Expected Utility Theory Notes

Section 6.B: Expected Utility Theory

• Expected Utility Theory:

  • Fundamental framework in decision theory and economics that evaluates risky prospects.
  • Utility refers to the satisfaction or value derived from outcomes.

• Key Steps in Establishing Expected Utility:

  • Step 1: Rearranging inequalities involving lotteries to derive preference relations.
  • Step 2: Establishing relationships between different lotteries based on preferences and probabilities.
  • Step 3: Existence of a unique parameter ( \alpha ) that satisfies conditions for any lottery ( L ).
  • Existence is implied by continuity of preferences and the extremity conditions (best and worst lotteries).
  • Step 4: Defining the utility function ( U(L) = \alpha ) for lotteries ( L ).
  • Step 5: Proving that the function ( U ) satisfies the linearity condition of expected utility.

• Independence Axiom:

  • The preference between lotteries does not depend on the outcomes of common components.
  • Impacts how people evaluate mixed lotteries, which leads to the structure of expected utility.

• Advantages of Expected Utility:

  • Technical Convenience: Enables straightforward analysis and derivation of results.
  • Normative Guidance: Helps decision-makers adhere to rational choice axioms in uncertain situations.

• Examples of Application:

  • Concepts illustrated through decision-making scenarios involving lotteries (probabilistic outcomes).
  • Example: Decision maker evaluating preferences of closely valued lotteries can infer indirect preferences, such as through comparisons.

• Challenges and Paradoxes:

  • Allais Paradox: A scenario demonstrating inconsistencies in expected utility theory, showcasing deviations in expected utility when faced with certain versus probabilistic outcomes.
  • Illustrates how risk preferences can vary contextually and lead to outcomes that challenge expected utility assumptions.

• Utility Functions and Risk Aversion:

  • Defining utility functions can reveal risk preferences and help model decision-making behavior.
  • Concave utility functions indicate risk-averse behavior, important in many economic scenarios.

Discussion of the Theory

• Expected Utility as a Guide to Action:

  • Satisfying the axioms yields norms for decision-making under risk, clarifying how to evaluate uncertain outcomes.

• Inductive Reasoning through Expected Utility:

  • Preference orderings and derived outcomes help inform choices even when conditions are complex.

• Illustrative Examples:

  • Various experiments (like the Allais and Machina paradoxes) showcasing potential deviations from expected utility.

Implications of Expected Utility Theory

• Provides foundational support for making rational decisions under uncertainty.
• Important for developing models in economic theory, finance, and other decision-making fields.
• Insights stress the utility of understanding mathematical structures within preferences to model behavior more effectively in probabilistic environments.