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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u0959101.png" /> is a complete transitive binary relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u0959102.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u0959103.png" />; it is represented by a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u0959104.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u0959105.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u0959106.png" /> is called a utility function if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u0959107.png" /> it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u0959108.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u0959109.png" />, 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]]].
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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]]].
  
It is obvious that a utility function exists in the case of a finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591010.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591011.png" />, i.e. for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591013.png" />, there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591014.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591017.png" /> is a strong preference relation (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591018.png" /> and not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591019.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591020.png" /> is a convex set in a vector space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591021.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591022.png" /> and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591024.png" />, and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591026.png" />, it is true that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591027.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591028.png" /> is a vector space, if it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591029.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591030.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591031.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591033.png" />, the function is single-valued and piecewise linear.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591034.png" />), as well as with generalizations to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095910/u09591035.png" />-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]]].
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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====
 
====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>
+
<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>
  
  
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====References====
 
====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></table>
+
<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>
 +
</table>
 +
 
 +
{{TEX|done}}

Revision as of 18:41, 17 October 2014

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 [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
How to Cite This Entry:
Utility theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Utility_theory&oldid=33727
This article was adapted from an original article by E.I. Vilkas (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article