Expected utility hypothesis

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 the subjective value associated with an individual's gamble is the statistical expectation of that individual's valuations of the outcomes of that gamble, where these valuations may differ from the dollar value of those outcomes. 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 the subjective value associated with an individual's gamble is the statistical expectation of that individual's valuations of the outcomes of that gamble, where these valuations may differ from the dollar value of those outcomes. The introduction of St. Petersburg Paradox by Daniel Bernoulli in 1738 is considered the beginnings of the hypothesis. This hypothesis has proven 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. Bernoulli came across expected utility by playing the St Petersburg paradox. This paradox involves you flipping a coin until you get to heads. The number of times it took you to get to heads is what you put as an exponent to 2 and receive that in dollar amounts. This game helped to understand what people were willing to pay versus what people were expected to gain from this game. 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. When the entity x {displaystyle x} whose value x i {displaystyle x_{i}} affects a person’s utility takes on one of a set of discrete values, the formula for expected utility, which is assumed to be maximized, is where the left side is the subjective valuation of the gamble as a whole, x i {displaystyle x_{i}} is the ith possible outcome, u ( x i ) {displaystyle u(x_{i})} is its valuation, and p i {displaystyle p_{i}} is its probability. There could be either a finite set of possible values x i , {displaystyle x_{i},} in which case the right side of this equation has a finite number of terms; or there could be an infinite set of discrete values, in which case the right side has an infinite number of terms. When x {displaystyle x} can take on any of a continuous range of values, the expected utility is given by where f ( x ) {displaystyle f(x)} is the probability density function of x . {displaystyle x.} 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 $1.00, and a gamble in which the probability of getting a $100 payment is 1 in 80 and the alternative, far more likely outcome (79 out of 80) is receiving $0. The expected value of the first alternative is $1.00 and the expected value of the second alternative is $1.25. According to expected value theory, people should choose the $100-or-nothing gamble; however, as stressed by expected utility theory, some people are risk averse enough to prefer the sure thing, despite its lower expected value. People with less risk aversion would choose the riskier, higher-expected-value gamble. This is precedence for utility theory.