Von Neumann–Morgenstern Utility Theorem

Published Mar 22, 2024

Definition of the Von Neumann–Morgenstern Utility Theorem

The Von Neumann–Morgenstern utility theorem is a foundational concept in the field of expected utility theory, which underpins much of modern economics and decision theory. It establishes that under certain axioms of rational behavior, an individual’s preferences over uncertain prospects (or “lotteries”) can be represented by a utility function. This utility function then enables the individual to rank these prospects based on their expected utilities, essentially quantifying the desirability of various outcomes.

Conceptual Example

To illustrate the Von Neumann–Morgenstern utility theorem, imagine an individual faced with choosing between two games:
1. Game A guarantees them $100.
2. Game B offers a 50% chance of winning $200 and a 50% chance of winning nothing.

According to the theorem, the individual can assign utility values to the outcomes of these games based on their preferences. If the individual is risk-neutral, they may assign a utility value of 1 to $100, 1.5 to $200, and 0 to $0, making the expected utility of Game B (0.5 * 1.5 + 0.5 * 0) higher than that of Game A. Thus, the theorem allows for a rational choice under uncertainty by comparing the expected utilities of different options.

Why the Von Neumann–Morgenstern Utility Theorem Matters

The Von Neumann–Morgenstern utility theorem is paramount in economics and decision theory for several reasons. Firstly, it provides a rigorous mathematical framework for understanding and predicting human behavior under risk and uncertainty. Through this framework, economists can analyze choices in various contexts, from investment decisions to policy-making.

Moreover, the theorem introduces the concept of expected utility, which has become a cornerstone in the theories of rational choice and welfare economics. It allows for the comparison of different policies or economic conditions based on their impact on the welfare or utility of individuals and societies as a whole. The theorem’s influence extends beyond economics into psychology, political science, and even bioethics, wherever decision-making under uncertainty is studied.

Frequently Asked Questions (FAQ)

What are the axioms of rational behavior underlying the theorem?

The Von Neumann–Morgenstern utility theorem rests on four axioms of rational preference ordering: completeness, transitivity, continuity, and independence. These axioms assume that individuals can consistently rank all possible choices, that their preferences are internally consistent, they prefer certain averages of lotteries to extreme ones, and their preferences between lotteries are invariant to irrelevant alternatives.

Can the utility function vary between individuals?

Yes, the utility function can vary significantly between individuals based on their risk tolerance. Some people are risk-averse, preferring certain outcomes over gambles with potentially higher payoffs but greater uncertainty. Others may be risk-seeking, favoring the excitement or potential for higher returns that uncertainty brings. The utility function captures these individual preferences and risk attitudes.

How does the theorem apply to real-world economic behavior?

In real-world economics, the theorem helps explain and predict individuals’ choices in situations involving risk, such as in financial investments, insurance markets, and gambling. It provides a theoretical foundation for the expected utility hypothesis, which asserts that individuals choose among risky projects to maximize their expected utility. This concept is applied in portfolio theory, the pricing of insurance, and in evaluating the economic effectiveness of public policies under uncertainty.

What are the limitations of the Von Neumann–Morgenstern utility theorem?

While influential, the theorem is not without critics. One limitation is its assumption of rational behavior, which may not always hold in real-world decision-making due to cognitive biases, lack of information, or other psychological factors. Additionally, the axioms of the theorem, especially the independence axiom, have been subjected to empirical scrutiny and found wanting in certain situations, as demonstrated in the Allais paradox and Ellsberg paradox. These limitations have prompted the development of alternative models that attempt to accommodate observed deviations from the theorem’s predictions about rational behavior.

Conclusion

The Von Neumann–Morgenstern utility theorem remains a fundamental pillar of economic theory, providing a mathematical basis for understanding choices under uncertainty. Despite its limitations, the theorem’s influence pervades economic analysis, decision theory, and beyond, highlighting its enduring relevance and utility in navigating the complexities of human decision-making.