View source for 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 [[#References|[1]]].

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 [[#References|[3]]]. 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., [[#References|[3]]], [[#References|[4]]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  J. von Neumann,   O. Morgenstern,   "Theory of games and economic behavior" , Princeton Univ. Press  (1947)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  G. Birkhoff,   "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  P.S. Fishburn,   "Utility theory for decision making" , Wiley  (1970)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  P. Suppes,   J. Zines,   "Psychological measurements" , Moscow  (1967)  (In Russian; translated from English)</TD></TR>
</table>



====Comments====
Among other aspects of utility theory, one might mention transferable utility [[#References|[a4]]], incomplete preferences [[#References|[a1]]], non-standard utilities [[#References|[a7]]], as well as the large body of literature concerning the (non-) significance of utility representations for economic theory (see, e.g., [[#References|[a5]]] and [[#References|[a6]]], but in the special case of concavifiable preference orderings one gets some measure of cardinal utility functions [[#References|[a2]]], [[#References|[a3]]]).

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  R.J. Aumann,  , ''Human Judgement and Optimality'' , Wiley  (1964)  pp. 217–242</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  Y. Kannai,   "The ALEP definition of complementary and least concave utility functions"  ''J. Economic Th.'' , '''22'''  (1980)  pp. 115–117</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  Y. Kannai,  , ''Generalized Concavity in Optimization and Economics'' , Acad. Press  (1981)  pp. 543–611</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  R.D. Luce,   H. Raiffa,   "Games and decisions. Introduction and critical survey" , Wiley  (1957)</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  P.A. Samuelson,   "Foundations of economic analysis" , Harvard Univ. Press  (1947)</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  P.A. Samuelson,   ''J. Economic Literature'' , '''12'''  (1974)  pp. 1255–1289</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  H. Skala,   "Nonstandard utilities and the foundations of game theory"  ''Internat. J. Game Theory'' , '''3'''  (1974)  pp. 67–81</TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top">  R.M. Thrall (ed.)  C.H. Coombs (ed.)  R.L. Davis (ed.) , ''Decision processes'' , Wiley  (1954)</TD></TR>
<TR><TD valign="top">[a9]</TD> <TD valign="top">  K.J. Arrow,   F.H. Hahn,   "General competitive analysis" , Oliver &amp; Boyd  (1971)</TD></TR>
<TR><TD valign="top">[a10]</TD> <TD valign="top">  G. Debreu,   "Continuity properties of Paretian utility"  ''Int. Econ. Review'' , '''5'''  (1964)  pp. 285–293</TD></TR>
<TR><TD valign="top">[a11]</TD> <TD valign="top">  J.T. Rader,   "The existence of a utility function to represent preferences"  ''Rev. of Econ. Studies'' , '''XXX'''  (1963)  pp. 229–232</TD></TR>
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[[Category:Game theory, economics, social and behavioral sciences]]
[[Category:Order, lattices, ordered algebraic structures]]