MfF_2024_L7-8

Lectures 7 & 8: Choice under Uncertainty

Choice Under Uncertainty: Examines the processes through which individuals and institutions make decisions when the outcomes are not known. This field integrates concepts from economics, psychology, and decision theory to analyze how uncertainty affects choices and behaviors.
Contingent Commodities: These are assets or rights in financial contexts that derive their value from specific states of the world. For example, options and derivatives can be viewed as contingent commodities because their payoff relies on future events.
Lotteries and Their Classification: Various types of lotteries exist, including simple lotteries with fixed probabilities and complex compound lotteries that encompass a series of simple lotteries, influencing individual choice strategies based on psychological biases and preferences.

Expected Utility Theory (EUT): A cornerstone of decision-making under uncertainty that asserts individuals choose to maximize their expected utility rather than just monetary outcomes. EUT is based on several axioms that reflect rational behavior.
Risk Attitudes: These attitudes dictate how individuals perceive potential losses and gains in uncertain situations, encompassing:
- Risk Aversion: A preference to avoid uncertain outcomes in favor of known rewards.
- Risk Neutrality: Where individuals are indifferent between certain outcomes and uncertain ones with equivalent expected values.
- Risk Loving: An inclination to prefer uncertain options, even if they yield lower expected values, leading to behavior that seeks out risk.

Random Variables & Outcomes: All random events representable by a finite number of variables and outcomes, allowing for mathematical modeling of decision processes.
Monetary Wealth Focus: Decision-makers prioritize their financial wealth and the probability distribution across various states, viewing wealth changes as primary motivations.
Consequentialist Assumptions: Agents focus on the outcomes impacting their wealth, disregarding their current states if their wealth remains unchanged.

Definition of a Simple Lottery: A simple lottery is formally defined by a set of monetary prizes and their probabilities, represented mathematically as: L = ((p1,...,ps),(x1,...,xs)), where pi is the probability of attaining prize xi.
Consolidation of States: Instead of treating every state with identical outcomes as separate, they can be consolidated into a single state, simplifying the decision-making process.
Compound Lotteries: Multi-stage lotteries can be resolved into simpler components, facilitating choice through expected values.

Fair Gambles vs. Unfair Gambles:
- Fair Gamble: A gambling scenario where the expected monetary outcome is zero, implying that there is no statistical advantage to gambling.
- Unfair Gambles: Gambles where expected outcomes can be favorable or unfavorable, leading to strategic implications in betting behavior.
Paradox of Risk: The Saint-Petersburg Paradox illustrates how traditional expected value calculations can contradict rational decision-making, particularly in risky scenarios.
- Example: A lottery offering progressively larger payouts but with diminishing probabilities illustrates this paradox, highlighting discrepancies between mathematical expectations and human choice behavior.
- Solution: Daniel Bernoulli suggested viewing utility rather than absolute monetary value to resolve this paradox.

Axioms of EUT:
1. Rationality: Encompasses completeness and transitivity in preferences.
2. Continuity: Suggests existence of intermediate probabilities in preference rankings.
3. Independence of Irrelevant Alternatives: The preference between options should not be affected by the introduction of an irrelevant alternative.
Expected Utility Function: Defined mathematically as: U((p1,...,pS)(x1,...,xS)) = ∑ s∈S ps · v(xs), where v(xs) denotes the utility corresponding to every monetary outcome.
Utility Function (v): Represents individual preferences and satisfaction derived from different levels of wealth.

Risk Averse: Characterized by a concave utility function where the marginal utility decreases with increased wealth, incentivizing stable outcomes.
Risk Neutrals: Represented by a linear utility function implying constant marginal utility, leading to indifference between various risky options.
Risk Lovers: Have a convex utility function, indicating increased marginal utility with more wealth, leading them to prefer riskier choices.

Certainty Equivalent (CE): The guaranteed amount of wealth an agent would consider equivalent to a risky gamble, reflecting their risk preferences.
Risk Premium (RP): The contrast between expected monetary value and the certainty equivalent, representing the amount individuals would pay to steer clear of risk.

Utility Functions: Adjust based on individual risk attitudes:
- Risk Averse: Characterized by decreasing marginal utility leading to a concave shape.
- Risk Neutral: Consistent marginal utility resulting in a straight line.
- Risk Seeking: Increasing marginal utility resulting in a convex shape.
Arrow-Pratt Measures: These indicate the degree of risk aversion quantitatively, serving as important indicators for economic decision-making.

Definition: Graphical representations of combinations of wealth in different states, facilitating visualization of preferences.
Graphical Analysis: Shifts in indifference curves reflect changes in risk attitudes, with slopes indicating trade-offs between different states of wealth.

Characteristics: This line demonstrates all possible bundles of contingent commodities resulting from fair gambling scenarios, where the slope indicates the ratio of probabilities between two states, thereby framing decision-making strategies within uncertain contexts.

Behavior Towards Fair Gambles: Behavioral economics suggests that individual attitudes toward fair gambles can significantly influence decision-making processes regarding participation in risk-laden scenarios.
Suggested Readings for Further Understanding:
- Hal Varian, "Intermediate Microeconomics"
- Pindyck & Rubinfeld, essential literature on market dynamics and economic theory.