Expected Utility Theory and the Von Neumann–Morgenstern Framework
When faced with risky projects or uncertain outcomes, how do we determine the best course of action? Expected utility theory provides a mathematical framework for analyzing choices among risky options with multiple, and sometimes multidimensional, outcomes. Rather than simply looking at the average monetary value, this theory focuses on the "utility"—the subjective value or satisfaction—derived from those outcomes.
The Evolution of Utility Theory
The foundations of this theory were laid in response to the St. Petersburg paradox, first proposed by Nicholas Bernoulli in 1713. The paradox highlighted a discrepancy between the mathematical expectation of a game and the amount a rational person would actually pay to play it. In 1728, Gabriel Cramer suggested using the expectation of a square-root utility function of money to address this.
By 1738, Daniel Bernoulli solved the paradox by arguing that decision-makers exhibit risk aversion. He proposed a logarithmic cardinal utility function, suggesting that the value of money decreases as one acquires more of it. Modern 21st-century analysis of international survey data supports this, showing that if utility represents happiness (as in utilitarianism), it is indeed proportional to the logarithm of income.
The theory reached a new milestone when John von Neumann and Oskar Morgenstern integrated the assumption of expected utility maximization into their formulation of game theory, providing a rigorous basis for strategic decision-making.
Calculating Expected Utility
To find the expected utility (EU), one calculates the probability-weighted average of the utility derived from every possible outcome. For a choice with two possible outcomes, z and y, the formula is expressed as:
EU = Pr(z) ⋅ u(Value(z)) + Pr(y) ⋅ u(Value(y))
In this equation, Pr represents the probability of the outcome occurring, and u(Value) represents the utility of that specific outcome's value.
The Von Neumann–Morgenstern Framework
Von Neumann and Morgenstern expanded this theory to handle situations where outcomes are not certain but are associated with specific probabilities. They used the concept of a lottery to represent these choices. A simple lottery involving options A and B with probability p and 1 − p is written as a linear combination:
L = pA + (1 − p)B
For more complex lotteries with many options, the formula is L = ∑ piAi, where the sum of all probabilities (∑ pi) equals 1.
The Four Axioms of Preference
Von Neumann and Morgenstern demonstrated that if an agent's preferences follow four specific axioms, their desirability for any lottery can be computed as a linear combination of the utilities of its parts. These axioms apply to "simple lotteries" (those with only two options) and use the notation B ⪯ A to indicate that A is weakly preferred to B.
- Completeness: For any two lotteries L and M, either L ⪯ M, M ⪯ L, or both (meaning they are equally desirable).
- Transitivity: If L ⪯ M and M ⪯ N, then L ⪯ N.
- Convexity/Continuity (Archimedean property): If L ⪯ M ⪯ N, there exists a probability p (between 0 and 1) such that a lottery combining L and N (pL + (1 − p)N) is equally desirable as M.
- Independence: For any three lotteries L, M, and N, L ⪯ M if and only if pL + (1 − p)N ⪯ pM + (1 − p)N. This means the preference between L and M should not change if they are both combined with the same third option N.
While these axioms allow for the creation of a formal utility function (u : X → R) that assigns real numbers to outcomes, the independence axiom is the most frequently discarded or relaxed in generalized expected utility theories.
Key Facts
- Core Purpose: Analyzes choices among risky projects with multiple outcomes.
- Historical Root: Developed to solve the St. Petersburg paradox.
- Key Contributors: Nicholas Bernoulli, Daniel Bernoulli, Gabriel Cramer, John von Neumann, and Oskar Morgenstern.
- Risk Aversion: The concept that the utility of money is logarithmic rather than linear.
- Mathematical Basis: Uses probability-weighted averages of utility to determine the most desirable choice.
- The Four Axioms: Completeness, Transitivity, Convexity/Continuity, and Independence.
| Component | Description | Key Significance |
|---|---|---|
| Utility Function | A function mapping outcomes to real numbers | Represents an agent's subjective preferences |
| Lottery | A linear combination of outcomes and probabilities | Models uncertainty in decision-making |
| Risk Aversion | Preference for a certain outcome over a risky one with equal expected value | Explains the St. Petersburg paradox |
| Independence Axiom | Preference between two options is independent of a third common option | Most commonly relaxed in advanced theories |
Frequently Asked Questions
What is the St. Petersburg paradox?
It is a problem first proposed by Nicholas Bernoulli in 1713 that showed a conflict between the infinite mathematical expectation of a specific game and the finite amount people are actually willing to pay to play it.
How did Daniel Bernoulli solve the paradox?
Daniel Bernoulli argued that the paradox is resolved if we assume decision-makers are risk-averse and that the utility of money follows a logarithmic function rather than a linear one.
What is a "simple lottery" in this context?
In the von Neumann–Morgenstern framework, a simple lottery is a choice that consists of exactly two possible options, each with an associated probability.
Why is the independence axiom often discarded?
The independence axiom is the most restrictive of the four assumptions; many real-world decision-making behaviors do not strictly follow it, leading to the development of generalized expected utility theories.
How does expected utility differ from expected value?
Expected value calculates the average monetary outcome, whereas expected utility calculates the average subjective value (utility), accounting for factors like risk aversion.