The Most Famous Problem in Game Theory: The Prisoner's Dilemma

Introduction

• The Prisoner's Dilemma is a fundamental problem in game theory.
• It appears in various situations, from international conflicts to everyday chores.
• The game illustrates the challenges of cooperation and competition.

Historical Context

• On September 3, 1949, the US discovered that the Soviet Union had developed nuclear weapons.
• The US considered a preemptive strike against the Soviets.
• John von Neumann, founder of game theory, suggested immediate action.
• The RAND Corporation studied the implications using game theory.
• Two mathematicians at RAND invented the Prisoner's Dilemma in 1950.

The Prisoner's Dilemma Explained

• Two players can either cooperate or defect.
• Payoff matrix:
  • Both cooperate: 3 coins each.
  • One defects, the other cooperates: Defector gets 5 coins, cooperator gets none.
  • Both defect: 1 coin each.
• Rational choice leads both players to defect, resulting in a suboptimal outcome.
• Applied to US-Soviet relations: Both developed nuclear arsenals, wasting resources.

Repeated Prisoner's Dilemma

• Impalas grooming each other illustrate repeated interactions in nature.
• In repeated games, cooperation can emerge as a strategy.
• Robert Axelrod's 1980 computer tournament tested game strategies.

Axelrod's Tournament

• 14 strategies plus one random strategy competed in 200 rounds.
• Strategies included: Friedman (always defect after one defection), Joss (cooperate with random defections), Grass Camp (probe strategy), and Name Withheld (most elaborate).
• Tit for Tat, the simplest strategy, won the tournament.
• Tit for Tat: Cooperates first, then mirrors opponent's previous move.

Key Findings from Axelrod's Tournament

• Successful strategies shared four qualities:
  1. Nice: Do not defect first.
  2. Forgiving: Do not hold grudges.
  3. Retaliatory: Strike back immediately after a defection.
  4. Clear: Easy to understand and predict.
• Nice and forgiving strategies outperformed nasty ones.
• Tit for Tat's effectiveness came from balanced retaliation and forgiveness.

Second Tournament

• Axelrod held a second tournament with 62 entries.
• Strategies were adjusted based on first tournament results.
• Tit for Tat again performed well, highlighting the importance of being nice and forgiving.

Evolutionary Insights

• Axelrod's simulation showed that nice strategies can eventually dominate a population.
• Cooperation can emerge from self-interested behavior in repeated interactions.
• Real-world applications include international relations and evolutionary biology.

The Role of Noise

• Noise (random errors) affects strategy performance.
• Tit for Tat can fail in noisy environments due to retaliation loops.
• A slightly forgiving version of Tit for Tat performs better in noisy conditions.

Conclusion

• Axelrod's principles: Be nice, forgiving, retaliatory, and clear.
• Cooperation pays off in repeated interactions and can be a stable strategy in a population.
• Real-life implications for conflict resolution and understanding cooperation in nature.

Additional Resources

• The video sponsor, Brilliant, offers courses to improve problem-solving skills, including game theory and probability.