Table of Contents
What is Expected Utility?
Expected utility is a foundational concept in economics and decision theory that helps quantify the value of uncertain outcomes. Unlike simple expected value calculations that treat all dollars equally, expected utility accounts for the psychological reality that people's satisfaction (utility) from money isn't linear.
The theory was developed by Daniel Bernoulli in 1738 to resolve the St. Petersburg Paradox - a theoretical lottery with infinite expected value that no rational person would pay unlimited money to play. Bernoulli proposed that people evaluate outcomes based on utility (satisfaction) rather than raw monetary value.
Expected Utility Formula
The basic expected utility formula is:
For a single outcome:
Step-by-Step Calculation
- Determine the probability of each possible outcome
- Calculate the utility of each outcome using your chosen utility function
- Multiply each probability by its corresponding utility
- Sum all the weighted utilities to get expected utility
Example Calculation
Consider a gamble with:
- 50% chance of winning $100
- 50% chance of winning $0
Using square root utility function U(x) = √x:
- U($100) = √100 = 10
- U($0) = √0 = 0
- Expected Utility = 0.5 × 10 + 0.5 × 0 = 5 utils
Types of Utility Functions
Different utility functions model different attitudes toward risk:
1. Linear Utility (Risk Neutral)
Utility equals monetary value. A risk-neutral person only cares about expected value and is indifferent between a guaranteed amount and a gamble with the same expected value.
2. Square Root Utility (Risk Averse)
Exhibits diminishing marginal utility - each additional dollar provides less satisfaction than the previous one. This models typical human behavior where people prefer certainty.
3. Logarithmic Utility (Very Risk Averse)
Shows stronger diminishing marginal utility. Used in the Kelly Criterion for optimal bet sizing. Requires positive values only.
4. Quadratic Utility (Risk Seeking)
Exhibits increasing marginal utility - each additional dollar provides MORE satisfaction. Models risk-seeking behavior where people prefer gambles over certain outcomes.
Understanding Risk Attitudes
Risk Aversion
Most people exhibit risk aversion, especially for large stakes. A risk-averse person:
- Has a concave utility function (curves downward)
- Prefers a certain outcome over a risky one with the same expected value
- Willing to pay a "risk premium" to avoid uncertainty
- Experiences diminishing marginal utility of wealth
Risk Neutrality
A risk-neutral person:
- Has a linear utility function
- Cares only about expected value
- Indifferent between certain and risky options with same EV
- Often assumed in corporate finance decisions
Risk Seeking
A risk-seeking person:
- Has a convex utility function (curves upward)
- Prefers gambles over certain outcomes
- Willing to accept lower expected value for the chance of a big win
- Common in lottery players and problem gamblers
Certainty Equivalent and Risk Premium
Certainty Equivalent
The certainty equivalent is the guaranteed amount that provides the same utility as a risky gamble. It answers: "What certain amount would make you indifferent to this gamble?"
For a risk-averse person, the certainty equivalent is less than the expected value. For example, someone might be willing to accept a guaranteed $40 instead of a 50/50 chance at $100 or $0 (expected value = $50).
Risk Premium
The risk premium is the difference between expected value and certainty equivalent:
This represents the "cost" of risk - how much expected value someone is willing to give up to eliminate uncertainty.
Real-World Applications
Insurance Decisions
Expected utility explains why people buy insurance even though it has negative expected value (premiums exceed expected payouts). The utility loss from a catastrophic uninsured event outweighs the small utility cost of premiums.
Investment Portfolio Selection
Investors use utility functions to balance expected returns against risk. A risk-averse investor might accept lower expected returns for a more stable portfolio.
Medical Decision Making
Patients and doctors use expected utility (often implicitly) when choosing between treatments with different risk profiles.
Business Strategy
Companies evaluate projects not just by expected NPV but by risk-adjusted measures that account for the utility implications of potential outcomes.
Worked Examples
Example 1: Insurance Decision
Without Insurance:
- 99% chance: Keep $200,000 (utility = √200,000 = 447.21)
- 1% chance: Have $50,000 (utility = √50,000 = 223.61)
- Expected Utility = 0.99 × 447.21 + 0.01 × 223.61 = 444.98
With Insurance:
- 100% chance: Have $198,000 (utility = √198,000 = 445.0)
Decision: Buy insurance! Expected utility is slightly higher (445.0 vs 444.98), even though expected value is lower ($198,000 vs $198,500).
Example 2: Investment Choice
A) Guaranteed $10,000 return
B) 50% chance of $25,000, 50% chance of $0
Expected Values:
- Option A: $10,000
- Option B: $12,500
Expected Utilities (using √x):
- Option A: √10,000 = 100
- Option B: 0.5 × √25,000 + 0.5 × 0 = 79.06
Decision: A risk-averse person chooses Option A despite lower expected value, because it has higher expected utility.
Frequently Asked Questions
Can expected utility be negative?
Yes, expected utility can be negative when outcomes have negative utility values. This occurs when dealing with losses or when using utility functions defined for negative numbers. For example, losing $500 with a risk-averse utility function might yield a utility of -22.36 (using a modified square root function that handles negatives).
How do I calculate expected utility with three steps?
- Determine the probability of each outcome occurring
- Calculate the utility value for each outcome using your chosen utility function
- Multiply each probability by its utility and sum the results
What utility function should I use?
The choice depends on your risk preferences and the context. For most personal financial decisions, the square root or logarithmic functions work well as they capture typical risk aversion. For corporate decisions where diversification is possible, linear (risk-neutral) might be appropriate.
How does expected utility differ from expected value?
Expected value simply averages monetary outcomes weighted by probabilities. Expected utility first transforms monetary values into utility (satisfaction) units, then averages. This captures the psychological reality that $1,000,000 isn't twice as satisfying as $500,000.
Why is the certainty equivalent important?
The certainty equivalent translates utility back into monetary terms, making it easier to compare options and communicate decisions. It tells you the guaranteed amount that's equivalent (in terms of satisfaction) to a risky gamble.
What are the limitations of expected utility theory?
The theory assumes people are perfectly rational and consistent in their preferences. Real behavior often violates these assumptions (e.g., the Allais Paradox). Prospect theory offers an alternative that better describes actual human decision-making, including loss aversion and probability weighting.