2.2 Risk attitudes and expected utility theory
5 min read•july 30, 2024
Risk attitudes and expected 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
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- 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 )
- 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 reflects their risk attitude
- A concave utility function indicates risk aversion (diminishing 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 , 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 ()
Expected Utility Calculation
Steps to Calculate Expected Utility
- Identify the possible outcomes and their associated probabilities
- Each outcome should have a corresponding utility value that represents the subjective desirability of that outcome
- Multiply the utility of each outcome by its probability of occurrence
- This step calculates the expected utility contribution of each outcome
- 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
- 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: 0 with probability 0.4
- Investment B: 0 with probability 0.2
- Assuming a risk-neutral decision-maker, the expected utility of Investment A is 100 + 0.4 × 40 (0.8 × 0)
- The optimal choice for a risk-neutral decision-maker would be Investment A, as it has the higher expected utility
Key Terms to Review (18)
Allais Paradox: The Allais Paradox is a situation in decision theory that demonstrates a violation of the expected utility hypothesis, revealing inconsistencies in individuals' risk preferences when faced with choices involving probabilistic outcomes. This paradox highlights how people often make decisions that contradict their own perceived rationality, showing that real-world choices do not always align with the predictions of expected utility theory.
Certainty Equivalent: The certainty equivalent is the guaranteed amount of wealth that an individual considers equally desirable as a risky prospect with uncertain outcomes. It reflects a person's risk preferences and helps to understand how they perceive risk and make decisions under uncertainty. This concept connects to expected utility theory, where individuals evaluate their choices based on the expected outcomes and their associated probabilities.
Daniel Kahneman: Daniel Kahneman is a renowned psychologist known for his groundbreaking work in behavioral economics and decision-making, particularly regarding how people perceive risk and make choices under uncertainty. His research has profoundly influenced the understanding of human behavior, revealing that individuals often rely on cognitive shortcuts, leading to systematic biases in judgment and decision-making.
Expected Utility Formula: The expected utility formula is a mathematical representation used to calculate the anticipated satisfaction or benefit an individual derives from uncertain outcomes, factoring in both the probabilities of each outcome and the utility associated with them. This concept is central to understanding how people make choices under risk, as it allows for the comparison of different risky options by quantifying their expected values, thus providing a foundation for analyzing risk attitudes and decision-making behavior.
Expected Value: Expected value is a concept in probability that calculates the average outcome of a random event, taking into account all possible outcomes and their probabilities. This measure is crucial in understanding risk attitudes and decision-making under uncertainty, as it helps individuals weigh the potential benefits and losses of different choices. In scenarios involving expected utility theory, expected value serves as a foundational element for evaluating risky prospects, guiding rational behavior by focusing on maximizing anticipated returns.
Gamble: A gamble is an action or decision that involves taking a risk with the possibility of losing something of value, typically money, in hopes of achieving a greater reward. In the context of risk attitudes and expected utility theory, gambling illustrates how individuals assess potential outcomes based on their preferences and beliefs about risk, leading to different strategies for decision-making under uncertainty.
Insurance decisions: Insurance decisions refer to the choices individuals or organizations make regarding the purchase of insurance policies to manage potential financial risks. These decisions are influenced by various factors, including risk attitudes, perceived probabilities of adverse events, and the principles of expected utility theory, which helps individuals evaluate the potential outcomes of different choices in uncertain situations.
John von Neumann: John von Neumann was a Hungarian-American mathematician, physicist, and polymath who made significant contributions to many fields, including game theory, which he is often regarded as one of the founding figures. His work laid the groundwork for strategic decision-making and rational choice, revolutionizing how individuals and organizations analyze competitive situations.
Lottery: In the context of decision-making under uncertainty, a lottery refers to a probabilistic scenario where individuals can choose between different outcomes, each associated with a specific probability and utility. This concept is crucial for understanding how people assess risk and make choices when faced with uncertain payoffs, allowing us to analyze preferences and risk attitudes through expected utility theory.
Marginal Utility: Marginal utility refers to the additional satisfaction or benefit that a consumer derives from consuming one more unit of a good or service. It plays a crucial role in understanding how individuals make choices under conditions of scarcity, helping to explain consumer behavior and the allocation of resources. In particular, it connects to the assessment of risk attitudes and expected utility theory, as it illustrates how people evaluate the incremental benefits they receive from different options in uncertain situations.
Paradoxes of Choice: The paradoxes of choice refer to the phenomenon where having too many options can lead to negative outcomes, such as anxiety, paralysis in decision-making, and dissatisfaction with the choices made. This concept highlights the limitations of human cognitive processing when faced with an abundance of alternatives, suggesting that more choices do not necessarily lead to better decisions or greater satisfaction.
Portfolio Choice: Portfolio choice refers to the decision-making process individuals or institutions undergo when selecting a mix of assets to hold in their investment portfolios. This concept is closely linked to risk attitudes, as it reflects how an investor's preferences regarding risk and return influence their asset allocation decisions based on expected utility theory.
Prospect Theory: Prospect Theory is a behavioral economic theory that describes how individuals evaluate potential losses and gains when making decisions under risk. It suggests that people are more sensitive to potential losses than to equivalent gains, which leads to behaviors that deviate from traditional expected utility theory. This theory highlights how cognitive biases and emotional reactions can influence decision-making processes, particularly in uncertain situations.
Risk Aversion: Risk aversion is the preference for a sure outcome over a gamble with higher or equal expected value. This concept is crucial in understanding how individuals make choices under uncertainty and impacts various decision-making processes, especially when faced with potential losses. Recognizing that risk-averse individuals prefer safer options can help explain their behaviors in economic settings, as well as in game theory and experimental studies.
Risk seeking: Risk seeking refers to a behavioral attitude in which individuals prefer options that offer higher potential rewards despite the accompanying higher risks. This tendency is characterized by a willingness to gamble on uncertain outcomes for the chance at greater gains, contrasting with more risk-averse behaviors that prioritize safety over potential rewards. Risk seeking plays a crucial role in decision-making processes, especially when evaluating choices under uncertainty.
Subjective Probability: Subjective probability is an individual's personal estimate of the likelihood of an event occurring, based on their own experiences, beliefs, and information. This type of probability differs from objective probability, which is based on statistical or empirical data. It plays a significant role in understanding how people make choices under uncertainty and reflects varying risk attitudes influenced by personal perspectives.
Utility: Utility is a measure of the satisfaction or value that an individual derives from consuming goods and services or from engaging in specific actions. It plays a crucial role in decision-making processes, especially under conditions of uncertainty and varying risk levels, as individuals aim to maximize their utility based on their preferences. Understanding utility helps analyze how people make choices in strategic situations, such as those found in games and when faced with risks.
Utility Function: A utility function is a mathematical representation that assigns a real number to each possible outcome or choice, reflecting the level of satisfaction or preference that an individual derives from that outcome. This concept is crucial in understanding decision-making processes and is linked to how players evaluate their options in strategic interactions, providing insights into their preferences, strategies, and behaviors.