Lecture Notes: Game Theory PDF

**Lecture Notes: Game Theory** **Key Concepts and Definitions** **1. Prisoner's Dilemma** - **Setup:** - Two players decide independently whether to cooperate (R1/C1) or defect (R2/C2). - Payoff matrix example: **C1** **C2** -------- -------- -------- **R...

**Lecture Notes: Game Theory** **Key Concepts and Definitions** **1. Prisoner's Dilemma** - **Setup:** - Two players decide independently whether to cooperate (R1/C1) or defect (R2/C2). - Payoff matrix example: **C1** **C2** -------- -------- -------- **R1** 10,10 25,5 **R2** 5,25 20,20 - **Dominance:** - R1 strictly dominates R2 for the row player. - C1 strictly dominates C2 for the column player. - **Nash Equilibrium:** Both players defect (R2, C2), as this strategy maximises each player's outcome given the strategy of the other. **2. Nash Equilibrium** - **Definition:** A situation where no player can improve their payoff by unilaterally changing their strategy, assuming other players' strategies remain unchanged. - **Empirical Observations:** - Men are 24% more likely to defect than women. - Defection likelihood drops 33% with pre-play discussion. - 3rd-year students are 13% less likely to defect than 1st-years. - Economics students show a higher tendency to defect. **3. Weak Dominance** - Example matrix: C1 C2 -------- ------- ------ **R1** 10,10 5,10 **R2** 10,5 0,0 - 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. **4. Battle of the Sexes** - **Setup:** Two players choose between two outcomes (e.g., Opera vs Football). Preferences are asymmetrical. - Payoff matrix example: C1 C2 -------- ----- ----- **R1** 0,0 3,1 **R2** 1,3 0,0 - **Findings:** There are multiple pure strategy Nash equilibria (R1, C2 or R2, C1). - **Mixed Strategy:** Players randomise their choices to reach equilibrium (e.g., probabilities p and q for choosing R1/C1). **5. 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. **Applications** **1. Environmental Agreement** - **Scenario:** - N countries decide whether to implement a costly environmental policy. - Cost of implementation = £5. - If exactly m countries implement, benefit = B/N for all participants. - **Analysis:** - If B/N \> c, no player wants to deviate from the implementation strategy. **2. Market Entry** - **Scenario:** - Total fossil fuel market = £F. - Total renewable market = £R. - Total energy companies = E. - Players decide whether to enter fossil fuel or renewable markets. - **Key Question:** - How many firms (e) will enter the fossil fuel market? **3. Beauty Contest Game** - **Rules:** - Participants select a number between 0 and 100. - Winner is the one closest to 2/3 of the average. - **Key Insight:** Rational players iteratively reduce their guesses, leading to a Nash equilibrium of 0. **Assumptions and Extensions** **1. Common Knowledge of Rationality (CKR):** - Players are assumed to act rationally to maximise their payoffs. **2. Common Priors:** - All players have identical information about the game (Harsanyi-Aumann Doctrine). **3. Repeated and Sequential Play:** - **Repeated Games:** Cooperation may emerge in iterated versions of games like the Prisoner's Dilemma. - **Sequential Play:** Later players observe earlier players' actions before deciding. **Critical Exam Areas** 1. **Definitions:** - Nash equilibrium. - Strict vs weak dominance. 2. **Key Models:** - Prisoner's Dilemma. - Battle of the Sexes. - Mixed strategies and their calculations. 3. **Applications:** - Environmental agreements. - Market entry scenarios. - Beauty contest game. 4. **Potential Questions:** - Provide a payoff matrix and ask to find Nash equilibria. - Analyse strategies in environmental or market contexts. - Discuss the impact of assumptions like CKR and common priors. **Lecture Notes: How Unaided Decision Makers Make Decisions** **Key Concepts and Definitions** **1. Bounded Rationality** - Decision makers simplify choices due to limited time and cognitive capacity. - Example: Instead of evaluating all laptop brands, focus on a few options and prioritise key attributes (e.g., battery life vs weight). **2. Decoy Options** - Adding a less attractive option can make another option more appealing. - Example: Choosing a 7-day Amsterdam holiday over a 10-day Paris trip when a more expensive 7-day Amsterdam option is added. **3. Heuristics (Fast & Frugal)** - **Definition:** Simple decision rules for quick, low-stakes decisions. - **Recognition Heuristic:** Choose the option you recognise (e.g., Hewlett-Packard over unknown brands). - **Minimalist Heuristic:** Use one attribute to decide and disregard others. **4. Lexicographic Ordering** - **Process:** Rank attributes by importance and select the best option based on the most important attribute. - Example: Choosing a holiday where cost is the most important factor, followed by flight time. - **Strengths and Limitations:** Simple and easy but ignores trade-offs between attributes. **5. Elimination by Aspects (EBA)** - **Process:** 1. Set a performance threshold for the most important attribute. 2. Eliminate options below the threshold. 3. Repeat for the next most important attribute. - **Example:** Choosing a computer with a price limit (£45m), delivery time (\