Life is full of gambles. Should you take the stable job or the risky startup? Buy insurance or risk a large loss? How much would you pay to avoid a coin flip that could cost you $100? In this chapter, we’ll build the tools economists use to answer these questions—a simple way to think about choices when the future is uncertain.
The Big Picture#
Every money decision—saving, investing, buying a house—means swapping something certain today for an unknown tomorrow. To make sense of these choices, we need two things: a way to describe uncertainty, and a rule for ranking options. This chapter gives you the core ideas: how to describe what you want when things are uncertain, how to measure how much you dislike risk, and how to handle situations where even the odds are unclear.
Choices, Outcomes, and Lotteries#
First, we need a clear way to describe a situation where the future isn’t fixed. Imagine you can buy a stock that will either go up or down tomorrow. We call each possible future a state of the world. “Stock up” is one state, “stock down” is another.
State of the world: A complete description of one possible future, so that exactly one state will actually happen.
A prospect (or lottery) lists how much money you get in each state, along with the chance that state occurs. For a simple coin toss that pays
This is a simple lottery—the probabilities are known. Later we’ll relax that.
When you choose between “take $100 for sure” or “flip the coin”, you’re picking between a lottery with a single outcome and a lottery with two possible outcomes.
📝 Section Recap: We describe uncertain choices with states and lotteries. A lottery lists payoffs and their probabilities, giving us a clean language for what’s at stake.
The Expected Utility Idea#
How should a rational person compare lotteries? The von Neumann–Morgenstern (VNM) expected utility framework gives a simple answer: assign a utility number to each final amount of money, then take the probability‑weighted average of those numbers. Pick the lottery with the higher average utility.
Von Neumann–Morgenstern (VNM) expected utility: A way to rank lotteries. The value of a lottery is
. The function is called a Bernoulli utility function; it captures your attitude toward money.
This isn’t just a made‑up rule. It follows from a few common‑sense rules for consistent choice:
- Completeness: you can always rank two options.
- Transitivity: if you prefer A to B and B to C, then you must prefer A to C.
- Continuity: small changes in probabilities don’t flip your preferences.
- Independence: if you prefer A to B, then mixing each with the same third lottery in the same proportion shouldn’t reverse your preference. This axiom is the key.
Think of a game show. You’re offered a safe envelope with
The safe
📝 Section Recap: VNM expected utility ranks uncertain prospects by weighing the pleasure of each outcome by its probability. It rests on a handful of simple rules for making consistent choices.
Risk Aversion: Why We Dislike Uncertainty#
Most people dislike risk. A risk‑averse person always prefers a sure payoff equal to the average of a gamble over the gamble itself. In utility terms, that means the utility function is concave—it curves downward.
Risk aversion: A preference for a sure payoff equal to the expected value of a gamble rather than the gamble itself. Formally,
, with strict inequality for gambles that are not a sure thing.
The shape of
where
Absolute risk aversion (ARA): The measure
. It tells you how many dollars you need to be paid to take on a small risk, without worrying about how rich you are.
If your utility is
A more realistic pattern is that your worry about a gamble that is a fixed fraction of your wealth stays about the same as you get richer. That’s captured by relative risk aversion.
Relative risk aversion (RRA): The measure
. It tells you what fraction of your wealth you would pay to avoid a gamble that is proportional to your wealth.
A popular family is constant relative risk aversion (CRRA):
Here
A quick example: your friend offers a coin flip for
📝 Section Recap: Risk aversion is measured by how much the marginal utility of money falls as wealth rises. Absolute and relative risk aversion coefficients give us simple numbers to describe and compare attitudes toward risk.
Mean‑Variance Preferences: A Shortcut for Portfolios#
Working with full utility functions can be messy. When returns are bell‑shaped (normal) or if utility is a simple quadratic, we can boil a lottery down to just two numbers: the mean and the variance. This is the mean‑variance (MV) approach.
Mean‑variance preferences: A rule where you rank portfolios only by their expected return
and the variance of return , often as . Higher is good; higher is bad.
The parameter
Why does this matter? Because it lets us separate two tasks: first, estimate the means and variances of assets; second, pick the mix that gives the best trade‑off. The shortcut is exact if returns are normal, but it’s a handy approximation in many other settings.
