Decision-Making Under Uncertainty

You cannot see the future, yet you must choose. This chapter gives you a precise language to talk about choices when outcomes are uncertain. It builds the logical foundations that turn a gut feeling about risk into a powerful tool for analysis.

The Big Picture#

Imagine you are offered a safe job, or a risky startup that might make you rich or leave you broke. How do you compare them? Or you check the weather forecast before deciding whether to carry an umbrella—but the forecast is a probability, not a certainty. This chapter asks a simple question with a deep answer: what does it mean to make a rational decision when we do not know the outcome in advance? We will build the idea of expected utility, a way of assigning numbers to outcomes so that the best choice is the one that gives the highest average number, weighted by how likely each outcome is. Along the way we will meet the very rules – called axioms – that shape rational preferences, discover how to measure attitudes toward risk, and find out how we can still use the whole machinery when the probabilities themselves come from our own head, not from a data sheet.

Preferences: The Building Blocks of Choice#

Before we toss coins or spin wheels, we need a way to talk about what a person likes. The simplest starting block is a preference relation, usually written as . For any two outcomes and ,

means “ is at least as good as ”,
means “ is strictly better than ”,
means “ and are equally good”.

These are not about money yet – they could be about any items, such as a cup of coffee, a tea, or a glass of water.

For a preference to be called rational, we ask for three very natural properties.

Completeness: For any two alternatives and , either or (or both). In other words, you can compare everything; nothing leaves you in a “cannot decide” limbo.

Reflexivity: Every alternative is at least as good as itself: . This is trivial—a thing is as good as itself—but it ensures the relation behaves logically.

Transitivity: If and , then . No circular rankings. If you like pizza at least as much as pasta, and pasta at least as much as salad, you must like pizza at least as much as salad.

These three requirements together let us order everything on a single scale. Think of ranking films from 1 to 10: completeness means every film gets a number, reflexivity means the same film gets the same number, and transitivity stops cycles like “film A beats film B, B beats C, C beats A”. Without transitivity, someone could trick you into trading in circles until you lose everything of value.

📝 Section Recap: Rational preferences are summarized by a complete, reflexive, and transitive ranking—no gaps, no self-contradictions, no loops.

Lotteries: Turning Uncertainty into a List#

Real decisions rarely hand you a single outcome with certainty. Usually you face a gamble, a bundle of “what ifs” with attached probabilities. We call such a bundle a lottery.

A lottery is simply a list of possible outcomes together with how likely each one is. For example, “flip a fair coin; heads you get $100, tails you get nothing” can be written as:

Notice that the probabilities always sum to – something must happen.

Sometimes gambles come in stages: you spin a wheel first, and depending on where it stops you enter a second gamble. That is a compound lottery. We adopt a simple rule: the reduction of compound lotteries. A two‑stage gamble can be flattened into a single lottery by multiplying the probabilities along each path. If the first stage gives you a 50% chance of entering lottery and a 50% chance of entering lottery , then the overall probability of any final outcome is just . This rule treats the exact timing of the resolution of uncertainty as irrelevant; all that matters is the final chance of each outcome. It keeps things mathematically clean, and we will assume it throughout.

Reduction of compound lotteries: A multi-stage lottery is equivalent to the simple lottery whose probabilities are the combined path‑by‑path probabilities of the final outcomes.

📝 Section Recap: A lottery assigns a probability to every possible outcome. Compound lotteries can be reduced to an equivalent simple lottery by multiplying probabilities, so we can focus only on the final distribution of outcomes.

The von Neumann–Morgenstern Axioms#

Now we want preferences that work smoothly with lotteries. John von Neumann and Oskar Morgenstern proposed three extra rationality conditions—beyond completeness and transitivity—that give preferences a very useful shape.

Continuity. Suppose you like outcome better than , and is not worse than . Then there is some probability (not 0 or 1) that makes you exactly indifferent between getting for sure and a lottery that gives with probability and with probability . In plain terms, a middle‑ranked certainty can always be traded for a gamble on the best and worst outcomes, provided the odds are tilted appropriately. This rules out the idea that some outcome is “infinitely” better than another, where no gamble could ever make you indifferent; it keeps preferences from having sudden, unbridgeable gaps.

