Expected utility hypothesis

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[3] 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.[4] 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.[4]

The Savage representation theorem (Savage, 1954) A preference < satisfies P1–P7 if and only if there is a finitely additive probability measure P and a function u : C → R such that for every pair of acts f and g.[16] f < g ⇐⇒ Z Ω u(f(ω)) dP ≥ Z Ω u(g(ω)) dP [16] *If and only if all the axioms are satisfied when can used the information to reduce the uncertainty about the events that are out of your control. Additionally the theorem ranks the outcome according to utility function that reflects the personal preferences.

Key ingredients:

The key ingredients in Savage's theory are:

• States: The specification of every aspect of the decision problem at hand or  "A description of the world leaving no relevant aspect undescribed."[13]
• Events: A set of states identified by someone
• Consequences: A consequence is the description of all that is relevant to the decision maker's utility (e.g. monetary rewards, psychological factors, etc)
• Acts: An act is a finite-valued function that maps states to consequences.

There are four axioms of the expected utility theory that define a rational decision maker: completeness; transitivity; independence of irrelevant alternatives; and continuity.[17]

Completeness assumes that an individual has well defined preferences and can always decide between any two alternatives.

• Axiom (Completeness): For every ${\displaystyle A}$ and ${\displaystyle B}$ either ${\displaystyle A\succeq B}$ or ${\displaystyle A\preceq B}$ or both.

This means that the individual prefers ${\displaystyle A}$ to ${\displaystyle B}$, ${\displaystyle B}$ to ${\displaystyle A}$ or is indifferent between ${\displaystyle A}$ and ${\displaystyle B}$.

Transitivity assumes that, as an individual decides according to the completeness axiom, the individual also decides consistently.

• Axiom (Transitivity): For every ${\displaystyle A,B}$ and ${\displaystyle C}$ with ${\displaystyle A\succeq B}$ and ${\displaystyle B\succeq C}$ we must have ${\displaystyle A\succeq C}$.

Independence of irrelevant alternatives pertains to well-defined preferences as well. It assumes that two gambles mixed with an irrelevant third one will maintain the same order of preference as when the two are presented independently of the third one. The independence axiom is the most controversial axiom..

• Axiom (Independence of irrelevant alternatives): For every ${\displaystyle A,B}$ such that ${\displaystyle A\succeq B}$, the preference ${\displaystyle tA+(1-t)C\succeq tB+(1-t)C,}$ must hold for every lottery ${\displaystyle C}$ and real ${\displaystyle t\in [0,1]}$.

Continuity assumes that when there are three lotteries (${\displaystyle A,B}$ and ${\displaystyle C}$) and the individual prefers ${\displaystyle A}$ to ${\displaystyle B}$ and ${\displaystyle B}$ to ${\displaystyle C}$, then there should be a possible combination of ${\displaystyle A}$ and ${\displaystyle C}$ in which the individual is then indifferent between this mix and the lottery ${\displaystyle B}$.

• Axiom (Continuity): Let ${\displaystyle A,B}$ and ${\displaystyle C}$ be lotteries with ${\displaystyle A\succeq B\succeq C}$. Then ${\displaystyle B}$ is equally preferred to ${\displaystyle pA+(1-p)C}$ for some ${\displaystyle p\in [0,1]}$.

If all these axioms are satisfied, then the individual is said to be rational and the preferences can be represented by a utility function, i.e. one can assign numbers (utilities) to each outcome of the lottery such that choosing the best lottery according to the preference ${\displaystyle \succeq }$ amounts to choosing the lottery with the highest expected utility. This result is called the von Neumann–Morgenstern utility representation theorem.

In other words, if an individual's behavior always satisfies the above axioms, then there is a utility function such that the individual will choose one gamble over another if and only if the expected utility of one exceeds that of the other. The expected utility of any gamble may be expressed as a linear combination of the utilities of the outcomes, with the weights being the respective probabilities. Utility functions are also normally continuous functions. Such utility functions are also referred to as von Neumann–Morgenstern (vNM) utility functions. This is a central theme of the expected utility hypothesis in which an individual chooses not the highest expected value, but rather the highest expected utility. The expected utility maximizing individual makes decisions rationally based on the axioms of the theory.

The von Neumann–Morgenstern formulation is important in the application of set theory to economics because it was developed shortly after the Hicks–Allen "ordinal revolution" of the 1930s, and it revived the idea of cardinal utility in economic theory. However, while in this context the utility function is cardinal, in that implied behavior would be altered by a non-linear monotonic transformation of utility, the expected utility function is ordinal because any monotonic increasing transformation of expected utility gives the same behavior.

The utility function ${\displaystyle u(w)=\log(w)}$ was originally suggested by Bernoulli (see above). It has relative risk aversion constant and equal to one, and is still sometimes assumed in economic analyses. The utility function

${\displaystyle u(w)=-e^{-aw}}$

exhibits constant absolute risk aversion, and for this reason is often avoided, although it has the advantage of offering substantial mathematical tractability when asset returns are normally distributed. Note that, as per the affine transformation property alluded to above, the utility function ${\displaystyle K-e^{-aw}}$ gives exactly the same preferences orderings as does ${\displaystyle -e^{-aw}}$; thus it is irrelevant that the values of ${\displaystyle -e^{-aw}}$ and its expected value are always negative: what matters for preference ordering is which of two gambles gives the higher expected utility, not the numerical values of those expected utilities.

