Lecture 23: Judgment and Decision Making - More on Interesting Games

Overview of Surprise Exams and Backward Induction

  • Scenario of the Surprise Exam: A professor announces that there will be a surprise exam during a course with 10 lectures.
  • Reasoning Against a Surprise Exam:
    • Mary argues that the exam cannot happen in the last lecture (10) because it wouldn't be a surprise.
    • This reasoning is extended backward: if 10 is out, then so is 9, 8, etc., leading to the conclusion that there cannot be a surprise exam.
  • Disagreement on Reasoning: Students were polled on their opinion of Mary’s reasoning:
    • 22% thought her reasoning was sound.
    • 53% thought it was interesting but contained a glitch.
    • 13% found it absurd.
    • 11% were undecided.

The Paradox of the Surprise Exam

  • The Huge Question: How can a surprise exam be given if the reasoning suggests it is impossible?
  • The Surprise Exam as a Paradox: Introduced as the hanging paradox involving elements of backward induction, a key concept in game theory.
  • Backward Induction:
    • Focuses on analyzing the last move first, determining the best action based on the potential decisions of others.
    • It applies recursively to all previous moves until reaching the first, thus ‘solving’ the game in steps.

Theories on the Surprise Exam

  • Conditional Probability of Surprise:
    • Surprise defined mathematically as the probability that the exam happens at a given point, while recalculating major possible outcomes.
    • Surprises can still exist even with prior knowledge of potential dates based on the increase of uncertainty.
  • Logical Statements:
    • Statement (A): A surprise exam is scheduled, but its date cannot be predicted in advance.
    • Statement (B): Is this also self-referential? This invokes logical paradoxes and philosophical questioning of belief and knowledge.

Recursive Knowledge and Decision Making

  • Understanding Rationality:
    • The idea that the professor’s rationality and the students’ awareness of this creates layers of expectations.
    • Recursive statements about mutual knowledge can lead to unexpected outcomes, like a surprise exam happening based on odd/even levels of rational attribution (Cristina Bicchieri's theorem).
  • Example Drawing:
    • Students’ expectations change based on assumptions leading to different positive or negative utilities based on whether they study or not and whether the examination exists or not.

Game Theory Applications

  • Teaching Game Theory:
    • Explores strategies and outcomes based on rational play in games like mutual defection.
    • Introducing payoffs for both teachers and students shows varying strategies and outcomes depending on student's choices.

Experiments and Observations

  • Trust Games:
    • Classic example where players must decide to cooperate (play across) or defect (play down). Rationally, players should defect, but experiments shown preferences for cooperation exist despite rational outcomes.
  • The Disjunction Effect:
    • Individuals approached different gambles based on prior experiences of winning or losing, reflecting risk assessment and emotional responses rather than pure rationality.

Conclusion on Gambles and Trust

  • Risk and Emotion: Experiences before deciding on new equivalent gambles show the balance between risk and potential emotional weight influencing choices.
  • Underlying Principles:
    • The decisions reflect underlying principles in judgment and decision-making processes, highlighting the complex interplay of rationality, psychology, and game theory.

Key Takeaways

  • A surprise exam can be paradoxically possible despite logical reasoning against it.
  • Concepts of backward induction are crucial for understanding decision-making processes and outcomes in strategic interactions.
  • Rational players do not always act according to the principles of game theory in practical or experimental settings; emotions and previous experiences greatly influence decisions.