Monotonicity (or “more is better in a lottery”). If you strictly prefer outcome to outcome , then between two lotteries that differ only in the probability attached to versus , you must prefer the lottery that gives the higher probability to the better outcome. For example, a 70% chance of 0) is better than a 30% chance of 0) if you like 0. This seems almost too obvious, but it is explicitly needed.

Independence. This is the star of the show. Imagine two lotteries and that you consider equally good. Now create a new lottery by mixing each of them in the same way with a third lottery : say, a chance of the original lottery and a chance of . The independence axiom says you must be indifferent between the mixed versions: . Moreover, if you strictly preferred to , mixing both with in the same way preserves that strict preference. In essence, adding the same “noise” lottery to both alternatives does not change which one you like more; the common part cancels out in the decision.

These three axioms, together with completeness and transitivity, are the core of the expected utility hypothesis.

📝 Section Recap: Continuity says a middle‑ranked sure thing can be balanced by a gamble on the extremes; independence says that mixing two lotteries with a common third one does not reverse your preference; monotonicity says more weight on a better outcome is better. Together they lead to expected utility.

The vNM Utility Representation#

When your preferences over lotteries satisfy the five axioms (completeness, transitivity, continuity, monotonicity, independence), something magical happens: there exists a real‑valued utility function defined on the set of outcomes, such that for any lottery ,

This means you can assign a number to every possible outcome, and the number you assign to a whole lottery is simply the probability‑weighted average of those numbers. That average is called the expected utility of the lottery. The choice rule is then beautifully simple: pick the lottery that gives the highest expected utility.

Two crucial points about this function. First, it is cardinal in a specific, limited sense: the numbers themselves matter, not just their order. If you doubled all the values or added a constant, you would not change which lottery gives the highest expected utility, but stretching by a fixed positive amount and shifting by a constant is the only freedom you have. Formally, if works, then any other function that can be written as

with also represents exactly the same preferences over lotteries. We say is unique up to a positive affine transformation. This is like measuring temperature: switching from Celsius to Fahrenheit () changes the numbers but keeps the comparisons of differences and ratios of differences intact, which is what matters for trade‑offs under uncertainty.

Second, is defined on outcomes only; the linearity (weighted average) in probabilities is a consequence of the axioms, not an extra assumption. This linearity is often called the von Neumann–Morgenstern (vNM) utility representation.

vNM utility: A function on outcomes such that the value of any lottery is given by , and where any positive affine transformation (with ) represents the same preferences.

📝 Section Recap: If preferences obey the five axioms, they can be described by a utility function on outcomes, with the value of a lottery being the probability‑weighted average of those utilities. The function is unique up to a positive linear rescaling, giving it just the right amount of cardinal meaning for risky choices.

Risk Attitudes: What Your Utility Curve Says About You#

Once we have a vNM utility function, its shape tells us a lot about how a person feels about risk.

Risk neutrality: is a straight line, for instance . Then expected utility equals expected monetary value. A risk‑neutral person only cares about the average amount of money; a gamble and its sure‑thing average look equally good.
Risk aversion: curves downward, as a concave function (e.g., or ). Here the utility of a sure amount is larger than the expected utility of a gamble whose expected monetary value equals . Such a person would pay a premium to avoid risk; that is why people buy insurance.
Risk seeking: curves upward, as a convex function (e.g., ). The thrill of a big win pulls the expected utility above the utility of the safe average; these folks buy lottery tickets.

A simple example makes this concrete. Suppose you are offered a choice between a certain 100$200$0$100u(x) = xu(x) = \sqrt{x}0.5\sqrt{200} + 0.5\sqrt{0} \approx 0.5 \times 14.14 \approx 7.07\sqrt{100} = 10u(x) = x^20.5\times 40000 = 2000010000 $‑ ‑$ \sqrt{CE} = 7.07 \implies CE \approx 50 < 100$, revealing the value they place on avoiding the risk.

Risk aversion: A concave utility function, where the decision‑maker prefers a sure amount to a gamble with the same expected value.

Risk seeking: A convex utility function, where the gamble is preferred.

📝 Section Recap: The curvature of the vNM utility function captures risk attitude: straight lines mean indifference to risk, curves bent downward mean aversion, and curves bent upward mean a taste for risk.

