Lecture on Game Theory

Matching Pennies Game

• Game Setup: Two players each select a penny to show heads or tails.
• Outcomes:
  • If pennies match, Player A wins both.
  • If pennies don’t match, Player B wins both.
• Payoff Matrix: Illustrated through different scenarios of heads and tails.
• Nash Equilibrium: No fixed strategy Nash equilibrium because the loser always regrets their choice.
  • Mixed strategy equilibrium is used, involving random choices.
  • Best strategy: Be unpredictable, e.g., flipping the coin to decide.
• Pareto Optimality: All outcomes are Pareto optimal in a zero-sum game.

Advertising Game

• Scenario: Two companies decide whether to advertise.
• Outcomes:
  • Both don’t advertise: Split market 50/50, profit $50 each.
  • One advertises: Gains more market share, with adjusted profits after ad costs.
  • Both advertise: Cancel each other out, with reduced profits due to ad costs.
• Nash Equilibrium:
  • Dominant strategy: Always advertise.
  • Not Pareto optimal as both could be better off not advertising.
• Classic Prisoner’s Dilemma: Matches characteristics of this game type.

Rebellion and Coordination

• Oppressive Regime Scenario:
  • Successful rebellion vs. failed rebellion and punishment.
• Nash Equilibria: Exist in scenarios where everyone either rebels or does not.
• Pareto Optimality: Successful rebellion is the only Pareto optimal outcome.
• Historical Insight: Matches historical occurrences of rebellion.

Sequential Games

• Game Type: Decisions are made in sequence, unlike simultaneous choices.
• Employer-Employee Scenario:
  • Employer decides on salary; employee decides on effort.
• Nash Equilibrium: Low salary and low effort is the No Regret outcome.
• Pareto Optimality: All other outcomes except equilibrium are Pareto optimal.

Applications of Game Theory

• Biology:
  • Explains traits favored by natural selection.
  • Trees: Tall vs. short strategy as a prisoner’s dilemma.
• Nuclear Strategy:
  • Mutually assured destruction as a deterrent.
• Social Situations:
  • Concerts: Stand vs. sit dilemma similar to prisoner’s dilemma.

Conclusion

• Game theory helps to understand and navigate complex scenarios of competition and strategy in various fields and real-life situations.