KP

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.