# Expected utility hypothesis

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The expected utility hypothesis is a popular concept in economics, game theory and decision theory that serves as a reference guide for judging decisions involving uncertainty.[1] The theory recommends which option a rational individual should choose in a complex situation, based on his tolerance for risk and personal preferences.

The expected utility of an agent's risky decision is the mathematical expectation of his utility from different outcomes given their probabilities. If an agent derives 0 utils from 0 apples, 2 utils from one apple, and 3 utils from two apples, his expected utility for a 50-50 gamble between zero apples and two is .5u(0 apples) + .5u(2 apples) = .5(0 utils) + .5(3 utils) = 1.5 utils. Under the expected utility hypothesis, the consumer would prefer 1 apple with certainty (giving him 2 utils) to the gamble between zero and two.

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). In the example above, any function such that u(0) < (1) < u(2) would represent the same preferences; we could specify u(0)= 0, u(1) = 2, and u(2) = 40, for example. Under the expected utility hypothesis, setting u(2) = 3 requires if the agent is indifferent between one apple with certainty and a gamble with a 1/3 probability of no apple and a 2/3 probability of two apples, the utility of two apples must be set to u(2) = 2. This is because it requires that (1/3)u(0) + (2/3)u(2) = u(1), and 2 = (1/3)(0) + (2/3)(3).

The idea has antecedents in Daniel Bernoulli's 1738 St. Petersburg Paradox,[2] and was developed by Frank Ramsey and Leonard Jimmie Savage. 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. Until the mid-twentieth century, the standard term for the expected utility was the moral expectation, contrasted with "mathematical expectation" for the expected value.[3]

Although the expected utility hypothesis is standard in economic modelling, largely because of its simplicity and convenience, it has been found to be violated in psychology experiments. For many years, psychologists and economic theorists have been developing new theories to explain these deficiencies.[4] These include prospect theory, rank-dependent expected utility and cumulative prospect theory.

## Antecedents

### Limits of the Expected Value Theory

In the early days of the calculus of probability it was taken for granted that the value, and hence the ‘‘fair price’’ of a gamble, was the mathematical expectation of the gain.[2]  Classic utilitarians believed that the option who has the greatest utility will produce more pleasure or happiness for the agent and consequently must be chosen.[5] 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. 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.[6] 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.[6]

## 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[7][8] measure of absolute risk aversion:

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

where ${\displaystyle w}$ is wealth.

The Arrow–Pratt measure of relative risk aversion is:

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

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.[9] 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.

### The problem of interpersonal utility comparisons

Understanding utilities in term of personal preferences is really challenging as it face a challenge known as the Problem of Interpersonal Utility Comparisons or the Social Welfare Function. It is frequently pointed out that ordinary people usually make comparisons, however such comparisons are empirically meaningful because the interpersonal comparisons does not show the desire of strength which is extremely relevant to measure the expected utility of decision. In other words, beside we can know X and Y has similar or identical preferences (e.g. both love cars) we cannot determine which love it more or is willing to sacrifice more to get it.[30][31]

## Recommendations

In conclusion Expected Utility theories such as Savage and von Neumann-Morgenstern have to be improved or replaced by more general representations theorems.

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.[25][1] It is important the psychologist who study subjective Bayesian reasoning should carefully formulate the statement without ambiguity to avoid confusions.

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 there is heterogeneity in behaviour between individuals and within individuals. Applying a Mixture Model fits the data significantly better than either of the two preference functionals individually.[32] 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:[33]

In this model, Caplin(2001) expanded the standard prize space to include anticipatory emotions such suspense and anxiety influence on preferences and decisions. The author have replaced the standard prize space with a space of "psychological states," In this research, they open 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 this preceded emotions may change the result of choices, For example, this model founds 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.

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