utility representation theorem

Suppose a person is asked questions of the form "Do you prefer A or B?" (when A and B can be options, actions to take, states of the world, consumption bundles, etc.). If the agent prefers A to B, we write $A\backslash succ\; B$. The set of all such preference-pairs forms the person's preference relation.

Instead of recording the person's preferences between every pair of options, it would be much more convenient to have a single utility function - a function u that assigns a real number to each option, such that $u(A)>u(B)$ if and only if $A\backslash succ\; B$.

Not every preference-relation has a utility-function representation. For example, if the relation is not transitive (the agent prefers A to B, B to C, and C to A), then it has no utility representation, since any such utility function would have to satisfy $u(A)>u(B)\; >\; u(C)\; >\; u(A)$, which is impossible.

A utility representation theorem gives conditions on a preference relation, that are sufficient for the existence of a utility representation.

Often, one would like the representing function u to satisfy additional conditions, such as continuity. This requires additional conditions on the preference relation.

The set of options is a topological space denoted by X. In some cases we assume that X is also a metric space; in particular, X can be a subset of a Euclidean space R^m, such that each coordinate in {1,..., m} represents a commodity, and each m-vector in X represents a possible consumption bundle.

A preference relation is a subset of $X\backslash times\; X$. It is denoted by either $\backslash succ$ or $\backslash succeq$:

• The notation $\backslash succ$ is used when the relation is strict, that is, $A\backslash succ\; B$ means that option A is strictly better than option B. In this case, the relation should be irreflexive, that is, $A\backslash succ\; A$ does not hold. It should also be asymmetric, that is, $A\backslash succ\; B$ implies that not $B\backslash succ\; A$.
• The notation $\backslash succeq$ is used when the relation is weak, that is, $A\backslash succeq\; B$ means that option A is at least as good as option B (A may be equivalent to B, or better than B). In this case, the relation should be reflexive, that is, $A\backslash succeq\; A$ always holds.

Given a weak preference relation $\backslash succeq$, one can define its "strict part" $\backslash succ$ and "indifference part" $\backslash simeq$ as follows:

• $A\backslash succ\; B$ if and only if $A\backslash succeq\; B$ and not $B\backslash succeq\; A$.
• $A\; \backslash simeq\; B$ if and only if $A\backslash succeq\; B$ and $B\backslash succeq\; A$.

Given a strict preference relation $\backslash succ$, one can define its "weak part" $\backslash succeq$ and "indifference part" $\backslash simeq$ as follows:

• $A\backslash succeq\; B$ if and only if not $B\; \backslash succ\; A$;
• $A\; \backslash simeq\; B$ if and only if not $B\; \backslash succ\; A$ and not $A\; \backslash succ\; B$.

For every option $A\; \backslash in\; X$, we define the contour sets at A:

• Given a weak preference relation $\backslash succeq$, the weak upper contour set at A is the set of all options that are at least as good as A: $\backslash \{B\backslash in\; X\; :\; B\backslash succeq\; A\; \backslash \}$. The weak lower contour set at A is the set of all options that are at most as good as A: $\backslash \{B\backslash in\; X\; :\; A\; \backslash succeq\; B\; \backslash \}$.
• A weak preference relation is called continuous if its contour sets are topologically closed.
• Similarly, given a strict preference relation $\backslash succ$, the strict upper contour set at A is the set of all options better than A: $\backslash \{B\backslash in\; X\; :\; B\backslash succ\; A\; \backslash \}$, and the strict lower contour set at A is the set of all options worse than A: $\backslash \{B\backslash in\; X\; :\; A\; \backslash succ\; B\; \backslash \}$.
• A strict preference relation is called continuous if its contour sets are topologically open.

Sometimes, the above continuity notions are called semicontinuous, and a $\backslash succeq$ is called continuous if it is a closed subset of $X\backslash times\; X$.{{Cite journal |last1=Evren |first1=Özgür |last2=Ok |first2=Efe A. |date=2011-08-01 |title=On the multi-utility representation of preference relations |url=https://www.sciencedirect.com/science/article/pii/S0304406811000693 |journal=Journal of Mathematical Economics |language=en |volume=47 |issue=4 |pages=554–563 |doi=10.1016/j.jmateco.2011.07.003 |issn=0304-4068|url-access=subscription }}

A preference-relation is called:

• Countable - if the set of equivalence classes of the indifference relation $\backslash simeq$ is countable.
• Separable - if there exists a countable subset $Z\backslash subseteq\; X$ such that for every pair $A\backslash succ\; B$, there is an element $z\_i\backslash in\; Z$ that separates them, that is, $A\; \backslash succ\; z\_i\; \backslash succ\; B$ (an analogous definition exists for weak relations).

