Game Theory Notes

Overview of Game Theory Basics

  • Introduction to Nash Equilibrium

  • Concepts of Dominant Strategies

    • Explanation of dominant strategies

    • Implications for solving games

  • Cooperative Outcomes

    • Requirements for cooperation

    • Exploration of potential cooperative solutions

Simple Game Example: John and Sarah's Study Efforts

Players:

  • John

  • Sarah

Strategies Available for Study Efforts:

  • John can choose to study for an hour/day or four hours/day.

  • Sarah has the same options as John.

Example Context:

  • Context of grading on a curve in academia

  • Under graduate versus graduate student study expectations

Grading on a Curve: Discussion Points

Definition:

  • Grading on a curve means that students' grades are relative to the performance of their peers.

Real-Life Implications:

  • If everyone performs poorly, grades are inflated.

  • A professor might award grades based on top performers to uplift everyone's score.

Graphical Representation:

  • Bell Curve: Displaying IQ scores

    • Average IQ score = 100

    • Majority of individuals (around 68%) fall within one standard deviation of 100.

    • Extremes of bell curve (e.g., geniuses and disabilities)

    • Grades distribution example

    • Historical grading practices in schools using bell curves to assess performance

Implications of Grading on a Curve

  • Differences in grading outcomes based on effort:

    • If the whole class studies similarly, they all retain a C.

    • If students exert less effort, they face potential grade penalties.

    • Discussion on whether students prefer to exert minimal effort for a guaranteed B grade.

Game Setup: Two Effort Levels and Outcomes

  • Both students choose not to exert effort (sixty grade) → Both receive a B (due to a curve).

  • Both work hard and achieve high grades (eighty-five) → Receive Bs.

  • Individual discrepancies lead to different letter grades based on average performance.

Dominant Strategies Analysis:

  • For John:

    • If Sarah doesn’t try, John should try to maximize his A grade.

    • If Sarah does try, John would still benefit from trying.

  • For Sarah:

    • Similar reasoning affirms Sarah also has a dominant strategy to try.

Cooperative Outcomes Exploring Nash Equilibrium

  • Nash Equilibrium occurs where both players have chosen optimal strategies given the other player's choice.

Repeated Games and Cooperation Requirements

When Is Cooperation Possible?

  1. Indefinite/Infinite Duration: Must anticipate future interactions.

  2. Pareto Improvement: At least one player must benefit without hurting others.

  3. Punishment System: Mechanisms must be enforced for compliance against non-cooperative behavior.

Real Life Example from Graduate Program

  • Discussion of cooperative attempts among a group of students not exerting effort.

  • Causes of unraveling and critical lesson from a professor's perspective.

  • Non-cooperative behavior undermines collective effort leading to overall decline in grades during final evaluations.

Concepts of Nash Equilibrium

Definition:

  • Nash Equilibrium is the optimal arrangement where no player can benefit by changing strategies while the other player keeps theirs constant.

Key Point:**

  • All dominant strategy equilibria are Nash equilibria, but not all Nash equilibria are dominant strategy equilibria.

Example of Game: No Dominant Strategy

Game Setup for Players A and B - High and Low Outputs

  • Payoff matrix illustrates potential earnings based on strategies chosen by each player.

Steps to Identify Nash Equilibrium:

  1. Each player evaluates possible outcomes and strategic benefits.

  2. Identify if a player would prefer to shift strategies.

  3. Confirm Nash Equilibrium through conditions of no incentive to change strategies.

Conclusion and Transition to Utility Theory

  • Transitioning to utility theory to measure satisfaction and derive choices.

  • Introduction of marginal utility as a concept to assess consumer choices based on price and satisfaction.

Important Mentions

  • Discussions of individual differences in preferences and implication for utility modeling.

  • Homework assignment involving cereal-related preferences, reflecting the complexities of utility measurement in practical scenarios.