Lecture Notes: Game Theory and Nash Equilibrium

1. Prisoner's Dilemma Recap

• Main concept: A game where both players have a dominant strategy of not cooperating.
• Result: Both players end up worse off than if they had cooperated.

2. Dominant Strategies

• Definition: A strategy that is the best response no matter what the other player does.
• Prevalence: Rarely exist in many games.

3. Simultaneous Move Game: Brianna and Carlos

• Scenario:
  • Two study partners, Brianna and Carlos.
  • Weekdays: Decide to study or slack off.
  • Weekend: Work together.

3.1 Payoff Matrix

• Both study: Payoff = 4 each.
• Brianna studies & Carlos slacks: Brianna = 3, Carlos = 5.
• Brianna slacks & Carlos studies: Brianna = 5, Carlos = 3.
• Both slack: Payoff = -2 each (fail exam).

3.2 Strategy Analysis

• Brianna's best responses:
  • If Carlos studies: Slack off (Payoff: 5 vs. 4 for studying).
  • If Carlos slacks: Study (Payoff: 3 vs. -2 for slacking).
• Dominant Strategy: None for Brianna; it depends on Carlos.

4. Nash Equilibrium

• Definition: A set of strategies where no player can benefit by changing their strategy alone.
• Brianna and Carlos Equilibria:
  • Brianna studies, Carlos slacks (arrows point here).
  • Carlos studies, Brianna slacks (arrows point here).
• Characteristics:
  • Best response to the other’s best response.
  • More than one Nash equilibrium can exist.

5. Real-World Implications

• Who slacks/studies depends on real-world dynamics (e.g., intimidation).

6. Other Game Scenarios

• Games Without Nash Equilibria:
  • Players always have an incentive to change.
  • Arrows in payoff matrix move players around.
  • More complex to calculate best responses.

References

• John Nash: Nobel laureate, Nash Equilibrium concept.
• "A Beautiful Mind" movie about John Nash.

These notes summarize key concepts from the discussion on game theory, focusing on strategies, Nash equilibria, and decision-making in simultaneous move games.