Nash Equilibrium

Introduction

• Named after John Nash, portrayed by Russell Crowe in "A Beautiful Mind."
• Concept from game theory, which studies strategic decision-making in games.
• Relevant to scenarios like the Prisoner's Dilemma.

Definition of Nash Equilibrium

• A stable state in a system with multiple participants.
• No participant can gain by changing their strategy if others' strategies remain unchanged.
• Originates from Princeton, where John Nash spent a significant part of his career.

Analysis of Different States in the Prisoner's Dilemma

State 1

• Participants: Al and Bill both deny.
• Outcome: Not a Nash equilibrium.
  • Al can gain by changing strategy from deny to confess (improves from two years to one year)
  • Bill can improve by changing from deny to confess (improves from two years to one year)

State 2

• Participants: Al confesses, Bill denies.
• Outcome: Not a Nash equilibrium.
  • Al gains nothing by changing from confess to deny (worsens from one year to two years).
  • Bill can improve by changing from deny to confess (improves from ten years to three years).

State 3

• Participants: Bill confesses, Al denies.
• Outcome: Not a Nash equilibrium.
  • Al can improve by changing from deny to confess (improves from ten years to three years).
  • Bill cannot improve by changing from confess to deny (worsens from three years to ten years), but Al's potential to change invalidates equilibrium.

State 4

• Participants: Both confess.
• Outcome: Nash equilibrium.
  • Al cannot gain by changing from confess to deny (worsens from three years to ten years).
  • Bill cannot gain by changing from confess to deny (worsens from three years to ten years).
  • No participant can gain by unilateral change; stable state.

Conclusion

• Nash equilibrium signifies a state where no player benefits from changing their strategy alone.
• In the Prisoner's Dilemma, only state 4 represents a Nash equilibrium, where both confess.