📝 Section Recap: Mean‑variance preferences reduce a complex lottery to its expected return and risk (variance), making portfolio choice simple. The coefficient
again encodes how much you dislike risk.
When Utility Depends on the State Itself#
So far, your happiness only depended on the amount of money you end up with. But often the situation you find yourself in changes how much you value a dollar. For example, an umbrella gives more pleasure when it rains. This is state‑dependent utility.
State‑dependent utility: A utility function
where how much you enjoy consumption depends on which state occurs. The same dollar can bring different happiness in different states.
Think about health insurance. If you get sick,
The best insurance or portfolio will now reflect both the odds and how much you value consumption in each state.
One subtle point: when utility is state‑dependent, the simple definitions of risk aversion need to be reinterpreted. Someone might look risk‑averse in money terms, but they are really just smoothing happiness across states that differ in “background pain.” That insight helps explain why people buy insurance against losses that would be extra painful.
📝 Section Recap: Allowing utility to vary with the state itself explains why we insure against specific disasters and why we hold assets that pay off in bad times. It enriches the expected utility model without changing its core structure.
Beliefs without Objective Odds#
In real life, we rarely know the exact probability a stock will rise. Instead, we form personal judgments—subjective probabilities.
Subjective probability: A degree of belief, expressed as a number between 0 and 1, that an event will happen. It’s not an objective frequency; it reflects your own information and confidence.
Just as VNM showed how objective probabilities and preferences give expected utility, Leonard Savage showed that if a person’s choices among uncertain options follow certain consistency rules, we can figure out both their utility function and their subjective probabilities from those choices. In other words, we can model them as if they are maximizing expected utility using their own personal beliefs.
How does this work? Suppose an investor thinks there is a 40% chance of a recession next year. She can’t point to a long‑run frequency, but she feels 40% is right given the news, her intuition, and her past experience. Using that belief, she computes an expected utility and compares investments. The key is that her choices must be logically consistent—no set of bets that would lead to a sure loss.
This foundation is essential because every financial decision involves judgments about an uncertain future for which no objective die is rolled. The whole idea of asset pricing—what is a fair price for a risky stock?—rests on the idea that investors have, and act on, their own subjective probabilities.
📝 Section Recap: Subjective probabilities let us model decisions when odds aren’t given by nature. Savage’s framework shows that rational choice reveals both personal beliefs and utility, extending expected utility to the world of Knightian uncertainty.
Summary#
We’ve built the mental tools economists use to think about choice under uncertainty. Starting from a simple lottery, we built the expected utility model, measured how much we dislike risk, and then expanded the model to handle situations where happiness depends on the state and where the odds are just our personal beliefs. These ideas are the grammar of financial economics: every price, portfolio, and insurance contract traces back to someone’s preferences for risky outcomes.
| Key idea | What it means (plain English) | Why it matters |
|---|---|---|
| Expected utility | A rule that ranks uncertain prospects by weighing the pleasure of each outcome by its probability. | Gives a consistent way to compare gambles and underlies all modern asset pricing. |
| Risk aversion | A tendency to prefer a sure thing over a gamble with the same average payoff. | Explains why investors demand higher returns for riskier assets and why insurance exists. |
| Absolute risk aversion (ARA) | How many dollars you would pay to avoid a fixed‑size gamble. | Tells us how attitudes toward risk change with wealth for small dollar risks. |
| Relative risk aversion (RRA) | The fraction of your wealth you would sacrifice to avoid a gamble proportional to your wealth. | Useful for comparing risk‑taking across people with different wealth; constant RRA is a common modelling choice. |
| Mean‑variance preferences | Evaluating a portfolio by its expected return and the variance (risk) alone. | Makes portfolio choice simple and is the backbone of the Capital Asset Pricing Model. |
| State‑dependent utility | The enjoyment of money depends on which state of the world occurs (e.g., health vs. sickness). | Captures the extra value of insurance or assets that pay off in bad personal circumstances. |
| Subjective probability | A personal degree of belief about an event’s likelihood, not an objective frequency. | Allows decision‑making when true probabilities are unknown; forms the basis for most real‑world economic choices. |