Square-integrable function

Example of a preference relation

A simple example of a preference order over three goods

In economics and other social sciences, preference refers to the set of assumptions related to ordering some alternatives, based on the degree of happiness, satisfaction, gratification, enjoyment, or utility they provide, a process which results in an optimal "choice" (whether real or theoretical). Although economists are usually not interested in choices or preferences in themselves, they are interested in the theory of choice because it serves as a background for empirical demand analysis.

History

Ragnar Frisch was the first to write preference relations using the mathematics of axioms, in 1926. Up to then, economists had developed an elaborated theory of demand that omitted primitive characteristics of people. This changed at the end of the 19th and the beginning of the 20th century, when logical positivism started to affect economics through the notion that any theoretical concept used in a theory should be related to observables. Whereas economists in the 18th and 19th centuries felt comfortable theorizing about utility, with the advent of logical positivism in the 20th century, they felt that it needed more of an empirical structure. Because binary choices are directly observable, it instantly appealed to economists. The search for observables in microeconomics is taken even further by revealed preference theory.

Since the pioneer efforts of Frisch in the 1920s, one of the major issues which has pervaded the theory of preferences is the representability of a preference structure with a real-valued function. This has been achieved by mapping it to the mathematical index called utility. Gérard Debreu, influenced by the ideas of the Bourbaki group, championed the axiomatization of consumer theory in the 1950s, and the tools he borrowed from the mathematical field of binary relations have become mainstream since then. Even though the economics of choice can be examined either at the level of utility functions or at the level of preferences, to move from one to the other can be useful. For example, shifting the conceptual basis from an abstract preference relation to an abstract utility scale results in a new mathematical framework, allowing new kinds of conditions on the structure of preference to be formulated and investigated.

Another historical turnpoint can be traced back to 1895, when Georg Cantor, proved in a theorem that if a binary relation is linearly ordered, then it is also isomorphically embeddable in the ordered real numbers. This notion would become very influential for the theory of preferences in economics: by the 1940s prominent authors such as Paul Samuelson, would theorize about people actually having weakly ordered preferences.

Basic premises

In consumer theory, economic actors are thought of as being confronted with a set of possible consumption bundles or commodity space. Of all the available bundles of goods and services, only one is ultimately chosen. The theory of preferences examines the problem of getting to this optimal choice using a system of preferences within a budgetary limitation.

In reality, people do not necessarily rank or order their preferences in a consistent way. In preference theory, some idealized conditions are regularly imposed on the preferences of economic actors. One of the most important of these idealized conditions is the axiom of transitivity:

If alternative $A\!$ is preferred to alternative $B\!$, and $B\!$ to $C\!$, then $A\!$ is preferred to $C\!$.

The language of binary relations allow one to write down exactly what is meant by "ranked set of preferences", and thus gives an unambiguous definition of order. A preference relation should not be confused with the order relation $\geqslant \!$ used to indicate which of two real numbers is larger. Order relations satisfy an extra condition:

$A\geqslant B\!$, and $B\geqslant A\!$, implies $A=B\!$

which does not always hold in preference relations; hence, an indifference relation is used in its place (the symbol $\sim \!$ denotes this kind of relation).

A system of preferences or preference structure refers to the set of qualitative relations between different alternatives of consumption. For example, if the alternatives are:

• Apple
• Orange
• Banana

In this example, a preference structure would be:

"The apple is at least as preferred as the orange", and "The orange is as least as preferred as the Banana". One can use $\succsim \!$ to symbolize that some alternative is "at least as preferred as" another one, which is just a binary relation on the set of alternatives. Therefore:

• Apple $\succsim \!$ Orange
• Orange $\succsim \!$ Banana

The former qualitative relation can be preserved when mapped into a numerical structure, if we impose certain desirable properties over the binary relation: these are the axioms of preference order. For instance: Let us take the apple and assign it the arbitrary number 5.Then take the orange and let us assign it a value lower than 5, since the orange is less preferred than the apple. If this procedure is extended to the banana, one may prove by induction that if $u\!$ is defined on {apple, orange} and it represents a well-defined binary relation called "at least as preferred as" on this set, then it can be extended to a function $u\!$ defined on {apple, orange, banana} and it will represent "at least as preferred as" on this larger set.

