Editing Expected utility hypothesis (section)

==== The von Neumann–Morgenstern axioms ====
There are [[Von Neumann–Morgenstern utility theorem|four axioms]] of the expected utility theory that define a ''rational'' decision maker: completeness; transitivity; independence of irrelevant alternatives; and continuity.<ref>{{cite book | vauthors = von Neumann J, Morgenstern O |title=Theory of Games and Economic Behavior |url=https://archive.org/details/theoryofgameseco00vonn |url-access=registration |location=Princeton, NJ |publisher=Princeton University Press |orig-year=1944 |edition=Third |year=1953 }}</ref>

[[Completeness (order theory)|''Completeness'']] assumes that an individual has well-defined preferences and can always decide between any two alternatives.
* Axiom (Completeness): For every <math>A</math> and <math>B</math> either <math>A \succeq B</math> or <math>A \preceq B</math> or both.
This means that the individual prefers <math>A</math> to <math>B</math>, <math>B</math> to <math>A</math>, or is indifferent between <math>A</math> and <math>B</math>.

[[Transitive relation|''Transitivity'']] assumes that, as an individual decides according to the completeness axiom, the individual also decides consistently.
* Axiom (Transitivity): For every <math>A, B</math> and <math>C</math> with <math>A \succeq B</math> and <math> B \succeq C</math> we must have <math> A \succeq C</math>.

''[[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.{{Citation needed|date=September 2016}}.
* Axiom (Independence of irrelevant alternatives): For every <math>A, B</math> such that <math>A \succeq  B</math>, the preference <math>tA+(1-t)C \succeq t B+(1-t)C,</math> must hold for every lottery <math>C</math> and real <math>t \in [0, 1]</math>.

''Continuity'' assumes that when there are three lotteries (<math>A, B</math> and <math>C</math>) and the individual prefers <math>A</math> to <math>B</math> and <math>B</math> to <math>C</math>. There should be a possible combination of <math>A</math> and <math>C</math> in which the individual is then indifferent between this mix and the lottery <math>B</math>.
* Axiom (Continuity): Let <math>A, B</math> and <math>C</math> be lotteries with <math>A \succeq B \succeq C</math>. Then <math>B</math> is equally preferred to <math>pA+(1-p)C</math> for some <math>p\in [0,1]</math>.

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 <math>\succeq</math> amounts to choosing the lottery with the highest expected utility. This result is the [[Von Neumann–Morgenstern utility theorem|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|ordinal]] revolution" of the 1930s, and it revived the idea of [[cardinal utility]] in economic theory.{{Citation needed|date=August 2008}} 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.