As an example, the strict order ">" on real numbers is separable, but not countable.

A utility function is a function $u:\; X\; \backslash to\; \backslash mathbb\{R\}$.

• A utility function u is said to represent a strict preference relation $\backslash succ$, if $u(A)\; >\; u(B)\; \backslash iff\; A\backslash succ\; B$.
• A utility function u is said to represent a weak preference relation $\backslash succeq$, if $u(A)\; \backslash geq\; u(B)\; \backslash iff\; A\; \backslash succeq\; B$.

{{Main|Debreu's representation theorems}}Debreu{{cite book |author=Debreu, Gerard |url=https://archive.org/details/decisionprocesse033215mbp/page/n172/mode/1up |title=Representation of a preference ordering by a numerical function |year=1954}}{{Cite book |last=Debreu |first=Gerard |url=https://www.worldcat.org/oclc/25466669 |title=Mathematical economics : twenty papers of Gerard Debreu; introduction by Werner Hildenbrand. |date=1986 |publisher=Cambridge University Press |isbn=0-521-23736-X |edition=1st pbk. |location=Cambridge [Cambridgeshire] |chapter=6. Representation of a preference ordering by a numerical function |oclc=25466669}} proved the existence of a continuous representation of a weak preference relation $\backslash succeq$ satisfying the following conditions:

1. Reflexive and Transitive;
2. Complete, that is, for every two options A, B in X, either $A\backslash succeq\; B$ or $B\backslash succeq\; A$ or both;
3. For all $A\; \backslash in\; X$, both the upper and the lower weak contour sets are topologically closed;
4. The space X is second-countable. This means that there is a countable set S of open sets, such that every open set in X is the union of sets of the class S.{{cite journal |author=Debreu, Gerard |year=1964 |title=Continuity properties of Paretian utility |journal=International Economic Review |volume=5 |pages=285–293 |doi=10.2307/2525513 |number=3|jstor=2525513 }} Second-countability is implied by the following properties (from weaker to stronger):
5. * The space X is separable and connected.
6. * The relation $\backslash succeq$ is separable.
7. * The relation $\backslash succeq$ is countable.

Jaffray gives an elementary proof to the existence of a continuous utility function.{{Cite journal |last=Jaffray |first=Jean-Yves |date=1975 |title=Existence of a Continuous Utility Function: An Elementary Proof |url=https://www.jstor.org/stable/1911340 |journal=Econometrica |volume=43 |issue=5/6 |pages=981–983 |doi=10.2307/1911340 |jstor=1911340 |issn=0012-9682 |archive-date=2023-02-27 |access-date=2023-02-27 |archive-url=https://web.archive.org/web/20230227135931/https://www.jstor.org/stable/1911340 |url-status=live |url-access=subscription }}

Preferences are called incomplete when some options are incomparable, that is, neither $A\backslash succeq\; B$ nor $B\; \backslash succeq\; A$ holds. This case is denoted by $A\; \backslash bowtie\; B$. Since real numbers are always comparable, it is impossible to have a representing function u with $u(A)\; \backslash geq\; u(B)\; \backslash iff\; A\; \backslash succeq\; B$. There are several ways to cope with this issue.

Peleg defined a utility function representation of a strict partial order $\backslash succ$ as a function $u:\; X\; \backslash to\; \backslash mathbb\{R\}$ such that$A\; \backslash succ\; B\; \backslash implies\; u(A)>u(B)$, that is, only one direction of implication should hold.{{Cite journal |last=Peleg |first=Bezalel |date=1970 |title=Utility Functions for Partially Ordered Topological Spaces |url=https://www.jstor.org/stable/1909243 |journal=Econometrica |volume=38 |issue=1 |pages=93–96 |doi=10.2307/1909243 |jstor=1909243 |issn=0012-9682 |archive-date=2023-02-23 |access-date=2023-02-23 |archive-url=https://web.archive.org/web/20230223093008/https://www.jstor.org/stable/1909243 |url-status=live |url-access=subscription }} Peleg proved the existence of a one-dimensional continuous utility representation of a strict preference relation $\backslash succ$ satisfying the following conditions:

1. Irreflexive and transitive (which implies that it is asymmetric, that is, is a strict partial order);
2. Separable;
3. For all $A\; \backslash in\; X$, the lower strict contour set at A is topologically open;
4. Spacious: if $A\backslash succ\; B$, then the lower strict contour set at A contains the closure of the lower strict contour set at B.
5. * This condition is required for incomplete preference relations. For complete preference relations, every relation in which all lower and upper strict contour sets are open, is also spacious.

