Editing Expected utility hypothesis (section)

== Formula for expected utility ==
When the entity <math>x</math> whose value <math> x_i</math> affects a person's utility takes on one of a set of [[discrete and continuous variables|discrete values]], the formula for expected utility, which is assumed to be maximized, is
:<math>\operatorname E[u(x)]=p_1 \cdot u(x_1)+p_2 \cdot u(x_2)+\cdots</math>
where the left side is the subjective valuation of the gamble as a whole, <math>x_i</math> is the ''i''th possible outcome, <math>u(x_i)</math> is its valuation, and <math>p_i</math> is its probability. There could be either a finite set of possible values <math>x_i,</math>, in which case the right side of this equation has a finite number of terms, or there could be an infinite set of discrete values, in which case the right side has an infinite number of terms.

When <math>x</math> can take on any of a continuous range of values, the expected utility is given by
:<math>\operatorname E[u(x)] = \int_{-\infty}^\infty u(x)f(x) \, dx,</math>
where <math>f(x)</math> is the [[probability density function]] of <math>x.</math>

The [[certainty equivalent]], the fixed amount that would make a person indifferent to it vs. the distribution {{tmath|f(x)}}, is given by <math>\mathrm{CE} = u^{-1}(\operatorname E[u(x)])\,.</math>

===Measuring risk in the expected utility context===
Often, people refer to "risk" as a potentially quantifiable entity. In the context of [[modern portfolio theory|mean-variance analysis]], [[variance]] is used as a risk measure for portfolio return; however, this is only valid if returns are [[normal distribution|normally distributed]] or otherwise [[elliptical distribution|jointly elliptically distributed]],<ref>{{cite journal| vauthors = Borch K |date=January 1969|title=A note on uncertainty and indifference curves|journal=Review of Economic Studies|volume=36|issue=1|pages=1–4|doi=10.2307/2296336|jstor=2296336}}</ref><ref>{{cite journal| vauthors = Chamberlain G |year=1983|title=A characterization of the distributions that imply mean-variance utility functions|journal=Journal of Economic Theory|volume=29|issue=1|pages=185–201|doi=10.1016/0022-0531(83)90129-1}}</ref><ref>{{cite journal| vauthors = Owen J, Rabinovitch R |year=1983|title=On the class of elliptical distributions and their applications to the theory of portfolio choice|journal=Journal of Finance|volume=38|issue=3|pages=745–752|doi=10.2307/2328079|jstor=2328079}}</ref> or in the unlikely case in which the utility function has a quadratic form—however, David E. Bell proposed a measure of risk that follows naturally from a certain class of von Neumann–Morgenstern utility functions.<ref>{{cite journal| vauthors = Bell DE |date=December 1988|title=One-switch utility functions and a measure of risk|journal=Management Science|volume=34|issue=12|pages=1416–24|doi=10.1287/mnsc.34.12.1416}}</ref>  Let utility of wealth be given by

:<math> u(w)= w-be^{-aw}</math>

for individual-specific positive parameters ''a'' and ''b''.  Then, the expected utility is given by
:<math>
\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}
</math>
Thus the risk measure is <math>\operatorname{E}(e^{-a(w-\operatorname{E}w)})</math>, which differs between two individuals if they have different values of the parameter <math>a,</math> allowing other people to disagree about the degree of risk associated with any given portfolio. Individuals sharing a given risk measure (based on a 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, one representing the expected value of the variable in question and the other representing its risk.