The class of constant relative risk aversion utility functions contains three categories. Bernoulli's utility function

${\displaystyle u(w)=\log(w)}$

has relative risk aversion equal to 1. The functions

${\displaystyle u(w)=w^{\alpha }}$

for ${\displaystyle \alpha \in (0,1)}$ have relative risk aversion equal to ${\displaystyle 1-\alpha \in (0,1)}$. And the functions

${\displaystyle u(w)=-w^{\alpha }}$

for ${\displaystyle \alpha <0}$ have relative risk aversion equal to ${\displaystyle 1-\alpha >1.}$

See also the discussion of utility functions having hyperbolic absolute risk aversion (HARA).

Often people refer to "risk" in the sense of a potentially quantifiable entity. In the context of mean-variance analysis, variance is used as a risk measure for portfolio return; however, this is only valid if returns are normally distributed or otherwise jointly elliptically distributed,[18][19][20] or in the unlikely case in which the utility function has a quadratic form. However, David E. Bell proposed a measure of risk which follows naturally from a certain class of von Neumann–Morgenstern utility functions.[21] Let utility of wealth be given by

${\displaystyle u(w)=w-be^{-aw}}$

for individual-specific positive parameters a and b. Then expected utility is given by

${\displaystyle {\begin{aligned}\operatorname {E} [u(w)]&=\operatorname {E} [w]-b\operatorname {E} [e^{-aw}]\\&=\operatorname {E} [w]-b\operatorname {E} [e^{-a\operatorname {E} [w]-a(w-\operatorname {E} [w])}]\\&=\operatorname {E} [w]-be^{-a\operatorname {E} [w]}\operatorname {E} [e^{-a(w-\operatorname {E} [w])}]\\&={\text{expected wealth}}-b\cdot e^{-a\cdot {\text{expected wealth}}}\cdot {\text{risk}}.\end{aligned}}}$

Thus the risk measure is ${\displaystyle \operatorname {E} (e^{-a(w-\operatorname {E} w)})}$, which differs between two individuals if they have different values of the parameter ${\displaystyle a,}$ allowing different people to disagree about the degree of risk associated with any given portfolio. Individuals sharing a given risk measure (based on given value of a) may choose different portfolios because they may have different values of b. See also Entropic risk measure.

For general utility functions, however, expected utility analysis does not permit the expression of preferences to be separated into two parameters with one representing the expected value of the variable in question and the other representing its risk.

Psychologists have discovered systematic violations of probability calculations and behavior by humans. This have been evidenced with examples such as the Monty Hall problem where it was demonstrated that people do not revise their degrees on belief in line with experimented probabilities and also that probabilities cannot be applied to single cases. On the other hand, in updating probability distributions using evidence, a standard method uses conditional probability, namely the rule of Bayes. An experiment on belief revision has suggested that humans change their beliefs faster when using Bayesian methods than when using informal judgment.[24]

According to the empirical results there has been almost no recognition in decision theory of the distinction between the problem of justifying its theoretical claims regarding the properties of rational belief and desire. One of the main reasons is because people's basic tastes and preferences for losses cannot be represented with utility as they change under different scenarios.[25]

Behavioral finance has produced several generalized expected utility theories to account for instances where people's choices deviate from those predicted by expected utility theory. These deviations are described as "irrational" because they can depend on the way the problem is presented, not on the actual costs, rewards, or probabilities involved. Particular theories include prospect theory, rank-dependent expected utility and cumulative prospect theory are considered insufficient to predict preferences and the expected utility.[26] Additionally, experiments have shown systematic violations and generalizations based on the results of Savage and von Neumann–Morgenstern. This is because preferences and utility functions constructed under different contexts are significantly different. This is demonstrated in the contrast of individual preferences under the insurance and lottery context shows the degree of indeterminacy of the expected utility theory. Additionally, experiments have shown systematic violations and generalizations based on the results of Savage and von Neumann–Morgenstern.

In practice there will be many situations where the probabilities are unknown, and one is operating under uncertainty. In economics, Knightian uncertainty or ambiguity may occur. Thus one must make assumptions about the probabilities, but then the expected values of various decisions can be very sensitive to the assumptions. This is particularly a problem when the expectation is dominated by rare extreme events, as in a long-tailed distribution. Alternative decision techniques are robust to uncertainty of probability of outcomes, either not depending on probabilities of outcomes and only requiring scenario analysis (as in minimax or minimax regret), or being less sensitive to assumptions.

Bayesian approaches to probability treat it as a degree of belief and thus they do not draw a distinction between risk and a wider concept of uncertainty: they deny the existence of Knightian uncertainty. They would model uncertain probabilities with hierarchical models, i.e. where the uncertain probabilities are modelled as distributions whose parameters are themselves drawn from a higher-level distribution (hyperpriors).

Starting with studies such as Lichtenstein & Slovic (1971), it was discovered that subjects sometimes exhibit signs of preference reversals with regard to their certainty equivalents of different lotteries. Specifically, when eliciting certainty equivalents, subjects tend to value "p bets" (lotteries with a high chance of winning a low prize) lower than "$ bets" (lotteries with a small chance of winning a large prize). When subjects are asked which lotteries they prefer in direct comparison, however, they frequently prefer the "p bets" over "$ bets".[27] Many studies have examined this "preference reversal", from both an experimental (e.g., Plott & Grether, 1979)[28] and theoretical (e.g., Holt, 1986)[29] standpoint, indicating that this behavior can be brought into accordance with neoclassical economic theory under specific assumptions.

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.[31] 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.

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.