If we are given a weak preference relation $\backslash succeq$, we can apply Peleg's theorem by defining a strict preference relation: $A\backslash succ\; B$ if and only if $A\backslash succeq\; B$ and not $B\backslash succeq\; A$.

The second condition ($\backslash succ$ is separable) is implied by the following three conditions:

• The space X is separable;
• For all $A\; \backslash in\; X$, both lower and upper strict contour sets at A are topologically open;
• If the lower countour set of A is nonempty, then A is in its closure.

A similar approach was taken by Richter.{{Cite journal |last=Richter |first=Marcel K. |date=1966 |title=Revealed Preference Theory |url=https://www.jstor.org/stable/1909773 |journal=Econometrica |volume=34 |issue=3 |pages=635–645 |doi=10.2307/1909773 |jstor=1909773 |issn=0012-9682|url-access=subscription }} Therefore, this one-directional representation is also called a Richter-Peleg utility representation.{{Cite journal |last1=Alcantud |first1=José Carlos R. |last2=Bosi |first2=Gianni |last3=Zuanon |first3=Magalì |date=2016-03-01 |title=Richter–Peleg multi-utility representations of preorders |url=https://doi.org/10.1007/s11238-015-9506-z |journal=Theory and Decision |language=en |volume=80 |issue=3 |pages=443–450 |doi=10.1007/s11238-015-9506-z |issn=1573-7187|hdl=11368/2865746 |s2cid=255110550 |hdl-access=free }}

Jaffray defines a utility function representation of a strict partial order $\backslash succ$ as a function $u:\; X\; \backslash to\; \backslash mathbb\{R\}$ such that both $A\; \backslash succ\; B\; \backslash implies\; u(A)>u(B)$, and $A\backslash approx\; B\; \backslash implies\; u(A)=u(B)$, where the relation $A\backslash approx\; B$ is defined by: for all C, $A\backslash succ\; C\; \backslash iff\; B\backslash succ\; C$ and $C\backslash succ\; A\; \backslash iff\; C\backslash succ\; B$ (that is: the lower and upper contour sets of A and B are identical).{{Cite journal |last=Jaffray |first=Jean-Yves |date=1975-12-01 |title=Semicontinuous extension of a partial order |url=https://dx.doi.org/10.1016/0304-4068%2875%2990005-1 |journal=Journal of Mathematical Economics |language=en |volume=2 |issue=3 |pages=395–406 |doi=10.1016/0304-4068(75)90005-1 |issn=0304-4068|url-access=subscription }} He proved that, for every partially-ordered space $(X,\; \backslash succ)$ that is perfectly-separable, there exists a utility function that is upper-semicontinuous in any topology stronger than the upper order topology.{{Rp|location=Sec.4}} An analogous statement states the existence of a utility function that is lower-semicontinuous in any topology stronger than the lower order topology.

Sondermann defines a utility function representation similarly to Jaffray. He gives conditions for existence of a utility function representation on a probability space, that is upper semicontinuous or lower semicontinuous in the order topology.{{Cite journal |last=Sondermann |first=Dieter |date=1980-10-01 |title=Utility representations for partial orders |url=https://dx.doi.org/10.1016/0022-0531%2880%2990004-6 |journal=Journal of Economic Theory |language=en |volume=23 |issue=2 |pages=183–188 |doi=10.1016/0022-0531(80)90004-6 |issn=0022-0531|url-access=subscription }}

