Expected Utility: Understanding, Calculating, and Applying It

Definition

Expected utility is a measure of the anticipated satisfaction (utility) an individual or group assigns to uncertain outcomes. Rather than valuing outcomes by their monetary amounts, expected utility weights each possible outcome by its probability and by how much utility that outcome provides.

Calculation

The expected utility (EU) of a set of possible outcomes x_i with probabilities p_i is:
EU = Σ p_i · u(x_i)
where u(x) is the decision-maker’s utility function (captures preferences and risk attitude). Because utility is typically concave in wealth, additional money usually yields diminishing marginal utility.

Origins: Bernoulli and the St. Petersburg Paradox

Daniel Bernoulli introduced expected utility to resolve the St. Petersburg Paradox, a theoretical lottery with an infinite expected monetary payoff. Bernoulli showed that people value outcomes by utility, not raw dollars, which explains why they would pay a finite amount to play despite an infinite expected monetary value.

How Expected Utility Shapes Decision-Making

• Decisions under uncertainty: Individuals compare the expected utility of available actions and choose the action with the highest EU.
• Risk attitudes: A risk-averse person has a concave utility function and prefers a sure amount with the same EU as a risky prospect. A risk-seeking person has a convex utility function.
• Real-world behavior: Expected utility explains why people buy insurance (trade a small certain loss for avoiding a large, low-probability loss) and why wealth changes influence willingness to take risks.

Relation to Marginal Utility

Expected utility is closely tied to marginal utility of wealth. Because marginal utility typically decreases as wealth rises:
– Smaller increases in wealth yield less utility for richer people, which can make them more willing to sell a risky claim for a large sure amount.
– Conversely, poorer individuals may prefer the risky prospect if the potential gain substantially increases their utility.

Examples

• Lottery ticket: Buying a ticket involves a small cost vs. a small probability of a large payoff. The decision depends on the EU calculation and the buyer’s utility function.
• Selling a ticket: If a person holding a ticket with an expected payout of $1,000,000 is offered $500,000, a less wealthy holder might sell (safer, large marginal gain), while a wealthy person might keep it (additional $500,000 yields relatively little extra utility).
• Insurance: Paying a premium reduces expected monetary value but can increase expected utility by protecting against large losses that would sharply reduce utility.

Critiques and Limits

• Empirical concerns: Some research (e.g., Matthew Rabin’s work) argues expected utility theory can give unrealistic predictions for modest stakes unless utility functions are implausibly concave.
• Behavioral deviations: Real decision-makers exhibit biases and heuristics (loss aversion, probability weighting) that can diverge from the EU model.

Key Takeaways

• Expected utility evaluates uncertain choices by averaging utilities of outcomes weighted by probabilities (EU = Σ p_i · u(x_i)).
• It explains risk-averse behavior, insurance purchases, and why money’s subjective value differs from its nominal amount.
• The theory is foundational in economics and finance but has known empirical and behavioral limitations that motivate alternative models.