Chapter 2: Expected Utility and Risk Preferences#

Imagine you are offered a coin toss: heads you win 40 right now? The answer tells us a lot about your attitude toward risk. This chapter builds a simple way to think about risky choices — the expected utility model — and shows how being cautious, brave, or something in between shapes our decisions when the outcomes are uncertain.

The Big Picture#

Every day we make choices with uncertain outcomes. Should I buy insurance, invest in a new business, or stay in a safe job? The expected utility framework helps us understand not just what we decide, but why we decide it — whether we lean cautious, brave, or somewhere in between. It starts with a few simple rules of consistent choice and leads to a clear mathematical picture of “rational” decision‑making under risk. Even when real people don’t follow the model (as we will see later), knowing its basics is crucial — it’s the standard we compare real behaviour against.

Expected Utility: Putting Numbers on Happiness#

When an outcome is risky, we cannot simply compare the amounts of money involved, because a dollar gained in a good state feels different from a dollar lost in a bad state. The key insight is that people do not value money directly; they value the utility, or satisfaction, that money brings.

Suppose a lottery gives you outcomes with probabilities (each and ). The expected utility of that lottery is

where is your personal utility function — a scale that turns money into your own satisfaction score. According to the model, you choose the option with the highest expected utility.

But why should preferences follow this weighted‑average formula? The answer lies in four basic requirements for consistent choice, often called axioms. You do not need to memorise their names, but understanding the ideas will make the whole theory feel natural.

1. Completeness: given any two gambles, you can always say which you prefer or that you are indifferent. No shrugging “I don’t know” forever.
2. Transitivity: if you prefer A over B and B over C, then you must prefer A over C. Your tastes do not cycle.
3. Continuity: if A is better than B and B is better than C, there is some probability mixture of A and C that you find exactly as good as B. Intuitively, a small chance of a great prize can be traded off against a large chance of a mediocre one.
4. Independence: your preference between two lotteries should not change if you mix each of them with the same third lottery in identical proportions. If you like apples more than bananas, that does not change when I offer you a fruit salad that contains the same extra ingredient in both cases.

If your choices obey these rules, they act as if you are maximizing expected utility. The utility function can be stretched or shifted — adding a constant or multiplying by a positive number leaves the ranking of lotteries unchanged. So the exact happiness numbers don’t matter; only the comparisons do.

Expected utility: The sum of the utilities of each possible outcome, weighted by the probability that outcome occurs. A decision‑maker who follows the axioms will always pick the gamble with the highest expected utility.

📝 Section Recap: Expected utility theory turns risky choices into a simple weighted‑average rule, supported by a few logical conditions on preferences. Once we accept those conditions, the only thing that distinguishes one person from another is the shape of their utility function.

Risk Aversion and the Shape of Utility#

Here is where personality enters the picture. Suppose your utility function for money is . You are offered a coin toss that pays either 0, each with 50% probability. The expected monetary value is $50. The expected utility is

What sure amount of money would give you exactly the same utility? Solving gives . So you would be indifferent between the gamble and 50. You would happily swap the gamble for any sure amount above $25. In other words, you are risk‑averse: you value the gamble less than its expected monetary value.

The root of risk aversion is diminishing marginal utility — each extra dollar brings less extra happiness as you get richer. A concave utility function is one that gets flatter as wealth grows. If you draw a straight line between two points on a concave curve, the line lies below the curve. This means the utility of the average dollar amount is greater than the average utility of the possible outcomes. The gap is the pain of risk.

Risk‑averse: Preferring a sure thing over a gamble that has the same expected dollar value. Formally, for any gamble that isn’t a sure thing — a property that follows from Jensen’s inequality for a concave .

A risk‑neutral person cares only about expected monetary value: is a straight line, so . A risk‑loving person would pay a premium to gamble: their utility is convex, bending upwards.

Concave utility: A utility function where the extra happiness from an additional dollar shrinks as you get richer. The curve looks like a flattened hill. Concavity is the mathematical signature of risk aversion.

📝 Section Recap: Risk aversion is not a mysterious trait; it falls directly out of a utility function that displays diminishing sensitivity to gains. The more concave the function, the more a person dislikes variability in outcomes.

Certainty Equivalent and Risk Premium#

To measure how much someone dislikes a particular gamble, we can ask for its certainty equivalent (CE) — the guaranteed amount of money that makes you exactly as happy as the gamble. The difference between the gamble’s expected value and its certainty equivalent is the risk premium:

If you are risk‑averse, the risk premium is positive. For our coin toss, and , so the risk premium is . That is the “insurance value” — the amount you would willingly sacrifice to avoid the risk altogether.

A larger risk premium signals stronger risk aversion. For the same person, the risk premium typically grows when the stakes become larger or the probabilities shift toward more extreme outcomes.

Let’s see a more concrete example. Your utility is . You face a 50–50 gamble between and . Expected value = . Expected utility:

Certainty equivalent: . Risk premium . So you would give up $13.40 from the average just to walk away with a sure thing.

