Module 12.2 Nash Equilibrium Lecture

Overview of Game Theory Concepts

• Dominant Strategy: A strategy that is the best response regardless of what the other player does.

  • Example: In the prisoner's dilemma, both players have a dominant strategy of not cooperating.

Brianna and Carlos Study Game

• Players: Brianna and Carlos.

• Strategies: Each player can either study or slack off during the week, then work together on weekends.

Payoff Matrix

• If both study: Payoff = 4 (Brianna), 4 (Carlos)

• If Brianna studies, Carlos slacks off: Payoff = 3 (Brianna), 5 (Carlos)

  • Carlos benefits from Brianna’s efforts, having slacked off during the week.

• If Brianna slacks off, Carlos studies: Payoff = 5 (Brianna), 3 (Carlos)

  • Same reasoning as above, benefits from Brianna’s knowledge.

• If both slack off: Payoff = -2 (Brianna), -2 (Carlos)

  • They will fail the exam, leading to a negative payoff.

Evaluating Best Responses

• Brianna assesses her best responses:

  • If Carlos studies: Payoff = 5 (slacking off) vs. 4 (studying)

    • Best response: Slack off

  • If Carlos slacks off: Payoff = 3 (studying) vs. -2 (slacking off)

    • Best response: Study

• Conclusion on Dominant Strategy:

  • Brianna does not have a dominant strategy; her best response depends on Carlos's action.

Nash Equilibrium

• Definition: A set of strategies where no player has an incentive to change their choice, being the best response to each other's choices.

  • In this game:

  • If Brianna studies and Carlos slacks off: neither player wants to change their action.

  • Conversely, Carlos will choose to slack off if Brianna studies.

  • Both have incentives that align with each other’s strategies.

• Potential Outcomes:

  • Two Nash equilibria are possible:

  • Brianna studies, Carlos slacks off.

  • Brianna slacks off, Carlos studies.

Real-World Implications

• Determination of who studies and who slacks off may be influenced by contextual factors beyond the payoff matrix.

• Examples include intimidation, motivation, or social dynamics in varying real-world scenarios.

More Complex Games

• Some games may not have Nash equilibria, leading players to constantly change their strategies in response to one another.

• More advanced game theory courses can delve deeper into strategies where equilibrium does not exist, requiring more complex calculations.