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Choice under uncertainty is the branch of microeconomics that studies how individuals and organizations make decisions when the consequences of their actions are not known in advance. Unlike decisions made under certainty, where each action leads to a known outcome, decisions under uncertainty involve actions that yield a range of possible outcomes, each with some probability of occurring. The field asks two intertwined questions: How do people actually choose in such situations? And how should they choose, if they are to be rational? The tension between these descriptive and normative questions has shaped the field’s development and remains its central intellectual engine.
To analyze choice under uncertainty, economists model a decision as a choice among lotteries (also called gambles or prospects). A lottery is a list of possible outcomes, each attached to a probability. The outcomes can be monetary amounts, consumption bundles, health states, or any other objects of value. The decision-maker is assumed to have preferences over lotteries, and the goal is to represent those preferences in a tractable way that allows prediction.
The central concept is the expected utility of a lottery. If a lottery offers outcomes \(x1, x2, \dots, xn\) with probabilities \(p1, p2, \dots, pn\), its expected utility is \(\sum pi u(xi)\), where \(u\) is a utility function that assigns a number to each outcome. The expected utility hypothesis states that a rational decision-maker chooses the lottery with the highest expected utility. This formulation separates two components: the probabilities, which describe the objective or subjective chance of each outcome, and the utility function, which describes how the decision-maker values outcomes.
The utility function is not simply the monetary value of an outcome. A key insight is that utility is typically concave in money: the difference in utility between $100 and $200 is larger than the difference between $1,100 and $1,200. This concavity captures risk aversion—the willingness to accept a sure amount lower than the expected monetary value of a gamble to avoid its uncertainty. The degree of concavity measures how risk-averse an individual is. A linear utility function describes a risk-neutral person who cares only about expected value, while a convex function describes a risk-seeking person who prefers the gamble to its expected value.
The modern theory of choice under uncertainty was consolidated in the mid-twentieth century, most influentially by John von Neumann and Oskar Morgenstern in their 1944 work on game theory, and refined by Leonard Savage in 1954. Von Neumann and Morgenstern showed that if a decision-maker’s preferences over lotteries satisfy a small set of axioms, then those preferences can be represented by an expected utility function. The axioms are the field’s foundational assumptions:
Savage extended this framework to situations where probabilities are not given objectively but must be inferred from the decision-maker’s beliefs. His subjective expected utility theory shows that, under a different set of axioms, a rational decision-maker behaves as if they assign subjective probabilities to events and maximize expected utility with respect to those probabilities. This unified the treatment of risk (known probabilities) and ambiguity (unknown probabilities) under a single framework, though the latter distinction would later prove important.
The expected utility framework is not merely a descriptive tool. It is also a normative standard: a set of principles that a rational agent ought to follow. The axioms are presented as requirements of consistency. Violating them, the argument goes, leaves the decision-maker open to exploitation—for example, through a "money pump" in which a series of trades, each acceptable by the decision-maker’s preferences, leads them in a circle back to their starting point while losing money.
Beginning in the 1950s and accelerating through the following decades, experimental economists and psychologists documented systematic ways in which real people’s choices violate the expected utility axioms. The most famous is the Allais paradox, first presented by Maurice Allais in 1953. In a choice between a sure $1 million and a gamble offering a 10% chance of $5 million, an 89% chance of $1 million, and a 1% chance of nothing, many people choose the sure million. But in a second choice between a 10% chance of $5 million and an 11% chance of $1 million, many of the same people choose the gamble with the chance at $5 million. This pair of choices violates the independence axiom, because the two situations differ only by a common component that should cancel out.
Other documented violations include the certainty effect (people overweight outcomes that are certain relative to those that are merely probable), loss aversion (losses loom larger than gains of the same magnitude), and probability weighting (people tend to overweight small probabilities and underweight moderate and large ones). These findings do not merely show that people make mistakes; they show that the mistakes are systematic and predictable.
The most influential response to these findings is prospect theory, developed by Daniel Kahneman and Amos Tversky in 1979. Prospect theory modifies expected utility in several ways. First, it posits that people evaluate outcomes relative to a reference point—typically their current wealth or an expectation—rather than in absolute terms. Second, the value function is steeper for losses than for gains, capturing loss aversion. Third, people transform objective probabilities through a probability weighting function that overweights small probabilities and underweights large ones. Prospect theory was originally formulated for choices involving risk (known probabilities); a later version, cumulative prospect theory, extended it to handle ambiguity and multiple outcomes in a way that satisfies certain consistency requirements.
Prospect theory is a descriptive theory: it aims to predict actual behavior, not to prescribe rational choice. It has become the dominant framework in behavioral economics for understanding decisions under uncertainty. However, it is not a single unified theory but a family of models, and its parameters (the shape of the value function, the weighting function, the location of the reference point) are not fixed by theory but estimated from data. This flexibility is a strength for prediction but a weakness for explanation: almost any pattern of choices can be accommodated by suitable parameter choices.
