Title: Expected Utility — What It Is, How It Works, and How to Use It in Decisions Under Uncertainty
Key takeaways
– Expected utility (EU) is the probability-weighted average of an agent’s utility across possible outcomes; it’s the decision rule of expected-utility theory for choices under uncertainty.
– EU separates monetary amounts from their subjective value (utility), allowing for risk aversion, diminishing marginal utility, and tradeoffs that pure expected-value calculations miss.
– The concept originates with Daniel Bernoulli’s solution to the St. Petersburg paradox; it remains the baseline model for many economic and finance problems but has known empirical and theoretical limitations (e.g., Rabin’s critique).
– Practical use requires choosing a utility function, assigning probabilities, computing expected utilities, and doing sensitivity checks. Alternatives such as prospect theory should be considered when behavior departs from EU prescriptions.
What expected utility is
– Definition: Expected utility is the sum over all possible outcomes of (probability of outcome) × (utility of outcome). Symbolically:
EU = Σ p(i) · u(x(i))
where p(i) is the probability of outcome i, x(i) is the monetary or consumption outcome, and u(·) is the decision maker’s utility function.
– Rationale: People often care about the satisfaction or welfare associated with money (utility), not raw dollars. Because marginal utility typically falls as wealth rises, the same dollar change matters more to a poorer person than to a richer person. Expected utility formalizes that intuition when choices involve risk.
Historical background
– Daniel Bernoulli (1738) introduced the idea when resolving the St. Petersburg paradox: a gamble with infinite expected monetary value produced intuitively small prices people would pay. Bernoulli proposed that people maximize expected utility (e.g., logarithmic utility), not expected money, which resolves the paradox.
– Later formalizations and axioms (von Neumann–Morgenstern) gave expected-utility theory a rigorous foundation for rational choice under risk.
– Critiques and refinements: Empirical findings and theoretical observations (e.g., Matthew Rabin’s calibration objection) show that simple expected-utility specifications can be implausible for modest stakes, spurring alternative models (prospect theory, rank-dependent utility, and models for ambiguity aversion).
Expected utility vs. marginal utility (and expected value)
– Expected value (EV) = Σ p(i) · x(i) — the probability-weighted average of dollar outcomes.
– Expected utility = Σ p(i) · u(x(i)) — the probability-weighted average of subjective value.
– Marginal utility: u′(x) is the extra utility from one more unit of wealth; diminishing marginal utility (u′′(x) W0).
– If utility is logarithmic u(W) = ln(W):
EU = 0.9·ln(1,200) + 0.1·ln(900) ≈ 0.9·7.0901 + 0.1·6.8024 ≈ 7.0613.
Baseline utility u(W0) = ln(1,000) ≈ 6.9078.
Since EU > baseline utility, a person with log utility would take the gamble. This example shows how EU can confirm or reject risky options depending on the utility shape.
Practical steps to apply expected utility in decision-making
1. Define the decision and enumerate outcomes
– List all materially different outcomes that could result from each alternative.
2. Estimate probabilities
– Use objective frequencies, historical data, models, or subjective assessments. Be explicit about uncertainty and assumptions.
3. Choose a utility function that reflects preferences
– Common parametric families:
– Log utility: u(w) = ln(w) (implies decreasing absolute risk aversion)
– CRRA (constant relative risk aversion): u(w) = w^(1−γ)/(1−γ) for γ ≠ 1 (γ is risk-aversion parameter)
– CARA (constant absolute risk aversion), exponential utility: u(w) = −exp(−αw)
– Calibration: choose parameters (γ or α) based on observed behavior, stated risk aversion, or normative judgments.
4. Compute expected utilities
– For each action: EU(action) = Σ p(i) u(x(i)). Compare EUs across actions.
5. Apply the decision rule
– Choose the action with the highest EU (or reject if no action improves EU relative to a status quo).
6. Conduct sensitivity analysis
– Vary probabilities and utility parameters to test robustness. Small errors in probabilities or the utility specification can change decisions.
7. Consider constraints, multi-period effects, and liquidity
– If outcomes affect future options, incorporate dynamic utility or discounting. Include liquidity constraints and survival thresholds explicitly (large losses may be unacceptable regardless of EU).
8. Check for behavioral anomalies
– If observed or expected behavior deviates from EU (e.g., loss aversion, reference dependence), consider alternative models (prospect theory, rank-dependent utility).
9. Document assumptions and update with evidence
– Keep records of probability estimates and utility parameters; update as new data appear.
Applications and examples
– Insurance: EU explains why risk-averse individuals buy insurance even if expected monetary value is negative — insurance reduces the chance of catastrophic utility loss.
– Investment choice: Risk-return tradeoffs can be framed by EU with a utility that captures risk aversion; mean-variance analysis is a quadratic approximation to EU for small risks.
– Business decisions: Project evaluations under uncertainty can use expected utility to reflect managerial or firm-level risk preferences.
– Public policy: Social welfare functions can be treated as utility over income or consumption aggregated across people (with ethical implications for weighting).
Limitations and when to use alternatives
– Calibration problems: Rabin’s critique argues that the degree of risk aversion implied by some utility functions over small stakes leads to implausible behavior with large stakes — indicating a mismatch between simple EU models and observed choices.
– Behavioral departures: People display loss aversion, probability weighting, and reference dependence (prospect theory phenomena) that EU does not capture.
– Ambiguity and model uncertainty: EU presumes known probabilities; when probabilities are ambiguous, alternative frameworks (maxmin expected utility, ambiguity-averse models) may be preferable.
– Practical rule: Use expected utility when preferences are reasonably consistent with risk-averse, probabilistic decision-making, when probabilities are credible, and when you can reasonably specify or calibrate a utility function. Otherwise, complement EU with robustness checks and alternative decision models.
Quick checklist for practitioners
– Have you enumerated all meaningful outcomes?
– Are the probability estimates defensible? Have you tested alternate estimates?
– Is your chosen utility function appropriate and calibrated?
– Did you compute EU for each option and do a sensitivity analysis?
– Have you considered liquidity, ruin events, and multi-period dynamics?
– If behavior or stakes look inconsistent with EU, have you considered prospect theory or models of ambiguity?
Further reading and sources
– Investopedia: “Expected Utility” (overview and examples) — https://www.investopedia.com/terms/e/expectedutility.asp
– Daniel Bernoulli, “Specimen of a New Theory on the Measurement of Risk” (often cited as Bernoulli’s 1738 solution to the St. Petersburg paradox).
– Matthew Rabin, “Risk Aversion and Expected-Utility Theory: A Calibration Theorem” (critique of standard EU calibration).
– UC Berkeley materials on “Risk Aversion and Expected-Utility Theory: A Calibration Theorem” (discussion/lecture notes).
Bottom line
Expected utility is a foundational tool for making choices under risk: it replaces raw dollars with subjective value and incorporates risk aversion. Use it by specifying outcomes, probabilities, and a utility function, computing EU, and choosing the option with the highest EU. But also test robustness, be aware of empirical departures, and consider alternative models when probabilities are ambiguous or observed behavior departs systematically from EU predictions.