Foundations of Choice Under Uncertainty

Imagine someone offers you a coin flip: heads you win 40 right now? Most people would take the sure 50. Why? Because most of us dislike uncertainty. This chapter builds the core tools economists use to understand exactly that kind of decision — how rational people make choices when outcomes are not guaranteed.

The Big Picture#

Every day we make choices without knowing exactly what will happen. Investing in the stock market, buying insurance, choosing a career, even crossing the street — all involve uncertainty. This chapter introduces the standard framework for thinking about such decisions: expected utility theory. We will learn how to represent preferences over risky outcomes, what it really means to be "risk averse," and why a curved utility function captures our natural tendency to prefer a sure thing over a gamble with the same average payoff. By the end, you will have a precise language for describing attitudes toward risk, and you will see why this matters for everything from insurance markets to asset pricing.

Preferences Over Gambles#

Before we can talk about risk, we need a way to describe uncertain outcomes. Economists use the idea of a lottery or gamble — a set of possible outcomes, each with a probability attached.

Think of a lottery ticket. It might pay 0 with probability 0.5. We write this as a list of pairs: each outcome and its chance. More generally, a lottery gives outcome with probability , where all the probabilities are non-negative and sum to one:

The outcomes are usually amounts of money or consumption. The probabilities are known — we are not talking about uncertainty where you do not even know the odds. Here, the odds are clear.

Now, the central question: given a choice between two gambles, which one does a person prefer? We need a way to represent those preferences mathematically. We want a function that takes a lottery and spits out a single number — higher numbers for preferred lotteries — so we can rank them.

📝 Section Recap: A gamble lists possible dollar outcomes and their known probabilities. We need a single number to rank such risky choices, so we can compare them clearly.

Expected Utility: The Basic Idea#

The most widely used approach is expected utility. The idea is beautifully simple. Instead of evaluating the raw outcomes directly, you first pass each outcome through a utility function that measures how much satisfaction or "utility" that outcome gives you. Then you take the probability-weighted average of those utilities:

This number is the expected utility of the lottery. When comparing two lotteries, the one with higher expected utility is preferred.

Notice what this does not say. It does not say you simply compare the expected dollar value. It says you compare the expected utility of the dollars. The utility function transforms money into happiness, and only then do you compute the average. This transformation is the key to understanding risk attitudes.

Why is this so useful? It reduces a complicated choice under uncertainty to a simple number. We can compare any two lotteries by computing two numbers and seeing which is larger. It also separates two things: your beliefs about the odds (the 's) and your tastes over outcomes (the shape of ). This split is powerful.

Expected Utility: The probability-weighted average of the utilities of all possible outcomes of a gamble. It provides a single number that ranks lotteries according to a decision-maker's preferences.

📝 Section Recap: We rank gambles by the expected value of a utility function, not by expected money alone. This separates personal taste (the utility curve) from beliefs (the probabilities).

Risk Aversion and Concave Utility#

Here is where things get interesting. Why do people buy insurance? Why do they prefer a sure 100? The answer lies in the shape of .

Consider a simple gamble: you start with some wealth , and you are offered a fair coin flip that adds with probability 0.5 and subtracts with probability 0.5. The expected dollar gain is zero. If you decline this gamble, you are risk averse. You prefer to keep your current wealth for sure rather than take a fair gamble.

A person is risk averse if and only if their utility function is concave. A concave function curves downward — it bends toward the horizontal axis. What does that mean intuitively? It means that gaining an extra dollar adds less happiness than losing a dollar takes away. The joy of a win is smaller than the pain of an equal-sized loss.

Picture a concave utility function: it rises as wealth increases (more is better), but each additional dollar adds a little less extra utility. This is diminishing marginal utility. The slope of gets smaller as gets larger.

Now here is the critical link: for a concave function, the utility of the expected value of a gamble is greater than the expected utility of the gamble itself. In symbols:

for any random variable that is not constant (so there is real uncertainty). This inequality is called Jensen's inequality, and it is the mathematical heart of risk aversion.

Risk Aversion: A preference for a certain outcome over a gamble with the same expected value. This is the same as having a concave utility function.