Herdendefines a utility function representation of a weak preorder $\backslash succeq$ as an isotone function $u:\; (X,\; \backslash succeq)\; \backslash to\; (\backslash mathbb\{R\},\; \backslash geq)$ such that $A\; \backslash succ\; B\; \backslash implies\; u(A)>u(B)$. Herden{{Cite journal |last=Herden |first=G. |date=1989-06-01 |title=On the existence of utility functions |url=https://dx.doi.org/10.1016/0165-4896%2889%2990058-9 |journal=Mathematical Social Sciences |language=en |volume=17 |issue=3 |pages=297–313 |doi=10.1016/0165-4896(89)90058-9 |issn=0165-4896|url-access=subscription }}{{Rp|location=Thm.4.1}} proved that a weak preorder $\backslash succeq$ on X has a continuous utility function, if and only if there exists a countable family E of separable systems on X such that, for all pairs $A\backslash succ\; B$, there is a separable system F in E, such that B is contained in all sets in F, and A is not contained in any set in F. He shows that this theorem implies Peleg's representation theorem. In a follow-up paper{{Cite journal |last=Herden |first=G. |date=1989-10-01 |title=On the existence of utility functions ii |url=https://dx.doi.org/10.1016/0165-4896%2889%2990041-3 |journal=Mathematical Social Sciences |language=en |volume=18 |issue=2 |pages=107–117 |doi=10.1016/0165-4896(89)90041-3 |issn=0165-4896|url-access=subscription }} he clarifies the relation between this theorem and classical utility representation theorems on complete orders.

A multi-utility representation (MUR) of a relation $\backslash succeq$ is a set U of utility functions, such that $A\; \backslash succeq\; B\; \backslash iff\; \backslash forall\; u\backslash in\; U:\; u(A)\backslash geq\; u(B)$. In other words, A is preferred to B if and only if all utility functions in the set U unanimously hold this preference. The concept was introduced by Efe Ok.{{Cite journal |last=Ok |first=Efe |date=2002 |title=Utility Representation of an Incomplete Preference Relation |url=https://econpapers.repec.org/article/eeejetheo/v_3A104_3Ay_3A2002_3Ai_3A2_3Ap_3A429-449.htm |journal=Journal of Economic Theory |volume=104 |issue=2 |pages=429–449 |doi=10.1006/jeth.2001.2814 |issn=0022-0531 |archive-date=2023-02-23 |access-date=2023-02-23 |archive-url=https://web.archive.org/web/20230223091616/https://econpapers.repec.org/article/eeejetheo/v_3A104_3Ay_3A2002_3Ai_3A2_3Ap_3A429-449.htm |url-status=live |url-access=subscription }}

Every preorder (reflexive and transitive relation) has a trivial MUR.{{Rp|location=Prop.1}} Moreover, every preorder with closed upper contour sets has an upper-semicontinuous MUR, and every preorder with closed lower contour sets has a lower-semicontinuous MUR.{{Rp|location=Prop.2}} However, not every preorder with closed upper and lower contour sets has a continuous MUR.{{Rp|location=Exm.1}} Ok and Evren present several conditions on the existence of a continuous MUR:

• $\backslash succeq$ has a continuous MUR if-and-only-if (X,$\backslash succeq$) is a semi-normally-preordered topological space.{{Rp|location=Thm 0}}
• If X is a locally compact and sigma-compact Hausdorff space, and $\backslash succeq$ is a closed subset of $X\backslash times\; X$, then $\backslash succeq$ has a continuous MUR.{{Rp|location=Thm 1}} This in particular holds if X is a nonempty closed subset of a Euclidean space.
• If X is any topological space, and $\backslash succeq$ is a preorder with closed upper and lower contour sets, that satisfies strong local non-satiation and an additional property called niceness, then $\backslash succeq$ has a continuous MUR.{{Rp|location=Thm 2}}

All the representations guaranteed by the above theorems might contain infinitely many utilities, and even uncountably many utilities. In practice, it is often important to have a finite MUR - a MUR with finitely many utilities. Evren and Ok prove there exists a finite MUR where all utilities are upper[lower] semicontinuous for any weak preference relation $\backslash succeq$ satisfying the following conditions:{{Rp|location=Thm 3}}

1. Reflexive and Transitive (that is, $\backslash succeq$ is a weak preorder);
2. All upper[lower] contour sets are topologically closed;
3. The space X is second-countable, that is, it has a countable basis.
4. The width of $\backslash succeq$ (the largest size of a set in which all elements are incomparable) is finite.
5. * The number of utility functions in the representation is at most the width of $\backslash succeq$.

Note that the guaranteed functions are semicontinuous, but not necessarily continuous, even if all upper and lower contour sets are closed.{{Rp|location=Exm.2}} Evren and Ok say that "there does not seem to be a natural way of deriving a continuous finite multi-utility representation theorem, at least, not by using the methods adopted in this paper".

• Von Neumann-Morgenstern utility theorem
• Harsanyi's utilitarian theorem
• Arrow's impossibility theorem
• Revealed preference theory deals with representing the demand function of an agent by a preference relation, or by a utility function.

{{reflist}}

Category:Utility

Category:Economics theorems