Certainty equivalent (CE): The sure sum of money that gives the same utility as a risky gamble. It is the cash‑value, in your personal happiness terms, of the uncertainty.

Risk premium: The difference between a gamble’s expected monetary value and its certainty equivalent. It measures the “cost” of risk to the decision‑maker.

📝 Section Recap: By translating gambles into their certainty equivalents, we can gauge the intensity of risk aversion in concrete dollar terms. The risk premium is the simplest summary of how much a person would pay to eliminate uncertainty.

The Probability Triangle: A Picture of Preferences#

When outcomes can take only three values, we can draw a wonderfully simple diagram of risky choices. This is the probability triangle (often called the Marschak‑Machina triangle). It captures all lotteries over outcomes (say, 50, (p_1, p_3)p_1p_3p_2 = 1 - p_1 - p_3$.

• The bottom‑left corner corresponds to the sure thing of getting the middle outcome with certainty.
• The bottom‑right corner is the worst outcome for sure.
• The top corner is the best outcome for sure.

Lines of constant expected value are straight lines with slope equal to . But expected utility indifference curves — sets of lotteries that give the same — are also straight, parallel lines. Why? Because , which after substituting , becomes

This is the equation of a straight line in the space. The slope of any indifference line is , which is constant for a given utility function.

For a risk‑neutral person, is linear, so indifference lines exactly match the iso‑expected‑value lines. For a risk‑averse person, is concave, making the indifference lines steeper. This means that to accept a higher chance of the worst outcome, a risk‑averse person needs a larger increase in the chance of the best outcome to keep utility constant, compared to a risk‑neutral person. In the triangle, their indifference curves tilt upward more sharply.

The triangle is a powerful thinking tool. It lets us see how different models of decision‑making (which we will meet later) produce fanning‑out patterns or non‑linear indifference curves that classic expected utility cannot produce. But for pure expected utility, the straight, parallel indifference curves are the unmistakable fingerprint of the theory.

Probability triangle: A diagram where each point represents a lottery over three fixed outcomes, with the probabilities of the worst and best outcomes as axes. It reveals that expected utility implies straight, parallel indifference curves.

📝 Section Recap: The probability triangle translates abstract preferences into a visual map. The straight, parallel indifference curves of expected utility are a stark prediction that we can test with simple experiments.

Measuring Risk Attitudes: The Arrow‑Pratt Measures#

Knowing that someone is risk‑averse is a start, but we often need to compare the degree of aversion across people or across wealth levels. The economists Kenneth Arrow and John Pratt independently developed handy local measures that use the curvature of the utility function.

The Arrow‑Pratt measure of absolute risk aversion (ARA) is defined as

The minus sign makes it positive for a risk‑averse person (since is negative). We divide by so the measure stays the same even if you stretch or shift the utility scale. ARA tells us how much a person dislikes a small, additive risk at a given wealth level. A larger ARA means the person will demand a bigger risk premium for the same small gamble.

For example, if , then , , so . Absolute risk aversion falls as wealth rises — rich people get more comfortable with a fixed‑size gamble. This seems plausible.

If instead (with ), then , , so , a constant. This is called constant absolute risk aversion (CARA). Such a person would pay the same dollar premium to avoid a given gamble regardless of whether they are rich or poor.

Often we care about risks that are proportional to wealth, like losing 10% of your savings. For that, we use relative risk aversion (RRA):

If , , constant. This is the constant relative risk aversion (CRRA) family, which also includes for , where . These functions are the workhorses of many economic models because they capture the idea that people’s attitude toward percentage gambles is stable across different wealth scales.

To see the measures in action, imagine two people. Anna has ; Ben has . At a wealth of , Anna’s ARA = , while Ben’s ARA = . Ben is more risk‑averse locally, meaning he would pay a larger premium to avoid a small additive bet at that wealth level. The Arrow‑Pratt formulas let us distill that difference into a single number.

Arrow‑Pratt absolute risk aversion (ARA): A number that tells us how much a person dislikes small, fixed‑size gambles at their current wealth. Larger ARA = more cautious about fixed‑dollar risks.

Relative risk aversion (RRA): ARA multiplied by wealth; it measures caution about risks that are proportional to wealth. Constant RRA implies that a person’s attitude toward a 10% loss is the same whether they have 1,000,000.

📝 Section Recap: Arrow‑Pratt measures give us a precise, comparable way to talk about how much risk someone is willing to bear. They distill the curvature of utility into a single local number, either for additive or proportional gambles.

Summary#

We started with a simple coin toss and ended up with a whole toolkit for thinking about risky choices. Expected utility theory gives a clear standard of consistency: if your choices follow a few logical rules, you can be described as if you’re trying to maximize a weighted sum of happiness points. Once we accept that, the shape of your utility function tells us everything about your taste for risk. A concave function means caution; we can measure that caution with the certainty equivalent, the risk premium, and the Arrow‑Pratt measures. The probability triangle gives a visual test of the theory — and, as we’ll see later, a way to spot where real people depart from pure logic. This foundation is not just an abstract exercise; it’s the lens for understanding insurance, investing, and all the everyday gambles that shape our lives.