Concave Utility Function: A utility function where the graph bends downward, so that . It implies diminishing marginal utility of wealth and generates risk-averse behavior.

📝 Section Recap: Risk aversion means preferring the sure average over the gamble. A concave utility curve captures this: gains give less extra happiness than equal losses take away. Jensen’s inequality links concavity to the preference for certainty.

Jensen's Inequality: The Formal Link#

Let us state Jensen's inequality clearly. If is a concave function and is a random variable, then:

with strict inequality when is not constant (i.e., there is genuine uncertainty). The left side is the utility you would get if you received the average payoff for sure. The right side is the utility you expect, on average, from the random gamble.

This inequality tells us that a risk-averse person strictly prefers receiving the expected value for sure over facing the gamble. That is exactly the definition of risk aversion.

Let us make this concrete. Suppose , a classic concave function. Your current wealth is . You are offered a gamble: win 21 with probability 0.5, lose 19 with probability 0.5. The expected dollar payoff is . So the expected wealth after the gamble is 101. The utility of the expected wealth is:

Now compute expected utility. The two possible wealth levels are (if you win) and (if you lose), each with probability 0.5:

Expected utility is 10.0, which is less than 10.0499. You would reject the gamble even though it has a positive expected dollar payoff of 1. The concavity of the square root function makes you value the downside loss more than the upside gain.

This example shows something deeper: a risk-averse person might reject even some gambles with positive expected value. They need to be paid extra to take on risk.

Jensen's Inequality: For a concave function , the function evaluated at the expected value exceeds the expected value of the function. This is the formal reason concave utility implies risk aversion.

📝 Section Recap: Jensen’s inequality says for concave , so a risk‑averse person strictly prefers the sure average over a gamble. It is the math behind the avoidance of fair bets.

Risk Neutrality and Risk Loving#

Not everyone is risk averse — or at least, the theory allows for other attitudes. If is linear, so that with , then the person is risk neutral. For a linear function, Jensen's inequality holds with equality:

The decision-maker cares only about the expected dollar payoff. A 50-50 gamble for 0 is exactly as good as $50 for sure. They ignore the spread of outcomes entirely.

If is convex (curving upward, ), the person is risk loving. They get a thrill from the gamble itself. For them,

They would strictly prefer a fair gamble over its expected value for sure. Someone who buys lottery tickets with negative expected value might be exhibiting risk-loving behavior — though there are other explanations we will encounter later.

In practice, economists overwhelmingly assume risk aversion for most financial decisions. The entire insurance industry exists because people are willing to pay to avoid risk. Asset prices contain risk premiums because investors demand extra reward for bearing uncertainty. So concave utility is the workhorse assumption.

📝 Section Recap: Risk aversion means preferring a sure thing over a fair gamble and is equivalent to having a concave utility function. Jensen's inequality formalizes this — for concave , the utility of the expected value exceeds expected utility. Linear utility gives risk neutrality, and convex utility gives risk-loving behavior.

The Axioms of Expected Utility Theory#

Why should preferences over gambles take the expected utility form? It is not just a convenient assumption. It can be derived from a set of basic principles — axioms — that describe consistent, rational choice under uncertainty. If your preferences satisfy these axioms, then they must be representable by an expected utility function.

The standard set of axioms comes from John von Neumann and Oskar Morgenstern. Here are the key axioms, explained plainly.

1. Completeness#

For any two lotteries and , you can always compare them. Either you prefer to , prefer to , or you are indifferent between them. There are no choices you simply cannot rank. This seems reasonable — if forced to choose, you can always say which you like better, even if it is a hard call.

2. Transitivity#

Preferences are consistent across comparisons. If you prefer to and to , then you must prefer to . Indifference works the same way: indifferent between and and between and means indifferent between and . Without transitivity, your choices could cycle, and someone could take advantage of you in a "money pump" — a series of trades that leaves you worse off.