Example:

• Apple = 5
• Orange = 3
• Banana = 2

5 > 3 > 2 = u(apple) > u(orange) > u(banana)

and this is consistent with Apple $\succsim \!$ Orange, and with Orange $\succsim \!$ Banana.

Axioms of order

• Completeness: for all $A\!$ and $B\!$ we have $A\!\succsim \!B\!$ or $B\!\succsim \!A\!$ or both.

In order for preference theory to be useful mathematically, we need to assume continuity. Continuity simply means that there are no ‘jumps’ in people’s preferences: if we prefer very large oranges to apples, we will prefer large oranges to apples as well. In mathematical terms, if we prefer point A along a preference curve to point B, points very close to A will also be preferred to B. This allows preference curves to be differentiated. The continuity assumption is "too strong" in the sense that it indeed guarantees the existence of a continuous utility function representation. Continuity is, therefore, a sufficient condition, but not a necessary one.

Although some authors include reflexivity as one of the axioms required to obtain representability (this axiom states that $A\!\succsim \!A\!$), it is redundant inasmuch as the completeness axiom implies it already.

Most commonly used axioms

• Order-theoretic: acyclicity, transitivity, the semiorder property, completeness
• Topological: continuity, openness or closedness of the preference sets
• Linear-space: convexity, homogeneity, translation-invariance

Normative interpretations of the axioms

Everyday experience suggests that people at least talk about their preferences as if they had personal "standards of judgment" capable of being applied to the particular domain of alternatives that present themselves from time to time. Thus, the axioms are an attempt to model the decision maker's preferences, not over the actual choice, but over the type of desirable procedure (a procedure that any human being would like to follow). Behavioral economics investigates inconsistent behavior (i.e. behavior that violates the axioms) of people. Believing in axioms in a normative way does not imply that it is mandatory to behave according to them. Instead, they are a mode of behavior suggested; its what people would like to see themselves following.

Here is an illustrative example of the normative implications of the theory of preferences: Consider a decision maker who needs to make a choice. Assume that this is a choice of where to live or whom to marry and that the decision maker has asked an economist for advice. The economist, who wants to engage in normative science, attempts to tell the decision maker how she should make decisions.

Economist: I suggest that you attach a utility index to each alternative, and choose the alternative with the highest utility.

Decision Maker: You've been brainwashed. You think only in terms of functions. But this is an important decision, there are people involved, emotions, these are not functions!

Economist: Would you feel comfortable with cycling among three possible options? Preferring x to y, and then y to z, but then again z to x?

Decision Maker: No, this is very silly and counterproductive. I told you that there are people involved, and I do not want to play with their feelings.

Economist: Good. So now let me tell you a secret: if you follow these two conditions -making decision, and avoid cycling, then you can be described as if you are maximizing a utility function.

Consumers whose preference structures violate transitivity would get exposed to being milked by some unscrupulous person. For instance, Maria prefers apples to oranges, oranges to bananas, and bananas to apples. Let her be endowed with an apple, which she can trade in a market. Because she prefers bananas to apples, she is willing to pay, say, one cent to trade her apple for a banana. Afterwards, Maria is willing to pay another cent to trade her banana for an orange, and again the orange for an apple, and so on. There are other examples of this kind of "irrational" behaviour.

