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The expected utility hypothesis is the hypothesis in economics that the utility of an agent facing uncertainty is calculated by considering utility in each possible state and constructing a weighted average. The weights are the agent's estimate of the probability of each state. The expected utility is thus an expectation in terms of probability theory. To determine utility according to this method, the decision maker must rank their preferences according to the outcomes of various decision options. According to the theory, if someone prefers A to B and B to C, then weights for the weighted average must exist such that she is indifferent between receiving B outright and gambling-- with the specified weights-- between A and C.

Daniel Bernoulli (1738) gave the earliest known written statement of this hypothesis as a way to resolve the St. Petersburg Paradox. In the expected utility theorem, v. Neumann and Morgenstern proved that any "normal" preference relation over a finite set of states can be written as an expected utility. (Therefore, it is also called von-Neumann Morgenstern utility.) Von Neumann and Morgenstern published this in their Theory of Games and Economic Behavior in 1944. It is important because it was developed shortly after the Hicks-Allen “ordinal revolution” of the 1930's, and it revived the idea of cardinal utility in economic theory. Economics has not resolved whether (and in what cases) utility is cardinal or ordinal.

A related concept is the certainty equivalent of a gamble. The more risk-averse a person is, the more he will be prepared to pay to eliminate risk, for example accepting $1 instead of a 50% chance of $3, even though the expected value of the latter is more. People may be risk-averse or risk-loving depending on the amounts involved and on whether the gamble relates to becoming better off or worse off; this is a possible explanation for why the same person may buy both an insurance policy and a lottery ticket. However, expected utility as a descriptive model of decisions under risk has in recent years been replaced by more sophisticated variants that take irrational deviations from the expected utility model into account; compare Prospect theory and the general article on Behavioral finance.

The concept of risk-aversion comes into play in many gambling scenarios, such as poker strategy. A risk-neutral stance is generally the best strategy in most situations, as it attempts to maximize the expected value of each bet. However, there are situations where different strategies will be more beneficial. For example, many experts advocate a risk-averse strategy in the early stages of a poker tournament, when there are still many players left. As the tournament advances, a more risk-neutral or even risk-loving strategy becomes the more optimal play. This change in strategy is due to the difference between expected value and expected utility. If a player has only a small number of chips remaining, they should begin to make larger and more frequent bets, and consequently take on more risk, because this is the only approach that gives them a chance to quicky amass a large number of chips, which will be necessary in order to have success in the tournament. A risk-averse strategy may appear to be a good way of preserving a player's remaining chips, but due to the rising blinds, this approach generally decreases the player's chances of finishing "in the money" for the tournament. See M-ratio for more information on this concept as it relates to poker theory.

Preference Reversals over Uncertain Outcomes: Starting with studies such as Lichtenstein & Slovic (1971), it was discovered that subjects sometimes exhibit signs of preference reversals with regards to their certainty equivalents of different lotteries. Specifically, when eliciting certainty equivalents, subjects tend to value "p bets" (lotteries with a high chance of winning a low prize) lower than "$ bets" (lotteries with a small chance of winning a large prize). When subjects are asked which lotteries they prefer in direct comparison, however, they frequently prefer the "p bets" over "$ bets." Many studies have examined this "preference reversal," from both an experimental (e.g., Plott & Grether, 1979) and theoretical (e.g., Holt, 1986) standpoint, indicating that this behavior can be brought into accordance with neoclassical economic theory under certain assumptions.

• Bernoulli, D (1954) "Exposition of a New Theory on the Measurement of Risk" (original: 1738), "Econometrica" 22:23-36.
• Schoemaker PJH (1982) "The Expected Utility Model: Its Variants, Purposes, Evidence and Limitations", "Journal of Economic Literature", 20:529-563.
• P.Anand (1993) "Foundations of Rational Choice Under Risk", Oxford, Oxford University Press. ISBN 0198233035
• K.J. Arrow (1963) "Uncertainty and the Welfare Economics of Medical Care", American Economic Review, Vol. 53, p.941-73.
• Scott Plous (1993) "The psychology of judgment and decision making", Chapter 7 (specifically) and 8,9,10, (to show paradoxes to the theory).