Subjective Probability: When the Odds Are in Your Head#

So far we have assumed that the probabilities attached to outcomes are handed to us, like a casino wheel or a weather forecast backed by data. But in many real‑world decisions—whether to start a business, whom to trust, which policy to support—we do not have an objective, freestanding list of probabilities. Instead, we form personal beliefs. Can we still use the expected utility framework?

The answer, developed by Leonard Savage and others, is a beautiful yes. The idea is to start not with probabilities but with acts—actions you can take that lead to different outcomes depending on which state of the world turns out to be true. For example, carrying an umbrella yields one outcome if it rains and another if it stays dry. If your preferences over acts obey a set of rationality axioms similar in spirit to the vNM axioms (but now applied to state‑contingent consequences rather than lotteries with given numbers), then both a subjective probability for each state and a vNM utility function for outcomes can be deduced from your choices. That is, by observing which acts you consistently prefer, we can simultaneously infer your personal beliefs about how likely rain is and how much you value staying dry.

The decision rule then becomes the subjective expected utility maxim: pick the act that maximizes , where is your subjective probability for state and is your utility function. This is a remarkable extension: it tells us that rational choice under uncertainty—even when no objective probabilities exist—still takes the same mathematical form, just with probabilities that reflect your personal degree of belief. The probabilities are “in your head,” but they obey the same calculus and interact with utilities in the same clean, additive way. This unification is one of the great intellectual achievements behind modern game theory and decision science.

Subjective probability: A numerical degree of belief, inferred from consistent preferences, that obeys the rules of ordinary probability.

📝 Section Recap: Even without objective probabilities, consistent preferences over acts can be represented by a subjective expected utility formula, revealing both personal beliefs (subjective probabilities) and tastes (utility) from choice behavior.

Summary#

We have explored how to make rational choices when the future is uncertain. It all begins with preferences that are complete and transitive. We then added lotteries, which let us describe any gamble as a list of outcomes with probabilities. Three extra axioms—continuity, monotonicity, and the powerful independence axiom—force those lottery preferences to behave exactly as if you were calculating a probability‑weighted average of some utility numbers. That utility function is unique up to a positive affine transformation, giving it just enough flexibility to capture how much you love or hate a bit of extra money or a nice meal. Its shape tells us whether you are risk‑neutral, risk‑averse, or risk‑seeking. And the same framework can work even when the probabilities are not given by nature but are your own personal beliefs, so long as your choices obey a similar set of consistency rules. With these building blocks in place, you are ready to step into strategic situations where your payoff depends on what others do—the heart of game theory.

Key idea	What it means (plain English)	Why it matters
Preference relation	A ranking that satisfies completeness (you can compare any two options), reflexivity (every option is as good as itself), and transitivity (no cycles).	Without a clear ranking, rational choice is impossible. These three rules make sure your comparisons are consistent and can be turned into a utility scale.
Lottery	A list of possible outcomes, each with a probability that sums to 1. It captures any situation with a known chance structure.	Real decisions often involve risk; a lottery lets us describe them precisely before analyzing which one is best.
Reduction of compound lotteries	Flattening a multi‑stage gamble into a single lottery by multiplying probabilities along each path.	Simplifies the problem: only the final probability of each outcome matters, removing tricks that depend on the order of resolution.
Continuity axiom	A middle‑ranked sure thing can always be made indifferent to a gamble between a better and a worse outcome, given the right odds.	Prevents preferences from having “infinite” gaps, making it possible to assign real numbers to outcomes.
Independence axiom	Mixing two lotteries with a common third one does not change which one you prefer; the common part cancels out.	The key that forces preferences over lotteries to be linear in probabilities, leading directly to the expected utility formula.
vNM utility representation	A function on outcomes such that the value of a lottery is . It is unique up to with .	Gives a simple arithmetic rule for comparing lotteries and measuring how much better one outcome is than another, up to a consistent scaling.
Risk aversion	Concave utility function; the decision‑maker values a sure thing more than a gamble with the same expected value.	Explains insurance, investment caution, and why people demand a premium for bearing risk.
Subjective expected utility	When no objective probabilities exist, consistent preferences over acts reveal personal beliefs (subjective probabilities) and utility, and the best act still maximizes expected utility.	Extends rational decision‑making to the vast majority of real‑world situations where probabilities are not handed to us but must be formed from judgment.