# Expected utility hypothesis

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Expected utility hypothesis

In economics, game theory, and decision theory the expected utility hypothesis is a theory of utility in which "betting preferences" of people with regard to uncertain outcomes (gambles) are represented by a function of the payouts (whether in money or other goods), the probabilities of occurrence, risk aversion, and the different utility of the same payout to people with different assets or personal preferences. This theory has proved 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. Daniel Bernoulli initiated this theory in 1738. Until the mid twentieth century, the standard term for the expected utility was the moral expectation, contrasted with "mathematical expectation" for the expected value.

The von Neumann–Morgenstern utility theorem provides necessary and sufficient "rationality" axioms under which the expected utility hypothesis holds.

## Expected value and choice under risk

In the presence of risky outcomes, a decision maker could use the expected value criterion as a rule of choice: higher expected value investments are simply the preferred ones. For example, suppose there is a gamble in which the probability of getting a $100 payment is 1 in 80 and the alternative, and far more likely, outcome, is getting nothing. Then the expected value of this gamble is$1.25. Given the choice between this gamble and a guaranteed payment of $1, by this simple expected value theory people would choose the$100-or-nothing gamble. However, under expected utility theory, some people would be risk averse enough to prefer the sure thing, even though it has a lower expected value, while other less risk averse people would still choose the riskier, higher-mean gamble.

## Bernoulli's formulation

Nicolas Bernoulli described the St. Petersburg paradox (involving infinite expected values) in 1713, prompting two Swiss mathematicians to develop expected utility theory as a solution. The theory can also more accurately describe more realistic scenarios (where expected values are finite) than expected value alone.

In 1728, Gabriel Cramer, in a letter to Nicolas Bernoulli, wrote, "the mathematicians estimate money in proportion to its quantity, and men of good sense in proportion to the usage that they may make of it."

In 1738, Nicolas' cousin Daniel Bernoulli, published the canonical 18th Century description of this solution in Specimen theoriae novae de mensura sortis or Exposition of a New Theory on the Measurement of Risk.

Daniel Bernoulli proposed that a mathematical function should be used to correct the expected value depending on probability. This provides a way to account for risk aversion, where the risk premium is higher for low-probability events than the difference between the payout level of a particular outcome and its expected value.

Bernoulli's paper was the first formalization of marginal utility, which has broad application in economics in addition to expected utility theory. He used this concept to formalize the idea that the same amount of additional money was less useful to an already-wealthy person than it would be to a poor person.

## Infinite expected value — St. Petersburg paradox

The St. Petersburg paradox (named after the journal in which Bernoulli's paper was published) arises when there is no upper bound on the potential rewards from very low probability events. Because some probability distribution functions have an infinite expected value, an expected-wealth maximizing person would pay an infinite amount to take this gamble. In real life, people do not do this.

Bernoulli proposed a solution to this paradox in his paper: the utility function used in real life means that the expected utility of the gamble is finite, even if its expected value is infinite. (Thus he hypothesized diminishing marginal utility of increasingly larger amounts of money.) It has also been resolved differently by other economists by proposing that very low probability events are neglected, by taking into account the finite resources of the participants, or by noting that one simply cannot buy that which is not sold (and that sellers would not produce a lottery whose expected loss to them were unacceptable).

## Von Neumann–Morgenstern formulation

### The von Neumann-Morgenstern axioms

There are four axioms of the expected utility theory that define a rational decision maker. They are completeness, transitivity, independence and continuity.

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

• Axiom (Completeness): For every A and B either $A \succ B$, AB or $A \prec B$.

This means that the individual either prefers A to B, or is indifferent between A and B, or prefers B to A.

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

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

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

• Axiom (Independence): Let A, B, and C be three lotteries with $A \succeq B$, and let $t \in (0, 1]$; then $tA+(1-t)C \succeq t B+(1-t)C$ .

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

• Axiom (Continuity): Let A, B and C be lotteries with $A \succeq B \succeq C$; then there exists a probability p such that B is equally good as pA + (1 − p)C.

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 $\succeq$ 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 always chooses his/her most preferred alternative available, then the individual will choose one gamble over another if and only if there is a utility function such that 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.[citation needed] Note, however, that 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 utilty function is ordinal because any monotonic increasing transformation of it gives the same behavior.

### Risk aversion

The expected utility theory implies that rational individuals act as though they were maximizing expected utility, and allows for the fact that many individuals are risk averse,[citation needed] 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 first 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: $ARA(w) =-\frac{u''(w)}{u'(w)}$

The Arrow–Pratt measure of relative risk aversion is: $RRA(w) =-\frac{wu''(w)}{u'(w)}$

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 (Castagnoli and LiCalzi,1996; Bordley and LiCalzi,2000;Bordley and Kirkwood, ). 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.

### Examples of von Neumann-Morgenstern utility functions

The utility function 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 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 Ke aw gives exactly the same preferences orderings as does e aw; thus it is irrelevant that the values of 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

u(w) = log(w)

has relative risk aversion equal to unity. The functions

u(w) = wα

for $\alpha \in (0,1)$ have relative risk aversion equal to 1 − α. And the functions

u(w) = − wα

for α < 0 also have relative risk aversion equal to 1 − α.

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

### Measuring risk in the expected utility context

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. However, D. E. Bell proposed a measure of risk which follows naturally from a certain class of von Neumann-Morgenstern utility functions. Let utility of wealth be given by u(w) = wbe aw for individual-specific positive parameters a and b. Then expected utility is given by \begin{align} \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{align}

Thus the risk measure is $\operatorname{E}(e^{-a(w-\operatorname{E}w)})$, which differs between two individuals if they have different values of the parameter a, allowing different people to disagree about the degree of risk associated with any given portfolio.

## Behavioral decision science: Deviations from expected utility theory

Like any mathematical model, expected utility theory is an abstraction and simplification of reality. The mathematical correctness of expected utility theory and the salience of its primitive concepts do not guarantee that expected utility theory is a reliable guide to human behavior or optimal practice.

The mathematical clarity of expected utility theory has helped scientists design experiments to test its adequacy, and to distinguish systematic departures from its predictions.

### Conservatism in updating beliefs

It is well established that humans find logic hard, mathematics harder, and probability even more challenging[citation needed]. Psychologists have discovered systematic violations of probability calculations and behavior by humans.[citation needed]

In updating probability distributions using evidence, a standard method uses conditional probability, namely the rule of Bayes. Some experiments on belief revision have suggested that humans change their beliefs faster when using Bayesian methods than when using informal judgment.

### Irrational deviations

Behavioral finance has produced several generalized expected utility theories to account for instances where people's choice 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 and SP/A theory

### Preference reversals over uncertain outcomes

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." Many studies have examined this "preference reversal," from both an experimental (e.g., Plott & Grether, 1979) and theoretical (e.g., Holt, 1986) standpoint, indicating that this behavior can be brought into accordance with neoclassical economic theory under some specific assumptions.

### Uncertain probabilities

Applying expected value and expected utility to decision-making requires knowing the probability of various outcomes. However, this is unknown in practice: one is operating under uncertainty (in economics, one talks of Knightian uncertainty). Thus one must make assumptions, but then the expected value of various decisions is 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.

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