Difference between revisions of "Utility theory"
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| − | 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 $X$; 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]]]. | + | 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. | 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. | ||
Latest revision as of 09:14, 7 April 2018
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 [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 [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., [3], [4].
References
| [1] | J. von Neumann, O. Morgenstern, "Theory of games and economic behavior" , Princeton Univ. Press (1947) |
| [2] | G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973) |
| [3] | P.S. Fishburn, "Utility theory for decision making" , Wiley (1970) |
| [4] | P. Suppes, J. Zines, "Psychological measurements" , Moscow (1967) (In Russian; translated from English) |
Comments
Among other aspects of utility theory, one might mention transferable utility [a4], incomplete preferences [a1], non-standard utilities [a7], as well as the large body of literature concerning the (non-) significance of utility representations for economic theory (see, e.g., [a5] and [a6], but in the special case of concavifiable preference orderings one gets some measure of cardinal utility functions [a2], [a3]).
References
| [a1] | R.J. Aumann, , Human Judgement and Optimality , Wiley (1964) pp. 217–242 |
| [a2] | Y. Kannai, "The ALEP definition of complementary and least concave utility functions" J. Economic Th. , 22 (1980) pp. 115–117 |
| [a3] | Y. Kannai, , Generalized Concavity in Optimization and Economics , Acad. Press (1981) pp. 543–611 |
| [a4] | R.D. Luce, H. Raiffa, "Games and decisions. Introduction and critical survey" , Wiley (1957) |
| [a5] | P.A. Samuelson, "Foundations of economic analysis" , Harvard Univ. Press (1947) |
| [a6] | P.A. Samuelson, J. Economic Literature , 12 (1974) pp. 1255–1289 |
| [a7] | H. Skala, "Nonstandard utilities and the foundations of game theory" Internat. J. Game Theory , 3 (1974) pp. 67–81 |
| [a8] | R.M. Thrall (ed.) C.H. Coombs (ed.) R.L. Davis (ed.) , Decision processes , Wiley (1954) |
| [a9] | K.J. Arrow, F.H. Hahn, "General competitive analysis" , Oliver & Boyd (1971) |
| [a10] | G. Debreu, "Continuity properties of Paretian utility" Int. Econ. Review , 5 (1964) pp. 285–293 |
| [a11] | J.T. Rader, "The existence of a utility function to represent preferences" Rev. of Econ. Studies , XXX (1963) pp. 229–232 |
Utility theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Utility_theory&oldid=33729