In economics, a utility representation theorem shows that, under certain conditions, a preference ordering can be represented by a real-valued utility function, such that option A is preferred to option B if and only if the utility of A is larger than that of B. The most famous example of a utility representation theorem is the Von Neumann–Morgenstern utility theorem, which shows that any rational agent has a utility function that measures their preferences over lotteries.
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 . 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 if and only if .
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 , 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 Rm, 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 . It is denoted by either or :
Given a weak preference relation , one can define its "strict part" and "indifference part" as follows:
Given a strict preference relation , one can define its "weak part" and "indifference part" as follows:
For every option , we define the contour sets at A:
Sometimes, the above continuity notions are called semicontinuous, and a is called continuous if it is a closed subset of . [1]
A preference-relation is called:
As an example, the strict order ">" on real numbers is separable, but not countable.
A utility function is a function .
Debreu [2] [3] proved the existence of a continuous representation of a weak preference relation satisfying the following conditions:
Jaffray gives an elementary proof to the existence of a continuous utility function. [5]
Preferences are called incomplete when some options are incomparable, that is, neither nor holds. This case is denoted by . Since real numbers are always comparable, it is impossible to have a representing function u with . There are several ways to cope with this issue.
Peleg defined a utility function representation of a strict partial order as a function such that, that is, only one direction of implication should hold. [6] Peleg proved the existence of a one-dimensional continuous utility representation of a strict preference relation satisfying the following conditions:
If we are given a weak preference relation , we can apply Peleg's theorem by defining a strict preference relation: if and only if and not. [6]
The second condition ( is separable) is implied by the following three conditions:
A similar approach was taken by Richter. [7] Therefore, this one-directional representation is also called a Richter-Peleg utility representation. [8]
Jaffray defines a utility function representation of a strict partial order as a function such that both , and , where the relation is defined by: for all C, and (that is: the lower and upper contour sets of A and B are identical). [9] He proved that, for every partially-ordered space that is perfectly-separable, there exists a utility function that is upper-semicontinuous in any topology stronger than the upper order topology. [9] : 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. [10]
Herdendefines a utility function representation of a weak preorder as an isotone function such that . Herden [11] : Thm.4.1 proved that a weak preorder 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 , 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 [12] he clarifies the relation between this theorem and classical utility representation theorems on complete orders.
A multi-utility representation (MUR) of a relation is a set U of utility functions, such that . 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. [13]
Every preorder (reflexive and transitive relation) has a trivial MUR. [1] : 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. [1] : Prop.2 However, not every preorder with closed upper and lower contour sets has a continuous MUR. [1] : Exm.1 Ok and Evren present several conditions on the existence of a continuous MUR:
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 satisfying the following conditions: [1] : Thm 3
Note that the guaranteed functions are semicontinuous, but not necessarily continuous, even if all upper and lower contour sets are closed. [13] : 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".