subgame Perfect Nash Equilibria

• Refines Nash equilibrium for better predictions in sequential games.

• Requires a Nash equilibrium in every subgame.

• Solved using backwards induction:

  • Start at end and solve for best responses; move backwards.

Backwards Induction Method

• Analyze Player B's response after Player A's action, then identify Player A's best response.

• Outcome: Unique subgame perfect Nash equilibrium detected.

Summary of Game Theory Concepts

• Game Theory: Represents strategic interactions among players.

• Key Elements:

  • Players: Decide strategies impacting payoffs.

  • Game Rules: Specify timing and sequence of moves.

  • Dominant Strategy: Best choice regardless of others.

  • Nash Equilibrium: Best strategy considering other players' behaviors.

  • Subgame Perfection: Nash equilibrium applies in every subgame.

Expected Outcomes Post-Lecture

• Ability to represent strategic interactions as games in both extensive and normal forms.

• Understand differences between dominant strategy equilibria, Nash equilibria, and subgame perfect Nash equilibria.

• Derive equilibria for both sequential-move and simultaneous-move games.