Expected Utility and Risk Preference
Expected Utility and Risk Preferences · Why choices differ even when expected values are the same
Why do people make different choices even when the expected value is the same? To make the best choice under uncertainty, you have to consider not only the possible outcomes but also how much you value them. Getting $100,000 for certain and getting $200,000 with 50% probability or nothing with 50% probability have the same expected amount, but not everyone makes the same choice. Someone who values a steady income may prefer the sure amount, while someone willing to take risks for a bigger reward may prefer the lottery-like option. The concepts that explain this difference are expected utility and risk preference.
How is utility different from expected value?
Expected value is the probability-weighted average of the amounts or quantities of the outcomes, while utility expresses how valuable an outcome is to the decision maker. Doubling your money doesn’t double your satisfaction or subjective value. An extra $20,000 can be worth something different to a person who already has plenty of assets than to a person who is short on living expenses.
| Concept | What it evaluates | Question |
|---|---|---|
| Expected value | The amount or quantity of the outcome | How much do you get on average? |
| Utility | The value the outcome provides | How much do you prefer that outcome? |
| Expected utility | The probability-weighted average of utility | Taking probabilities into account, which choice do you prefer? |
EU(A): expected utility of alternative A · pi: probability of outcome i · xi: amount or state of the outcome · u(xi): utility assigned to that outcome
Choosing the alternative with the higher expected utility is the basic rule of expected utility theory. To apply it, though, you need an assumption about how utility is valued, that is, about the utility function.
Three types of risk preference
Given the same expected amount, prefers a certain outcome to an uncertain one.
Example: prefers a fixed interest rate to an investment with an uncertain return
Chooses based only on the expected amount and places no separate value on variability itself.
Example: indifferent between options with the same expected return
Under certain conditions, prefers an uncertain outcome to a certain one.
Example: prefers a lottery-like option with a chance of a big reward
Risk preference isn’t fixed by personality alone. It can change with the size of the amount, whether there’s a loss, current assets, and the context of the choice.
Expressing risk preference with a utility function
A utility function describes the relationship between the size of an outcome and the utility assigned to it. A typical example is the following function.
With this function, utility rises as the amount rises, but the gain in utility from adding the same amount gets smaller and smaller. This is called diminishing marginal utility.
In Figure 1, going from $2,000 to $4,000 raises utility by 0.586, but going from $20,000 to $22,000 raises it by only 0.218. Even though both are the same $2,000 increase, the first one is valued about 2.7 times as much. A concave utility function like this, rising with a flattening slope, is the standard way to express risk-averse preferences.
In Figure 2, the expected utility of a 50/50 lottery paying $0 or $200,000 (black dot) is 0.5 for all three functions. The utility of a sure $100,000 (colored dot) is 0.71 for the concave function, higher than the lottery; 0.5 for the straight line, the same; and 0.25 for the convex function, lower than the lottery. If the curve lies above the dashed line joining the two endpoints, that’s risk aversion; if below, risk seeking.
| Shape of the utility function | Usual meaning |
|---|---|
| Concave function | Risk aversion |
| Linear function | Risk neutrality |
| Convex function | Risk seeking |
This classification is based on the shape the utility function takes over the relevant range of outcomes. A large variance (spread) of outcomes doesn’t by itself determine risk preference; you have to consider the utility function and the distribution of outcomes together.
A worked comparison: a sure amount versus a lottery
| Alternative | Calculation | Expected amount |
|---|---|---|
| A | A sure $100,000 | $100,000 |
| B | 0.5 × $200,000 + 0.5 × $0 | $100,000 |
The two alternatives have the same expected amount, so the amount alone can’t decide it. Assume the utility function is u(x) = √x, and express amounts in thousands of dollars.
So a decision maker with this utility function prefers alternative A. The expected amounts are the same, but because the utility function is risk averse, the sure amount comes out with higher utility. With a different utility function, the preference could change.
Certainty equivalent and risk premium
The certainty equivalent (CE) is the sure amount that gives the same utility as an uncertain alternative, and the risk premium is the difference between the uncertain alternative’s expected amount and its certainty equivalent.
√CE ≈ 7.07 → CE = 50 ($ thousands) · Risk premium = 100 − 50 = 50 ($ thousands)
In Figure 3, lottery B’s expected utility of 7.07 sits on the dashed line joining the two endpoints, at the expected amount of 100. Find the same height (7.07) on the utility curve and drop down, and you get a certainty equivalent of $50,000. In other words, to this decision maker lottery B is worth the same as a sure $50,000, which means they’re willing to give up half the expected amount ($50,000) to avoid the risk.
Figure 4 shows how the certainty equivalent of the same lottery changes as you vary the exponent a of the utility function u(x) = xa. With a = 1 (risk neutral), the certainty equivalent equals the expected value of $100,000, and the smaller a is (the more concave), the smaller the certainty equivalent and the larger the risk premium. This example assumes a utility function and a range of amounts; a real person’s certainty equivalent or risk premium can only be known by measuring or estimating their preferences.
How is it used in real decisions?
Evaluates not only expected returns but also the chance of loss and the investor’s risk tolerance.
Analyzes the choice to pay a premium to reduce an uncertain large loss.
Among projects with similar expected profits, considers the risk of loss and how much the company can absorb.
Considers treatment effects, side effects, and the outcomes the patient cares about.
Insurance is a classic example that risk aversion can explain. In Figure 5, if there’s a 10% chance of losing $128,000 out of assets of $200,000, the expected loss is $12,800. But for a person with √x utility, the certainty equivalent of going uninsured is $184,320, so they’re willing to pay a premium of up to $15,680. That’s why it’s rational to buy insurance even at a premium above the expected loss, and it’s also why insurers can sell policies at premiums higher than the expected loss.
In practice, it’s hard to know the utility function exactly. In that case, you can compare several risk scenarios, set a loss limit or a minimum-return condition, or use sensitivity analysis to see how the conclusion changes as the assumptions change.
Limits of expected utility theory
Expected utility is a powerful framework for comparing uncertain choices systematically, but it doesn’t always predict what people actually choose. People can misunderstand probabilities, and they value the same amount differently depending on whether it’s a gain or a loss. For example, people who pick the sure option when facing gains are known to pick "a gamble that might avoid the loss" over "a sure loss" when facing losses. Current assets, experience, emotions, and how the choice is presented also affect judgment.
So use expected utility as a theoretical standard that explains the logic of decisions, but in real decisions, carefully evaluate realistic probabilities and outcome values and the decision maker’s preferences.
The two alternatives had the same expected amount of $100,000, but assuming the utility function u(x) = √x (in $ thousands), the utility of a sure $100,000 was 10.00 and the expected utility of the alternative paying $200,000 with 50% probability was 7.07. The latter’s certainty equivalent was $50,000, for a risk premium of $50,000, so a risk-averse decision maker prefers the sure alternative.
Even with the same expected value, choices split on how much you value the outcomes.
- Expected value and expected utility: Expected value is a probability-weighted average of amounts; expected utility is a probability-weighted average of the utilities assigned to outcomes.
- Risk preference: A concave utility function means risk aversion, a straight line means risk neutrality, and a convex one means risk seeking.
- Certainty equivalent: The sure amount that gives the same utility as an uncertain alternative; its difference from the expected amount is the risk premium.
- Applications: Choices like insurance, where you pay more than the expected loss, can be explained by risk aversion.
- Limits: People’s actual choices can differ from expected utility depending on gain · loss framing, how they perceive probabilities, and how the choice is presented.
Up next: 04 Multi-Criteria Decision Making. We’ll look at MCDM and the weighted-sum method, which compare alternatives on several criteria at once.