The experimental violations of expected utility prompted two distinct lines of response. One, exemplified by prospect theory, abandoned the normative project and focused on describing actual behavior. The other sought to relax the axioms while preserving a coherent normative standard. These non-expected utility theories maintain the idea that rational choice can be represented by some functional, but allow that functional to depart from the simple expected utility form.
The most important of these departures concerns ambiguity—situations where probabilities themselves are unknown. The classic thought experiment is the Ellsberg paradox, formulated by Daniel Ellsberg in 1961. An urn contains 30 red balls and 60 balls that are either black or yellow in unknown proportion. Most people prefer to bet on red (known probability 1/3) over black (unknown probability), and also prefer to bet on "not red" (known probability 2/3) over "not black" (unknown probability). This pattern violates Savage’s subjective expected utility theory, because it shows that people prefer known probabilities to unknown ones even when the unknown probabilities could, in principle, be anything.
The leading normative response to ambiguity is maxmin expected utility, developed by Itzhak Gilboa and David Schmeidler in 1989. In this model, the decision-maker does not have a single probability distribution over outcomes but a set of possible distributions. They evaluate each action by its worst-case expected utility across that set, and choose the action with the highest worst-case value. This captures ambiguity aversion: the decision-maker behaves as if nature is adversarial, choosing the worst distribution from their set. A related model, smooth ambiguity aversion (developed by Peter Klibanoff, Massimo Marinacci, and Sujoy Mukerji in 2005), allows a more graded response, where the decision-maker has a prior over the possible distributions and applies an additional utility function to the expected utilities they yield.
These models are normative in the sense that they satisfy axioms of consistency, but they are also motivated by descriptive evidence. They represent a middle ground: they preserve the idea that choice under uncertainty can be modeled as the maximization of a well-defined functional, but they allow that functional to encode a preference for certainty or for known probabilities that expected utility theory forbids.
The coexistence of descriptive and normative theories is not a sign of confusion but a reflection of the field’s dual purpose. Expected utility theory remains the standard tool in most economic applications—in finance, insurance, contract theory, and macroeconomics—because it is tractable and because its axioms provide a clear benchmark. When economists say that a person is "rational," they typically mean that the person maximizes expected utility. This is a working assumption, not an empirical claim about all people at all times.
Prospect theory and related behavioral models are used when the question is predictive: How will consumers respond to a new insurance product? Will investors hold too much risk? Will people save enough for retirement? In these contexts, the expected utility benchmark is known to be wrong in systematic ways, and behavioral models improve prediction.
The two approaches also inform each other. Behavioral findings have led to behavioral welfare economics, which asks how policy should be designed when people do not maximize expected utility. For example, if people overweight small probabilities, they may buy too much lottery insurance (insurance that pays off in unlikely disasters) and too little standard insurance. A policy maker who takes expected utility as the normative standard might conclude that such people are making mistakes; a policy maker who takes prospect theory as the normative standard might conclude that the people are expressing a legitimate preference for certainty. This debate is unresolved and is one of the field’s live frontiers.
The field today is characterized by a productive pluralism. Expected utility theory remains the workhorse for theoretical modeling and for most applied work. Prospect theory is the leading descriptive alternative, especially in behavioral economics and finance. Ambiguity models have become standard in decision theory, macroeconomics, and finance, where uncertainty about the true model of the world is often more important than risk within a known model.
Several developments have broadened the field’s scope. Dynamic choice extends the framework to situations where decisions are made over time and information arrives gradually. This raises questions about dynamic consistency—whether a plan made today will still be followed tomorrow—and about the value of information. Non-expected utility models often violate dynamic consistency, which creates both theoretical challenges and opportunities for modeling phenomena like procrastination or precommitment. Choice under uncertainty with multiple selves—where the same person makes different decisions at different times—has been modeled using hyperbolic discounting and related frameworks, though these are more often classified under intertemporal choice than under uncertainty proper.
Another important extension is the treatment of state-dependent utility, where the value of an outcome depends on the state of the world in which it occurs. This is crucial for understanding the value of health, where being alive in a sick state is different from being alive in a healthy state, and for environmental economics, where the state of the world affects the value of natural resources.
The field also connects to revealed preference theory, which asks what can be inferred about preferences from observed choices. Under uncertainty, this is subtle: a choice between a lottery and a sure thing reveals something about risk attitudes, but separating risk attitudes from beliefs about probabilities requires careful experimental design or strong assumptions. The elicitation of preferences—how to ask people questions that reveal their true utility functions and probability beliefs—is an active methodological area.
Finally, the field has engaged with neuroeconomics, which uses brain imaging and physiological measures to study the neural basis of decisions under uncertainty. This work has found correlates of risk aversion and probability weighting in brain activity, but its contribution to economic theory remains debated. Some see it as providing microfoundations for behavioral models; others see it as largely irrelevant to the economic questions of prediction and welfare.
The enduring questions of the field remain those with which it began: What does it mean to choose rationally under uncertainty? How do real people choose, and why do they depart from the rational benchmark? And what should a policy maker do when the two answers diverge? No single framework has settled these questions, and the field’s vitality comes from the ongoing tension among its normative, descriptive, and prescriptive strands.