Editing Utility (section)

==Expected utility==
{{Main|Expected utility hypothesis}}
Expected utility theory deals with the analysis of choices among '''risky''' projects with multiple (possibly multidimensional) outcomes.

The [[St. Petersburg paradox]] was first proposed by [[Nicolaus I Bernoulli|Nicholas Bernoulli]] in 1713 and solved by [[Daniel Bernoulli]] in 1738, although the Swiss mathematician [[Gabriel Cramer]] proposed taking the expectation of a square-root utility function of money in an 1728 letter to N. Bernoulli. D. Bernoulli argued that the paradox could be resolved if decision-makers displayed [[risk aversion]] and argued for a logarithmic cardinal utility function. (Analysis of international survey data during the 21st century has shown that insofar as utility represents happiness, as for [[utilitarianism]], it is indeed proportional to log of income.)

The first important use of the expected utility theory was that of [[John von Neumann]] and [[Oskar Morgenstern]], who used the assumption of expected utility maximization in their formulation of [[game theory]].

In finding the probability-weighted average of the utility from each possible outcome:

:<math>\text{EU} = \Pr(z) \cdot u(\text{Value}(z)) + \Pr(y) \cdot u(\text{Value}(y))</math>

===Von Neumann–Morgenstern===
{{Main|Von Neumann–Morgenstern utility theorem}}

Von Neumann and Morgenstern addressed situations in which the outcomes of choices are not known with certainty, but have probabilities associated with them.

A notation for a ''[[Lottery (probability)|lottery]]'' is as follows: if options A and B have probability ''p'' and 1&nbsp;−&nbsp;''p'' in the lottery, we write it as a linear combination:
:<math>
L = p A + (1-p) B
</math>

More generally, for a lottery with many possible options:

:<math>
L = \sum_i p_i A_i,
</math>

where <math>\sum_i p_i =1</math>.

By making some reasonable assumptions about the way choices behave, von Neumann and Morgenstern showed that if an agent can choose between the lotteries, then this agent has a utility function such that the desirability of an arbitrary lottery can be computed as a linear combination of the utilities of its parts, with the weights being their probabilities of occurring.

This is termed the ''expected utility theorem''. The required assumptions are four axioms about the properties of the agent's [[Preference (economics)|preference relation]] over 'simple lotteries', which are lotteries with just two options. Writing <math>B\preceq A</math> to mean 'A is weakly preferred to B' ('A is preferred at least as much as B'), the axioms are:

# completeness: For any two simple lotteries <math>L</math> and <math>M</math>, either <math>L\preceq M</math> or <math>M\preceq L</math> (or both, in which case they are viewed as equally desirable).
# transitivity: for any three lotteries <math>L,M,N</math>, if <math>L\preceq M</math> and <math>M\preceq N</math>, then <math>L\preceq N</math>.
# convexity/continuity (Archimedean property): If <math>L \preceq M\preceq N</math>, then there is a <math>p</math> between 0 and 1 such that the lottery <math>pL + (1-p)N</math> is equally desirable as <math>M</math>.
# independence: for any three lotteries <math>L,M,N</math> and any probability ''p'', <math>L \preceq M</math> if and only if <math>pL+(1-p)N \preceq pM+(1-p)N</math>. Intuitively, if the lottery formed by the probabilistic combination of <math>L</math> and <math>N</math> is no more preferable than the lottery formed by the same probabilistic combination of <math>M</math> and <math>N,</math> then and only then <math>L \preceq M</math>.

Axioms 3 and 4 enable us to decide about the relative utilities of two assets or lotteries.

In more formal language: A von Neumann–Morgenstern utility function is a function from choices to the real numbers:
:<math> u\colon X\to \R</math>
which assigns a real number to every outcome in a way that represents the agent's preferences over simple lotteries. Using the four assumptions mentioned above, the agent will prefer a lottery <math>L_2</math> to a lottery <math>L_1</math> if and only if, for the utility function characterizing that agent, the expected utility of <math>L_2</math> is greater than the expected utility of <math>L_1</math>:
:<math>L_1\preceq L_2 \text{ iff } u(L_1)\leq u(L_2)</math>.

Of all the axioms, independence is the most often discarded. A variety of [[generalized expected utility]] theories have arisen, most of which omit or relax the independence axiom.