Expected Utility
Introduction
Expected utility is a fundamental concept in decision theory, economics, and statistics, which provides a framework for understanding how rational agents make choices under uncertainty. It is based on the premise that individuals choose between risky or uncertain prospects by comparing their expected utilities, which are calculated as the weighted average of the utilities of all possible outcomes, with the probabilities of these outcomes serving as weights. This concept is pivotal in the study of rational choice theory, where it helps to predict and explain the behavior of agents in various economic and strategic settings.
Historical Background
The notion of expected utility can be traced back to the works of Daniel Bernoulli, who introduced it in the context of the St. Petersburg Paradox in 1738. Bernoulli's work laid the groundwork for the development of the expected utility hypothesis, which was later formalized by John von Neumann and Oskar Morgenstern in their seminal book, "Theory of Games and Economic Behavior" (1944). This formalization provided a rigorous mathematical foundation for the concept, which has since become a cornerstone of modern economic theory.
Theoretical Framework
Utility Function
At the heart of expected utility theory is the utility function, which represents an individual's preferences over a set of outcomes. The utility function is assumed to be a real-valued function that assigns a numerical value to each possible outcome, reflecting the level of satisfaction or preference the individual derives from that outcome. The utility function is typically assumed to be monotonic, meaning that higher levels of an outcome lead to higher utility.
Probability and Uncertainty
Expected utility theory incorporates the concept of probability to account for uncertainty in decision-making. Probabilities are used to weigh the utilities of different outcomes, reflecting the likelihood of each outcome occurring. In this framework, an individual's decision-making process involves calculating the expected utility of each available option and choosing the one with the highest expected utility.
Axioms of Expected Utility Theory
Expected utility theory is built upon a set of axioms that describe rational behavior under uncertainty. These axioms, known as the von Neumann-Morgenstern axioms, include:
1. **Completeness**: For any two outcomes, an individual can determine a preference or indifference between them. 2. **Transitivity**: If an individual prefers outcome A to B and B to C, then they must prefer A to C. 3. **Independence**: Preferences between outcomes should remain consistent, even when combined with a third outcome. 4. **Continuity**: If an individual prefers outcome A to B and B to C, there exists a probability mix of A and C that is equally preferred to B.
These axioms ensure that preferences can be represented by a utility function, allowing for the calculation of expected utility.
Applications in Economics
Expected utility theory is widely used in economics to model decision-making under risk and uncertainty. It provides a framework for analyzing choices in various contexts, including:
Risk Aversion
Risk aversion is a key concept in expected utility theory, describing an individual's preference for certain outcomes over uncertain ones with the same expected value. A risk-averse individual has a concave utility function, indicating diminishing marginal utility of wealth. This concept is crucial in understanding behavior in insurance markets, investment decisions, and other economic activities involving risk.
Portfolio Theory
In portfolio theory, expected utility is used to model investor behavior and optimize asset allocation. Investors are assumed to choose portfolios that maximize their expected utility, balancing the trade-off between risk and return. The Capital Asset Pricing Model (CAPM) and other asset pricing models are built on the principles of expected utility theory.
Game Theory
Expected utility theory is also integral to game theory, where it is used to model strategic interactions between rational agents. In games of incomplete information, players use expected utility to make decisions based on their beliefs about other players' strategies and payoffs.
Criticisms and Alternatives
Despite its widespread use, expected utility theory has faced criticism for its assumptions and limitations. Some of the main criticisms include:
Behavioral Anomalies
Empirical studies have identified several behavioral anomalies that violate the axioms of expected utility theory. These include the Allais Paradox and the Ellsberg Paradox, which demonstrate that individuals do not always behave in accordance with the theory's predictions.
Prospect Theory
In response to these anomalies, prospect theory was developed by Daniel Kahneman and Amos Tversky as an alternative to expected utility theory. Prospect theory accounts for observed deviations from rational behavior by incorporating psychological factors such as loss aversion and probability weighting.
Bounded Rationality
The concept of bounded rationality, introduced by Herbert Simon, challenges the assumption of fully rational decision-making. It suggests that individuals have cognitive limitations and rely on heuristics to make decisions, which may not align with the predictions of expected utility theory.
Mathematical Formulation
The mathematical formulation of expected utility involves calculating the expected value of a utility function over a set of possible outcomes. For a discrete set of outcomes \(\{x_1, x_2, \ldots, x_n\}\) with corresponding probabilities \(\{p_1, p_2, \ldots, p_n\}\), the expected utility \(E(U)\) is given by:
\[ E(U) = \sum_{i=1}^{n} p_i \cdot U(x_i) \]
where \(U(x_i)\) is the utility of outcome \(x_i\).
For continuous outcomes, the expected utility is calculated using an integral:
\[ E(U) = \int U(x) \cdot f(x) \, dx \]
where \(f(x)\) is the probability density function of the outcomes.
Extensions and Generalizations
Expected utility theory has been extended and generalized in various ways to address its limitations and broaden its applicability. Some notable extensions include:
Subjective Expected Utility
Subjective expected utility theory, developed by Leonard Savage, incorporates subjective probabilities to account for situations where objective probabilities are not available. This extension allows for the modeling of decision-making under uncertainty when individuals have personal beliefs about the likelihood of outcomes.
Rank-Dependent Utility
Rank-dependent utility theory modifies the expected utility framework by introducing a probability weighting function, which captures the decision-maker's attitude towards risk. This approach accounts for observed deviations from expected utility theory, such as the overweighting of small probabilities.
Cumulative Prospect Theory
Cumulative prospect theory, an extension of prospect theory, combines elements of rank-dependent utility and prospect theory to provide a more comprehensive model of decision-making under risk. It incorporates both probability weighting and loss aversion, offering a more accurate representation of observed behavior.
Conclusion
Expected utility theory remains a foundational concept in decision theory and economics, providing a robust framework for analyzing choices under uncertainty. Despite its limitations and the emergence of alternative models, expected utility continues to be a valuable tool for understanding and predicting rational behavior in various contexts. Its mathematical rigor and theoretical elegance have made it a cornerstone of economic analysis and a subject of ongoing research and debate.