# Utility theory

A theory dealing with individual preferences and the representation of these by numerical functions. A preference relation on a set of alternatives $X$ is a complete transitive binary relation $R$ on $X$ (cf. Pre-order); it is represented by a function $u(x)$ on $X$, and $u(x)$ is called a utility function if for any $x,y \in X$ it follows from $x R y$ that $u(x) \ge u(y)$, and vice versa. Therefore utility theory deals with ordered sets and their monotone mappings into a numerical space (usually one-dimensional). Utility theory arose from researches by economists in the 18th century; the basis of modern utility theory was laid in the 1940s by J. von Neumann and O. Morgenstern .
It is obvious that a utility function exists in the case of a finite set $X$. In the infinite case, a necessary and sufficient condition for the existence of a utility function is the existence of a utility-dense countable subset $A \subset X$, i.e. for any $x,y \in X \setminus A$ such that $x R^* y$, there exists a $z \in A$ such that $x R^* z$ and $z R^* y$, where $R^*$ is a strong preference relation ($x R^* y \Leftrightarrow x R y$ and not $y R x$). If $X$ is a convex set in a vector space, $R$ is continuous on $X$ and for any $x,y,z \in X$, $x R^* y$, and any $\alpha$, $0 < \alpha < 1$, it is true that $[\alpha x + (1-\alpha)z] R^* [\alpha y + (1-\alpha)z]$, then there exists a linear utility function that is unique, up to a positive linear transformation . Various combinations of weaker conditions lead to non-linear, discontinuous, or in some sense non-unique, utility functions. For example, if $X$ is a vector space, if it follows from $x R^* y$ that $[x+z] R^* [y+z]$ and if $[\alpha x] R^* [\alpha y]$ for all $z \in X$ and $\alpha > 0$, the function is single-valued and piecewise linear.
Utility theory also deals with stochastic ordering and ordering of the sums or differences of alternatives (in that case the utility function is constructed from a certain quaternary relation on $X$), as well as with generalizations to $n$-ary relations instead of binary ones, with the construction of a utility function at the same time with subjective probabilities, with the relation between the utility of multi-component alternatives and the utilities of the components, etc., , .