# Expected utility hypothesis

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The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on their risk appetite and preferences.

## Contents

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.  On a graph, the curvature of u will explain the agent's risk attitude.

For example, if an agent derives 0 utils from 0 apples, 2 utils from one apple, and 3 utils from two apples, their expected utility for a 50–50 gamble between zero apples and two is 0.5u(0 apples) + 0.5u(2 apples) = 0.5(0 utils) + 0.5(3 utils) = 1.5 utils. Under the expected utility hypothesis, the consumer would prefer 1 apple (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 and assuming 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, requires that the utility of one apple must be set to u(1) = 2. This is because it requires that (1/3)u(0) + (2/3)u(2) = u(1), and (1/3)(0) + (2/3)(3) = 2.

Although the expected utility hypothesis is standard in economic modelling, 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.  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.

### 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.  

## 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.  

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: 

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.

## Related Research Articles

As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosophers such as Jeremy Bentham and John Stuart Mill. The term has been adapted and reapplied within neoclassical economics, which dominates modern economic theory, as a utility function that represents a single consumer's preference ordering over a choice set but is not comparable across consumers. This concept of utility is personal and based on choice rather than on pleasure received, and so is specified more rigorously than the original concept but makes it less useful for ethical decisions. In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more certain outcome. Risk aversion explains the inclination to agree to a situation with a more predictable, but possibly lower payoff, rather than another situation with a highly unpredictable, but possibly higher payoff. For example, a risk-averse investor might choose to put their money into a bank account with a low but guaranteed interest rate, rather than into a stock that may have high expected returns, but also involves a chance of losing value. Prospect theory is a theory of behavioral economics and behavioral finance that was developed by Daniel Kahneman and Amos Tversky in 1979. The theory was cited in the decision to award Kahneman the 2002 Nobel Memorial Prize in Economics.

In mathematical optimization and decision theory, a loss function or cost function is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite, in which case it is to be maximized.

Decision theory is the study of an agent's choices. Decision theory can be broken into two branches: normative decision theory, which analyzes the outcomes of decisions or determines the optimal decisions given constraints and assumptions, and descriptive decision theory, which analyzes how agents actually make the decisions they do.

In decision theory, subjective expected utility is the attractiveness of an economic opportunity as perceived by a decision-maker in the presence of risk. Characterizing the behavior of decision-makers as using subjective expected utility was promoted and axiomatized by L. J. Savage in 1954 following previous work by Ramsey and von Neumann. The theory of subjective expected utility combines two subjective concepts: first, a personal utility function, and second a personal probability distribution. The Ellsberg paradox is a paradox of choice in which people's decisions produce inconsistencies with subjective expected utility theory. The paradox was popularized by Daniel Ellsberg in his 1961 paper “Risk, Ambiguity, and the Savage Axioms”, although a version of it was noted considerably earlier by John Maynard Keynes. It is generally taken to be evidence for ambiguity aversion, in which a person tends to prefer choices with quantifiable risks over those with unknown risks.

Revealed preference theory, pioneered by economist Paul Anthony Samuelson in 1938, is a method of analyzing choices made by individuals, mostly used for comparing the influence of policies on consumer behavior. Revealed preference models assume that the preferences of consumers can be revealed by their purchasing habits.

The Allais paradox is a choice problem designed by Maurice Allais (1953) to show an inconsistency of actual observed choices with the predictions of expected utility theory.

Stochastic dominance is a partial order between random variables. It is a form of stochastic ordering. The concept arises in decision theory and decision analysis in situations where one gamble can be ranked as superior to another gamble for a broad class of decision-makers. It is based on shared preferences regarding sets of possible outcomes and their associated probabilities. Only limited knowledge of preferences is required for determining dominance. Risk aversion is a factor only in second order stochastic dominance.

In decision theory and economics, ambiguity aversion is a preference for known risks over unknown risks. An ambiguity-averse individual would rather choose an alternative where the probability distribution of the outcomes is known over one where the probabilities are unknown. This behavior was first introduced through the Ellsberg paradox. Cumulative prospect theory (CPT) is a model for descriptive decisions under risk and uncertainty which was introduced by Amos Tversky and Daniel Kahneman in 1992. It is a further development and variant of prospect theory. The difference between this version and the original version of prospect theory is that weighting is applied to the cumulative probability distribution function, as in rank-dependent expected utility theory but not applied to the probabilities of individual outcomes. In 2002, Daniel Kahneman received the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel for his contributions to behavioral economics, in particular the development of Cumulative Prospect Theory (CPT).

In decision theory, the von Neumann–Morgenstern (VNM) utility theorem shows that, under certain axioms of rational behavior, a decision-maker faced with risky (probabilistic) outcomes of different choices will behave as if he or she is maximizing the expected value of some function defined over the potential outcomes at some specified point in the future. This function is known as the von Neumann–Morgenstern utility function. The theorem is the basis for expected utility theory.

In decision theory, economics, and finance, a two-moment decision model is a model that describes or prescribes the process of making decisions in a context in which the decision-maker is faced with random variables whose realizations cannot be known in advance, and in which choices are made based on knowledge of two moments of those random variables. The two moments are almost always the mean—that is, the expected value, which is the first moment about zero—and the variance, which is the second moment about the mean.

In finance, economics, and decision theory, hyperbolic absolute risk aversion (HARA) refers to a type of risk aversion that is particularly convenient to model mathematically and to obtain empirical predictions from. It refers specifically to a property of von Neumann–Morgenstern utility functions, which are typically functions of final wealth, and which describe a decision-maker's degree of satisfaction with the outcome for wealth. The final outcome for wealth is affected both by random variables and by decisions. Decision-makers are assumed to make their decisions so as to maximize the expected value of the utility function.

In economics and other social sciences, preference is the order that a person gives to alternatives based on their relative utility, a process which results in an optimal "choice". Preferences are evaluations, they concern matters of value, typically in relation to practical reasoning. Instead of the prices of goods, personal income, or availability of goods, the character of the preferences is determined purely by a person's tastes. However, persons are still expected to act in their best interest. Rationality, in this context, means that when individuals are faced with a choice, they would select the option that maximizes self interest. Further, in every set of alternatives, preferences arise.

In expected utility theory, a lottery is a discrete distribution of probability on a set of states of nature. The elements of a lottery correspond to the probabilities that each of the states of nature will occur, e.g.. Much of the theoretical analysis of choice under uncertainty involves characterizing the available choices in terms of lotteries.

Risk aversion is a preference for a sure outcome over a gamble with higher or equal expected value. Conversely, the rejection of a sure thing in favor of a gamble of lower or equal expected value is known as risk-seeking behavior.

In economics, the Debreu theorems are several statements about the representation of a preference ordering by a real-valued function. The theorems were proved by Gerard Debreu during the 1950s.

In decision theory, a multi-attribute utility function is used to represent the preferences of an agent over bundles of goods either under conditions of certainty about the results of any potential choice, or under conditions of uncertainty.

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