Risk attitudes and expected utility theory are key concepts in decision-making under uncertainty. They help us understand how people evaluate risky choices and make decisions based on their preferences and risk tolerance.

These concepts are crucial in utility theory, as they explain why different individuals may make different choices when faced with the same options. Understanding risk attitudes allows us to predict and analyze decision-making behavior in various real-world scenarios.

Risk Attitudes and Decision-Making

Factors Influencing Risk Attitudes

• Risk attitudes refer to an individual's willingness to take on risk or uncertainty in decision-making situations
• Risk attitudes are influenced by various factors such as personal preferences, past experiences, cultural background, and situational context
  • Personal preferences shape an individual's risk tolerance and risk-taking behavior (risk-seeking vs risk-averse)
  • Past experiences with risky situations can impact future risk attitudes (positive outcomes encourage risk-taking, negative outcomes promote risk aversion)
  • Cultural background influences risk perceptions and attitudes (collectivist cultures may be more risk-averse than individualistic cultures)
  • Situational context affects risk attitudes (high-stakes decisions may elicit different risk attitudes compared to low-stakes decisions)

Utility and Risk Attitudes

• The concept of utility is used to quantify an individual's preferences and risk attitudes
  • Utility represents the subjective value or satisfaction derived from a particular outcome
• The shape of an individual's utility function reflects their risk attitude
  • A concave utility function indicates risk aversion (diminishing marginal utility as wealth increases)
  • A linear utility function indicates risk neutrality (constant marginal utility across different levels of wealth)
  • A convex utility function indicates risk-seeking behavior (increasing marginal utility as wealth increases)
• Risk attitudes affect decision-making by influencing how individuals evaluate and compare different options with uncertain outcomes
  • Risk-averse individuals tend to prefer safer options with more certain outcomes
  • Risk-seeking individuals are more willing to take chances for potentially higher rewards

Expected Utility Theory

Normative Model for Decision-Making Under Risk

• Expected utility theory is a normative model that prescribes how rational individuals should make decisions under risk or uncertainty
  • It provides a framework for evaluating and comparing different options based on their expected utility
• The expected utility of an option is calculated by multiplying the utility of each possible outcome by its probability of occurrence and summing these products
  • This calculation takes into account both the desirability of the outcomes and their likelihood of occurring
• To apply expected utility theory, decision-makers need to assign utilities to different outcomes and estimate the probabilities of those outcomes occurring
  • These utilities and probabilities are subjective and can vary across individuals

Applications and Assumptions

• The option with the highest expected utility is considered the optimal choice according to expected utility theory
  • Rational decision-makers are assumed to choose the option that maximizes their expected utility
• Expected utility theory can be used to analyze various decision-making scenarios
  • Investment decisions (choosing between different investment options based on their expected returns and risks)
  • Insurance choices (deciding whether to purchase insurance based on the expected utility of being insured vs uninsured)
  • Gambling situations (evaluating the expected utility of different gambling strategies)
• Expected utility theory assumes that individuals have well-defined preferences, are able to assign utilities to outcomes, and can accurately estimate probabilities
  • These assumptions may not always hold in real-world decision-making situations

Risk Aversion vs Risk Seeking

Characteristics and Utility Functions

• Risk-averse behavior is characterized by a preference for certainty and a willingness to accept lower expected returns to avoid risk
  • Risk-averse individuals have a concave utility function, indicating diminishing marginal utility as wealth increases
• Risk-neutral behavior is characterized by indifference between options with the same expected value, regardless of the level of risk involved
  • Risk-neutral individuals have a linear utility function, indicating constant marginal utility across different levels of wealth
• Risk-seeking behavior is characterized by a preference for risk and a willingness to accept lower expected returns for the chance of higher potential rewards
  • Risk-seeking individuals have a convex utility function, indicating increasing marginal utility as wealth increases

Measuring Risk Aversion

• The Arrow-Pratt measure of absolute risk aversion can be used to quantify an individual's level of risk aversion
  • It measures the curvature of the utility function and provides a standardized way to compare risk attitudes across individuals
• Risk attitudes can vary depending on the context and the magnitude of the potential gains or losses involved
  • Individuals may exhibit different risk attitudes in different domains (financial decisions vs health-related choices)
  • The same individual may be risk-averse for small stakes but risk-seeking for large stakes (prospect theory)

Expected Utility Calculation

Steps to Calculate Expected Utility

1. Identify the possible outcomes and their associated probabilities

   • Each outcome should have a corresponding utility value that represents the subjective desirability of that outcome

2. Multiply the utility of each outcome by its probability of occurrence

   • This step calculates the expected utility contribution of each outcome

3. Sum the expected utility contributions of all outcomes to obtain the overall expected utility of the decision

   • This sum represents the average utility that can be expected from the decision, considering all possible outcomes and their probabilities

4. Repeat the expected utility calculation for each available option or decision alternative

   • This step allows for a comparison of the expected utilities across different choices

Determining the Optimal Choice

• The optimal choice is the option with the highest expected utility
  • Rational decision-makers should select the alternative that maximizes their expected utility based on their risk attitudes and the given probabilities and utilities
• Sensitivity analysis can be performed to assess how changes in probabilities or utilities affect the optimal choice
  • This analysis helps determine the robustness of the decision and identifies critical factors that influence the expected utility calculations
• Example: Consider a decision between two investments with different potential payoffs and probabilities
  • Investment A: $100 with probability 0.6, $0 with probability 0.4
  • Investment B: $50 with probability 0.8, $0 with probability 0.2
  • Assuming a risk-neutral decision-maker, the expected utility of Investment A is $60 (0.6 × $100 + 0.4 × $0) and the expected utility of Investment B is $40 (0.8 × $50 + 0.2 × $0)
  • The optimal choice for a risk-neutral decision-maker would be Investment A, as it has the higher expected utility