Editing Expected utility hypothesis (section)

== Risk aversion ==
{{main|Risk aversion}}
The expected utility theory takes into account that individuals may be [[Risk aversion|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 function|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, risk-seeking individuals have convex utility functions, and risk-averse individuals have concave utility functions. The curvature of the utility function can measure the degree of risk aversion.

Since the risk attitudes are unchanged under [[affine transformation]]s of ''u'', the second derivative ''u<nowiki>''</nowiki>'' 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<ref name=":4">{{cite book | vauthors = Arrow KJ | date = 1965 | chapter = The theory of risk aversion | title = Aspects of the Theory of Risk Bearing Reprinted in Essays in the Theory of Risk Bearing | veditors = Saatio YJ  | publisher = Markham Publ. Co. | location = Chicago, 1971 | pages = 90–109 }}</ref><ref name=":5">{{cite journal| vauthors = Pratt JW |date=January–April 1964|title=Risk aversion in the small and in the large|journal=Econometrica|volume=32|issue=1/2|pages=122–136|doi=10.2307/1913738|jstor=1913738}}</ref> measure of absolute risk aversion:

: <math>\mathit{ARA}(w) =-\frac{u''(w)}{u'(w)},</math>

where <math>w</math> is wealth.

The Arrow–Pratt measure of relative risk aversion is:

: <math>\mathit{RRA}(w) =-\frac{wu''(w)}{u'(w)}</math>

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.<ref>Castagnoli and LiCalzi, 1996; Bordley and LiCalzi, 2000; Bordley and Kirkwood</ref> 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.