WEEK 7 - ECN306 Lecture 7 Deviations from Nash equilibrium(2)
Lecture 7: Deviations from Nash Equilibrium
Page 1: Introduction
Lecture Title: Deviations from Nash Equilibrium
Course: ECN306 Game Theory for Economists
Page 2: Today's iSheffield code
[Specific iSheffield code provided in class]
Page 3: Plan for Today
Focus Areas:
Discussion of the Nash equilibrium concept.
When does play converge to Nash equilibrium?
Reasons for behavioral differences from predictions.
Methods to improve accuracy of NE predictions.
Exploration of games of conflict.
Discussion of beauty contest games.
Examination of games lacking pure-strategy equilibria.
Page 4: Critical Discussion of the Nash Equilibrium Concept
Examination of Nash equilibrium in various contexts and implications.
Page 5: Normative and Descriptive Interpretation of NE
Normative Interpretation:
Represents how rational players should behave in a game.
If payoffs denote utility and all players are rational, behavior aligns with NE predictions.
Applicable in non-cooperative game theory; cooperative game theory involves different solution concepts.
Descriptive Interpretation:
Reflects actual player behavior in a game, which often diverges from NE predictions.
Page 6: Why Does Behaviour Differ from NE?
Assumption of Selfishness:
NE assumes players are selfish expected utility maximizers.
This can be relaxed by modeling preferences considering altruism or spite.
Cognitive Limitations:
NE assumes unlimited cognitive abilities; players may make mistakes or learn over time.
Belief Accuracy:
NE assumes players have accurate beliefs; can be modeled to account for errors in belief.
Page 7: Reiteration of Behavioral Differences from NE
Re-emphasizes points from page 6 about selfishness, cognitive abilities, and belief accuracy.
Page 8: Preferences in Game Theory
Nash Equilibrium calculations usually assume players maximize monetary payoffs.
Real-world factors influencing utility include:
Inequality or concerns regarding others' earnings (altruism or spite).
Emotional responses (e.g., anger or guilt).
Social norms can also impact behaviors.
Recommendation: incorporate these factors into utility functions for realistic NE calculations.
Page 9: Illustration: Push-Pull Game
Structure of the Game:
Payoff matrix considering candy as the utility.
Equilibria Explanation:
Nash equilibrium under the candy assumption vs. alternative realistic utility models.
Page 10: Behavioral Differences from NE (Continued)
Repeats the reasons behavioral differences arise from NE from previous pages.
Page 11: Example: Traveller’s Dilemma
Scenario Description:
Lost bag reimbursement claim scenario outlining incentives to inflate claims.
Claims are independent and lead to penalties for the inflated higher claim.
Page 12: Best Response and NE in Traveler's Dilemma
Mathematical representation of claims and penalties leading to NE of (80, 80).
Page 13: Data from Traveller’s Dilemma Experiment
Claims were higher with a penalty of 5 vs. 80, illustrating NE predictions versus actual behavior.
Page 14: Laboratory Experiment Results
NE prediction insensitivity to penalty size demonstrated through experimental data.
Page 15: NE Sensitivity to Incentives
Discussing how NE predictions remain unchanged despite variable incentives.
Page 16: Quantal Response Equilibrium (QRE)
QRE incorporates player errors in decision-making by predicting best responses to beliefs about opponents' moves.
Page 17: Finding QRE in 2x2 Game
Explanation of sensitivity parameters and how to derive probabilities for players' strategies in equilibrium.
Page 18: QRE's Superior Data Explanation
QRE better explains observations in scenarios like the volunteer's dilemma as compared to NE.
Page 19: Case Study: Behavior in Contests
Experiment results show average effort exceeded NE predictions, reflected in investment distribution data.
Page 20: Reasons for Over-Expenditure in Contests
Factors influencing behavior include risk-seeking preferences, spiteful social preferences, and joy from winning—not purely monetary incentives.
Page 21: Cognitive Abilities and Convergence to NE
Discussion on players' abilities to converge to the NE through deliberation and learning from past actions.
Page 22: Learning and Feedback Mechanisms
Two main models:
Belief Learning: Based on observed past actions.
Reinforcement Learning: Players choose actions that historically yielded better payoffs.
Page 23: Non-equilibrium Models
Rational for using non-equilibrium models to simulate player behavior and learning dynamics rather than relying solely on NE predictions.
Page 24: Conditions for Convergence to NE
Identifying repeated interactions, effective feedback mechanisms, and simplified game structures as conducive to convergence.
Page 25: NE Convergence with Feedback
Emphasizes that players converge towards NE when given information about foregone payoffs.
Page 26: Reiteration of Reasons for Behavioral Differences from NE
Repeats previous points on behaviors diverging from NE regarding preferences, cognitive abilities, and beliefs.
Page 27: Another Game Example
Encouragement for students to log in for further resources related to the next game topic.
Page 28: Beauty Contest Game
Participants choose numbers between 0 and 100; the goal is to be closest to 2/3 of the average chosen number.
Page 29: Best Responses and NE of the Beauty Contest
Details that the best response is choosing 2/3 of the average, with potential NE outcomes discussed.
Page 30: Results in Beauty Contest Game
Emphasizes variable player experience influencing NE achievement, notably in trained versus untrained participants.
Page 31: Explanation of Beauty Contest Results
Level-k reasoning framework illustrating how players adjust their beliefs and responses based on perceived tendencies of other players.
Page 32: Belief Models for One-Shot Games
Discusses common belief modeling approaches in one-shot games and their implications for NE predictions.
Page 33: Games Without Pure Strategy Equilibria
Introduction to scenarios where traditional pure strategy equilibria do not exist.
Page 34: Penalty Kick Game Dynamics
Outlining strategic interactions between striker and goalkeeper in a penalty kick scenario, represented through probability-based payoffs.
Page 35: Lack of Pure Strategy Equilibrium in Penalty Kicks
Demonstrates mixed strategy equilibrium as the resolution in penalty kick scenarios.
Page 36: Idea Behind Mixed Strategies
Transition from discrete choices to developing continuous strategies using probabilities for player actions.
Page 37: Set Up of Expected Payoff Functions
Details calculating expected payoffs and determining conditions for mixed-strategy Nash equilibria.
Page 38: Best Responses for Striker
Mathematical conditions for the striker's best responses based on goalie probabilities.
Page 39: Best Responses for Goalkeeper
Determining optimal strategies for the goalkeeper based on striker's probabilities.
Page 40: Mixed-Strategy Nash Equilibrium Results
Summary of mixed-strategy equilibrium conclusions for the penalty kick scenario, highlighting critical probability thresholds.
Page 41: Next Time
Assignments for next class:
Worksheet 8
Engagement task 3
Reading: Chapter 5 focusing on NE discussions.