# Expected utility hypothesis

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The expected utility hypothesis is a foundational assumption in mathematical economics concerning human preference when decision making under uncertainty. It postulates that a rational agent maximizes utility, as formulated in the mathematics of game theory, based on their risk aversion. Rational choice theory, a cornerstone of microeconomics, builds upon the expected utility of individuals to model aggregate social behaviour.

The expected utility hypothesis states an agent chooses between risky prospects by comparing expected utility values (i.e. the weighted sum of adding the respective utility values of payoffs multiplied by their probabilities). The summarised formula for expected utility is $U(p)=\sum u(x_{k})p_{k}$ where $p_{k}$ is the probability that outcome indexed by $k$ with payoff $x_{k}$ is realized, and function u expresses the utility of each respective payoff. Graphically the curvature of the u function captures the agent's risk attitude.

Standard utility functions represent ordinal preferences. The expected utility hypothesis imposes limitations on the utility function and makes utility cardinal (though still not comparable across individuals).

Although the expected utility hypothesis is standard in economic modelling, it has been found to be violated in psychological experiments. For many years, psychologists and economic theorists have been developing new theories to explain these deficiencies. These include prospect theory, rank-dependent expected utility and cumulative prospect theory, and bounded rationality.

## Antecedents

### Limits of the expected value theory

In the early days of the calculus of probability, classic utilitarians believed that the option which has the greatest utility will produce more pleasure or happiness for the agent and therefore must be chosen The main problem with the expected value theory is that there might not be a unique correct way to quantify utility or to identify the best trade-offs. For example, some of the trade-offs may be intangible or qualitative. Rather than monetary incentives, other desirable ends can also be included in utility such as pleasure, knowledge, friendship, etc. Originally the total utility of the consumer was the sum of independent utilities of the goods. However, the expected value theory was dropped as it was considered too static and deterministic. The classical counter example to the expected value theory (where everyone makes the same "correct" choice) is the St. Petersburg Paradox. This paradox questioned if marginal utilities should be ranked differently as it proved that a "correct decision" for one person is not necessarily right for another person.

## Risk aversion

The expected utility theory takes into account that individuals may be risk-averse, meaning that the individual would refuse a fair gamble (a fair gamble has an expected value of zero). Risk aversion implies that their utility functions are concave and show diminishing marginal wealth utility. The risk attitude is directly related to the curvature of the utility function: risk neutral individuals have linear utility functions, while risk seeking individuals have convex utility functions and risk averse individuals have concave utility functions. The degree of risk aversion can be measured by the curvature of the utility function.

Since the risk attitudes are unchanged under affine transformations of u, the second derivative u'' is not an adequate measure of the risk aversion of a utility function. Instead, it needs to be normalized. This leads to the definition of the Arrow–Pratt measure of absolute risk aversion:

${\mathit {ARA}}(w)=-{\frac {u''(w)}{u'(w)}},$

where $w$  is wealth.

The Arrow–Pratt measure of relative risk aversion is:

${\mathit {RRA}}(w)=-{\frac {wu''(w)}{u'(w)}}$

Special classes of utility functions are the CRRA (constant relative risk aversion) functions, where RRA(w) is constant, and the CARA (constant absolute risk aversion) functions, where ARA(w) is constant. They are often used in economics for simplification.

A decision that maximizes expected utility also maximizes the probability of the decision's consequences being preferable to some uncertain threshold. In the absence of uncertainty about the threshold, expected utility maximization simplifies to maximizing the probability of achieving some fixed target. If the uncertainty is uniformly distributed, then expected utility maximization becomes expected value maximization. Intermediate cases lead to increasing risk aversion above some fixed threshold and increasing risk seeking below a fixed threshold.

## Recommendations

There are three components in the psychology field that are seen as crucial to the development of a more accurate descriptive theory of decision under risks.

1. Theory of decision framing effect (psychology)
2. Better understanding of the psychologically relevant outcome space
3. A psychologically richer theory of the determinants

### Mixture models of choice under risk

In this model Conte (2011) found that behaviour differs between individuals and for the same individual at different times. Applying a Mixture Model fits the data significantly better than either of the two preference functionals individually. Additionally it helps to estimate preferences much more accurately than the old economic models because it takes heterogeneity into account. In other words, the model assumes that different agents in the population have different functionals. The model estimate the proportion of each group to consider all forms of heterogeneity.

### Psychological expected utility model:

Caplin (2001) expanded the standard prize space to include the influence on preferences and decisions of anticipatory emotions such as suspense and anxiety. He replaced the standard prize space with a space of "psychological states". In this research he opens up a variety of psychologically interesting phenomena to rational analysis. This model explained how time inconsistency arises naturally in the presence of anticipations and also how emotions may change the result of choices. For example, this model finds that anxiety is anticipatory and that the desire to reduce anxiety motivates many decisions. A better understanding of the psychologically relevant outcome space will facilitate theorists to develop richer theory of determinants.