Completeness implies that some choice will be made, an assertion that is more philosophically questionable. In most applications, the set of consumption alternatives is infinite and the consumer is not conscious of all preferences. For example, one does not have to choose over going on holiday by plane or by train: if one does not have enough money to go on holiday anyway then it is not necessary to attach a preference order to those alternatives (although it can be nice to dream about what one would do if one would win the lottery). However, preference can be interpreted as a hypothetical choice that could be made rather than a conscious state of mind. In this case, completeness amounts to an assumption that the consumers can always make up their mind whether they are indifferent or prefer one option when presented with any pair of options.

Under some extreme circumstances there is no "rational" choice available. For instance, if asked to choose which one of one's children will be killed, as in Sophie's Choice, there is no rational way out of it. In that case preferences would be incomplete, since "not being able to choose" is not the same as "being indifferent".

The indifference relation ~ is an equivalence relation. Thus we have a quotient set S/~ of equivalence classes of S, which forms a partition of S. Each equivalence class is a set of packages that is equally preferred. If there are only two commodities, the equivalence classes can be graphically represented as indifference curves. Based on the preference relation on S we have a preference relation on S/~. As opposed to the former, the latter is antisymmetric and a total order.

Applications to theories of utility

In economics, a utility function is often used to represent a preference structure such that $u\left(A\right)\geqslant u(B)$ if and only if $A\succsim \!B$. When a preference order is both transitive and complete, then it is standard practice to call it a rational preference relation, and the people who comply with it are rational agents. A transitive and complete relation is called a weak order (or total preorder). The literature on preferences is far from being standardized regarding terms such as complete, partial, strong, and weak. Together with the terms "total", "linear", "strong complete", "quasi-orders", "pre-orders" and "sub-orders", which also have a different meaning depending on the author's taste, there has been an abuse of semantics in the literature.

According to Simon Board, a continuous utility function always exists if $\succsim \!$ is a continuous rational preference relation on $R^{n}$. For any such preference relation, there are many continuous utility functions that represent it. Conversely, every utility function can be used to construct a unique preference relation.

All the above is independent of the prices of the goods and services and of the budget constraints faced by consumers. These determine the feasible bundles (which they can afford). According to the standard theory, consumers chooses a bundle within their budget such that no other feasible bundle is preferred over it; therefore their utility is maximized.

Primitive equivalents of some known properties of utility functions

• An increasing utility function is associated with a monotonic preference relation.
• Quasi-concave utility functions are associated with a convex preference order. When non-convex preferences arise, the Shapley–Folkman lemma is applicable.
• Weakly separable utility functions are associated with the weak separability of preferences.

Lexicographic preferences

Lexicographic preferences are a special case of preferences that assign an infinite value to a good, when compared with the other goods of a bundle.

Strict versus weak

The possibility of defining a strict preference relation $\succ \!$ from the weaker one $\succsim \!$, and viceversa, suggest in principle an alternative approach of starting with the strict relation $\succ \!$ as the primitive concept and deriving the weaker one and the indifference relation. However, an indifference relation derived this way will generally not be transitive. According to Kreps "beginning with strict preference makes it easier to discuss noncomparability possibilities".

Aggregation

Under certain assumptions, individual preferences can be aggregated onto the preferences of a group of people. As a result of agreggation, Arrow's impossibility theorem states that voting systems sometimes can not convert individual preferences into desirable community-wide acts of choice.

Expected utility theory

Preference relations can also be applied to a space of simple lotteries, as in expected utility theory. In this case a preference structure over lotteries can also be represented by a utility function.

Criticism

Some critics say that rational theories of choice and preference theories rely too heavily on the assumption of invariance, which states that the relation of preference should not depend on the description of the options or on the method of elicitation. But without this assumption, one's preferences cannot be represented as maximization of utility.

See also

• Behavioral economics
• Convex preferences
• Economic subjectivism
• Gibbard-Satterthwaite theorem
• Lexicographic preferences
• Pairwise comparison
• Revealed preference
• Strict weak ordering
• Time preference

References

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• Kreps, David (1990). A Course in Microeconomic Theory. New Jersey: Princeton University Press. ISBN 0-691-04264-0