Game Theory: Prisoner's Dilemma and Nash Equilibrium

Prisoner's Dilemma

• Two players independently choose to cooperate (R1/C1) or defect (R2/C2).
• Payoff matrix example (C1, C2 on columns, R1, R2 on rows):
  • R1: [10,10, 25,5]; R2: [5,25, 20,20]
• R1 strictly dominates R2 for the row player.
• C1 strictly dominates C2 for the column player.
• Nash Equilibrium is when both players defect (R2, C2).

Nash Equilibrium

• Definition: No player can improve their payoff by unilaterally changing their strategy, assuming other players' strategies remain the same.
• Empirical Observations:
  • Men are 24% more likely to defect than women.
  • Defection drops 33% with pre-play discussion.
  • 3rd-year students are less likely to defect than 1st-year students.
  • Economics students show increased tendency to defect.

Weak Dominance

• A strategy weakly dominates another if it yields a payoff at least as good as the other in all scenarios and strictly better in at least one.
• Example matrix (C1, C2 on columns, R1, R2 on rows):
  • R1: [10,10, 5,10]; R2: [10,5, 0,0]

Battle of the Sexes

• Two players choose between two outcomes (e.g., Opera vs. Football).
• Preferences are asymmetrical.
• Payoff matrix example (C1, C2 on columns, R1, R2 on rows):
  • R1: [0,0, 3,1]; R2: [1,3, 0,0]

Mixed Strategies

• Players assign probabilities to their strategies.
• Example:
  • Row player chooses R1 with probability p.
  • Column player chooses C1 with probability q.
• Equilibrium is found by equating expected payoffs.

Environmental Agreements

• Scenario: N countries decide on implementing a costly environmental policy.
• Cost of implementation is £5.
• Benefit for all participants is B/N if exactly m countries implement.
• Analysis: If B/N > cost, no player deviates from implementing the policy.

Market Entry

• Scenario: Total fossil fuel market is £F, total renewable market is £R, total energy companies are E.
• Players decide if they enter the fossil fuel or renewable markets.
• Key Question: How many firms will enter the fossil fuel market?

Beauty Contest Game

• Rules: Participants select a number between 0 and 100.
• Winner is closest to 2/3 of the average.
• Key Insight: Rational players reduce guesses iteratively to reach a Nash equilibrium of 0.

Repeated and Sequential Play

• Repeated Games can show cooperation in prisoner's dilemma versions.
• Sequential Play is where later players observe earlier player actions before deciding.

Risk Homeostasis (Risk Thermostat)

• Definition: People adjust their behavior based on their perceived risk.
• Examples:
  • Safer equipment leads to riskier behavior
  • Drivers with ABS brakes drive faster.

Prospect Theory

• People's decision factors:
  • Low probabilities are over weighted.
  • Middle probabilities are under weighted.
  • Certain outcomes are given more weight (certainty effect).
  • Losses are perceived as more strongly than gains (loss aversion).
  • Gain risk aversion, loss risk-seeking.
• Examples:
  • Avoiding risk when a disease probability doubles (from 1% to 2%)
  • Sacrificing leisure when the chances of exam success increase (from 99% to 100%).

Incrementalism (Muddling Through)

• Definition: Small, incremental changes instead of radical shifts.
• Advantages: Mistakes are easier to correct.

Key Concepts and Definitions: Bounded Rationality

• Decision makers simplify choices due to limitations in time and cognitive capacity
• Examples: Instead of evaluating all laptop brands, focus on a few options and prioritise key attributes, e.g., battery life, weight.

Decoy Options

• Adding a less attractive choice can make another option more compelling.
• Example: A less expensive 7-day Amsterdam holiday being more attractive than a 10-day Paris trip if an expensive 7-day Amsterdam option is also available.

Heuristics (Fast & Frugal)

• Simple rules for quick decision-making in low-stakes decisions
• Recognition Heuristic: Choose the recognisable option
• Minimalist Heuristic: Use a single attribute to decide

Lexicographic Ordering

• Ranking attributes by importance, selecting the best option based on the most important attribute.
• Example: Cost is the most important factor when selecting a holiday, followed by flight time.

Elimination by Aspects (EBA)

• Setting a threshold for the most important attribute and eliminating options that don't meet it.
• Repeating for the next most important attribute
• Example: Choosing a computer with a price cap (£45m), delivery time (<40 months), and required training

Satisficing

• Stopping the search for options once a good enough option is found.
• Example: Choosing a house with a set price range, distance from work, and garden size, stopping once a suitable option is found

Explore the intricacies of game theory with this quiz on the Prisoner's Dilemma and Nash Equilibrium. Understand concepts such as weak dominance and the effects of gender and student status on defecting behavior. Test your knowledge of strategic decision-making in competitive scenarios.