Expected utility

In economics, game theory, and decision theory the expected utility hypothesis, concerning people's preferences with regard to choices that have uncertain outcomes (gambles), states that if specific axioms are satisfied, the subjective value associated with an individual's gamble is the statistical expectation of that individual's valuations of the outcomes of that gamble. Initiated by Daniel Bernoulli in 1738, this hypothesis has proved useful to explain some popular choices that seem to contradict the expected value criterion (which takes into account only the sizes of the payouts and the probabilities of occurrence), such as occur in the contexts of gambling and insurance. Until the mid-twentieth century, the standard term for the expected utility was the moral expectation, contrasted with "mathematical expectation" for the expected value.

The von Neumann–Morgenstern utility theorem provides necessary and sufficient conditions under which the expected utility hypothesis holds. From relatively early on, it was accepted that some of these conditions would be violated by real decision-makers in practice but that the conditions could be interpreted nonetheless as 'axioms' of rational choice.

In the presence of risky outcomes, a human decision maker does not always choose the option with higher expected value investments. For example, suppose there is a choice between a guaranteed payment of $1, and a gamble in which the probability of getting a $100 payment is 1 in 80 and the alternative, far more likely outcome is getting nothing (the expected value is then $1.25); according to expected value theory people should choose the $100-or-nothing gamble, but as stressed by expected utility theory, some people are risk averse enough to prefer the sure thing, even though it has a lower expected value, while other less risk averse people would still choose the riskier, higher-mean gamble.

Nicolas Bernoulli described the St. Petersburg paradox (involving infinite expected values) in 1713, prompting two Swiss mathematicians to develop expected utility theory as a solution. The theory can also more accurately describe more realistic scenarios (where expected values are finite) than expected value alone. In 1728, Gabriel Cramer, in a letter to Nicolas Bernoulli, wrote, "the mathematicians estimate money in proportion to its quantity, and men of good sense in proportion to the usage that they may make of it."

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