Expected utility hypothesis

 ( Belief Revision ) 

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."
• Events: A set of states identified by someone
• Consequences: A consequence describes everything 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.

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 $A\; \backslash succeq\; B$ or $A\; \backslash preceq\; B$ or both.

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 $A\; \backslash succeq\; B$ and $B\; \backslash succeq\; C$ we must have $A\; \backslash succeq\; C$.

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 (Independence of irrelevant alternatives): For every $A,\; B$ such that $A\; \backslash succeq\; B$, the preference $tA+(1-t)C\; \backslash succeq\; t\; B+(1-t)C,$ must hold for every lottery $C$ and real $t\; \backslash in\; 0,$.

Continuity assumes that when there are three lotteries ($A,\; B$ and $C$) and the individual prefers $A$ to $B$ and $B$ to $C$. 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 $A\; \backslash succeq\; B\; \backslash succeq\; C$. Then $B$ is equally preferred to $pA+(1-p)C$ for some $p\backslash in\; 0,1$.

If all these axioms are satisfied, then the individual is rational. A utility function can represent the preferences, i.e., one can assign numbers (utilities) to each outcome of the lottery such that choosing the best lottery according to the preference $\backslash succeq$ amounts to choosing the lottery with the highest expected utility. This result is 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 called von Neumann–Morgenstern (vNM). 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 theory's axioms.

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 utility 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 nonlinear monotonic transformation of utility, the expected utility function is ordinal because any monotonic increasing transformation of expected utility gives the same behavior.

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

It 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 $K-e^\{-aw\}$ gives 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

$u(w)\; =\; \backslash log(w)$

Has relative risk aversion equal to 1. The functions

$u(w)\; =\; w^\{\backslash alpha\}$

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

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

for have relative risk aversion equal to $1-\backslash alpha\; >1.$

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

$\backslash operatorname\; Eu(x)=p\_1\; \backslash cdot\; u(x\_1)+p\_2\; \backslash cdot\; u(x\_2)+\backslash cdots$

When $x$ can take on any of a continuous range of values, the expected utility is given by

$\backslash operatorname\; Eu(x)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; u(x)f(x)\; \backslash ,\; dx,$

The certainty equivalent, which is the fixed amount that would make a person indifferent to it versus the outcome distribution, is given by $\backslash mathrm\{CE\}\; =\; u^\{-1\}(\backslash operatorname\; Eu(x))\backslash ,.$

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

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

                   &=\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}.
     

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

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:

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

where $w$ is wealth.

The Arrow–Pratt measure of relative risk aversion is:

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

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. These functions are often used in economics to simplify.

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.