3. Continuity#

Preferences do not jump wildly. If you prefer outcome to to , then there exists some probability such that you are indifferent between getting for sure and a lottery that gives with probability and with probability . This ensures we can smoothly trade off probabilities. It rules out extreme preferences like "I would never accept any chance of death, no matter how small, for any amount of money." While such preferences exist, continuity is a useful approximation for most economic settings.

4. Independence (or Substitution)#

This is the most important and most debated axiom. Suppose you prefer lottery to . Now consider mixing each with a third lottery using the same probability weight. The independence axiom says your preference should not switch: you should prefer the mixture with over the mixture with .

Formally: If , then for any and any ,

Think of it this way. You prefer an apple to a banana. Now I offer you two bags. Each bag contains a lottery ticket: with probability you get the fruit (apple or banana), with probability you get a chocolate bar instead. The axiom says you should prefer the bag that gives you the apple when the fruit ticket is drawn. The common extra (the chocolate bar) should not affect your choice between the differing parts. This independence idea is why expected utility simply adds up probabilities times utilities.

Expected Utility Axioms: A set of rationality conditions (completeness, transitivity, continuity, independence) on preferences over lotteries. If preferences satisfy these axioms, they can be represented by an expected utility function.

If your preferences satisfy these four axioms, then there exists a utility function defined on outcomes such that you rank lotteries by their expected utility. This is the Expected Utility Theorem. It is a powerful result: from a handful of plausible-sounding rules, we get the entire expected utility framework.

It is worth noting that real people sometimes violate these axioms — especially independence. The famous Allais paradox shows systematic violations. But for much of financial economics, expected utility is the benchmark. It gives us a clear, tractable model of rational choice under uncertainty, and it remains the starting point for understanding risk and return.

📝 Section Recap: Expected utility is not just an assumption; it follows from basic axioms of rational choice: completeness, transitivity, continuity, and independence. Together they imply that preferences over gambles can be represented by an expected utility function, giving the framework a solid logical foundation.

Putting It All Together: A Worked Example#

Let us walk through a complete example to see how these ideas fit together. Suppose an investor has utility function , the natural logarithm. This is a concave function, so she is risk averse. Her current wealth is .

She faces a gamble: with probability 0.6 she gains 500, and with probability 0.4 she loses 400. Should she take it?

First, compute the possible final wealth levels:

• Win:
• Lose:

The expected dollar payoff is:

The expected wealth is . A risk-neutral person would take the gamble since the expected value is positive.

Now compute expected utility:

Using approximate values: , .

Now compare to the utility of not taking the gamble — staying with 1000 for sure:

The expected utility of the gamble (6.9467) is higher than the utility of staying put (6.9078). Even though she is risk averse, the gamble's expected gain is big enough to make up for the risk. She should take it.

What certain amount would make her exactly indifferent? That is called the certainty equivalent (CE) — the guaranteed wealth that gives the same utility as the expected utility of the gamble. We solve:

So receiving 1039.5 for sure makes her equally happy as taking the gamble. The difference between the expected wealth (1140) and the certainty equivalent (1039.5) — about 100.5 — is the risk premium she would pay to avoid the risk. It measures her strength of risk aversion in dollar terms.

Certainty Equivalent: The guaranteed amount of wealth that gives the same utility as the expected utility of a risky gamble. For a risk-averse person, the CE is less than the expected wealth.

📝 Section Recap: The certainty equivalent translates expected utility back into dollar terms, measuring the sure amount an investor values equally to a risky prospect. The difference between the expected value and the certainty equivalent is the risk premium — the extra reward required to bear risk.

Summary#

In this chapter, we built the foundations of choice under uncertainty from the ground up. The big idea is that people do not evaluate gambles by their expected dollar payoff alone. They transform outcomes through a utility function and then compute the expected utility. When that utility function is concave, it captures risk aversion — the extra weight we put on bad outcomes compared with good ones. Jensen’s inequality is the math that ties concavity to preferring a sure thing. And we saw that expected utility isn’t pulled out of thin air: it follows naturally from a short list of common‑sense rules about consistent preferences. These tools give us a precise way to talk about risk attitudes, certainty equivalents, and risk premiums — ideas that will underpin almost everything later in financial economics.