Preference (economics)

Add the Microeconomics is called a preference relation generally an order of precedence, in the two commodity bundles ( " alternatives ") are arranged according to how they prefer an individual or group of individuals to each other. Formally, for a preference relation is a binary relation. For example, R is a preference relation ( the so-called preference - indifference relation, also: weak preference relation ), indicating that its first component is better seen as strictly as or as good as the second. They preferred a person I for example an alternative (weak) against, then the tuple is included in the set (the index should indicate that it is the preferences of person I).

Other preference relations are strict preference relation P ( " strictly better than" ) as well as the indifference relation I ( " as good as "); on a separate definition of the inverse constellations ( "worse than or as good as " or " strictly less than" ) is usually omitted, since one can formulate the underlying preference structures by interchanging the components in the manner defined here.

It refers to a preference relation as a preference order if it meets certain minimum requirements (see the section # order of preference ).

• 4.1 Implications of R for P and I
• 4.2 Implications of P for R and I

Definition

It is initially based on an amount in which all existing bundle of goods ( " alternatives " ) are included:

The bundle of goods and are thus tuples, so that for example indicating the quantity of good 5 in the bundle of goods 3.

Preference relations are binary relations, that is, they are subsets of. Looking is to first discuss only the so-called preference - indifference relation R (as described). In the so- defined R all ordered pairs are included, for which it applies that is weak against preferred. Man decides to use the notation for. You can R without detours directly as " is better than or as good as " read.

By R be two other relations - again subsets of - induced. For one, the indifference relation I, on the other hand, the preference relation P. Its importance arises from the R: For two alternatives and is just then and when and at the same time. I can be considered " is as good as " read. The same applies to P: For two alternatives and is exactly then or when, but not at the same time. P to read as " is better than".

Instead of the letters R, P and I for the relations and symbols are common. It is then, and.

Will you express that one refers to the preference structure of a concrete person I can index the relation given; so then is, for example, that person I prefer the alternative strictly against.

Properties

Characteristics

Depending on their individual characteristics can examine the preference - indifference relation R, for example, the following properties for:

• Completeness: or (or both)

• Transitivity:

• Reflexivity:

• Consistency: For all true: The amounts ( upper contour set) and (lower contour set) are closed under
• ( Weak ) monotonicity:

• Strict monotonicity:

• Local unsaturation: A preference relation is not locally saturated if for any and every environment exists an alternative, so Formal:

• Convexity:

• Strict convexity:

Preference order

Of particular importance are the first two properties. With them, namely, the following applies:

Rationality of the preference - indifference relation: A preference - indifference relation R if and only rational if it is complete and transitive. They are then referred to as preference order.

It can be shown that the efficiency of R also has important implications with respect to the induced by their relations:

Implications for the preference and the indifference relation: Is the preference - indifference relation R rational, then for the resulting induced relations I and P:

• Transitive.

• Reflexive;
• Transitive and
• Symmetrical.

In support see the section # implications of R for P and I.

Implications for the utility function

For purposes of mathematical handling, it is recommended many times to represent the relations explained on the basis of utility functions. A function is then just a utility function with which the preferences can be mapped, though. However, it raises the question of whether there are any relations for a corresponding utility function. This is not the case, as the following theorem shows:

Representability by a utility function:

Lexicographic preference order

→ See also: Lexicographic order

Is given by with the bundle of goods / the alternative and the alternative (accordingly, for example, the amount of good 2 in the bundle of goods ).

Definition: The preferential indifference ratio R is referred to as lexicographically if if and only if either or both and.

In lexicographic preferences ie a bundle of goods is only weakly preferred to another ( that is, as equally good or better ), if it contains more of good 1 or, if both goods bundles contain the same amount of it if it in any case a of good 2 at least such a large amount includes - in any case, regardless of the quantities of other goods in the goods bundles. Lexicographic preference orderings can be normally do not represent by a utility function. Although they are rational, but not steadily.

Mathematical foundations; formal addenda

The following operational definitions ( partial repetition from above) for a general non-empty set X ( B a binary relation on X):

• Completeness: or (or both)
• Reflexivity:
• Irreflexivity:
• Symmetry:
• Asymmetry:
• Transitivity:
• Negative transitivity:

Implications of R for P and I

The outlined approach can be generalized, so that it is possible, among other things, to represent the three considered here relations in a formal context.

We agree that

• The preference - indifference relation R is a complete quasi-ordering, ie it is complete, reflexive and transitive ( economically, this corresponds to our definition of preference order ).
• P is the asymmetric part of the quasi-ordering R, ie we have:

• I is the symmetric part of the quasi-ordering R, ie we have:

Note the following key messages:

Implications of the property as quasi-ordering: Let R be a complete quasi-ordering on a non- empty X P and symmetric part I. Then with asymmetrical part:

Evidence to 1st and 2nd: (1a ) Be and arbitrary elements of X and is. Then by definition of P that and at the same time, a contradiction, so. (1b ) The asymmetry is clear from the definition of the asymmetric part. (1c ) Be such that; will be shown below, that then. From the assumptions and the definition of P follows, first, that as well. Since then, and is considered by transitivity of R and - consequently, it now suffices to show that to prove that it is not at the same time. Proof by contradiction: If, then, since according to the above conclusion from the definition of P, and thus also according to transitivity of R. This contradicts the upper realization that, consequently, and together with ( see above) follows in fact what was to be shown. (1d ) Be. It is to indicate that either or. Be well, then also ( asymmetry). Together with the original assumption follows but then: ( transitivity ), which was to be proved. (2a, b ) follows directly from the definition of the symmetric piece. (2c ) Be such that; will be shown below, that then. From the assumptions and the definition of I follows, first, that as well. According to transitivity of R it follows that (left side ) and (right side ). But this is true by definition of the symmetric part, that then, what was to be shown.

Implications of P for R and I

The third statement (3 ) of the Theoremkastens in the previous section ( " implication of the property as quasi-ordering " ) can be a significant relationship between P and R may be revealed. In the economic context, therefore, the preference - indifference relation is just the negation of a strict preference relation. However, this suggests that you also unlike in the past, the preferences determine starting from the strict preference relation, and not only, as described in the remainder of this article, starting from the preference - indifference relation. One might ask practical, whether it should not be possible, instead of " query " by consumers their weak preferences ( " Do you like ice cream at least as happy as pie? " And " Do you like cake at least as happy as ice? " ) But at to start the strict preferences ( " Do you prefer ice cream than cake ?") and derive from it, among other things, the preference - indifference relation. However, this is not always correspond to a quasi-ordering (or a preference ordering ), as the following example illustrates.

The following theorem forms the first part of the starting point for an initial technical considerations that last statement describes a specific criterion under which a given strict preference relation induces a preference - indifference relation that satisfies the rationality criterion (ie, a preference ordering is ).

Implications of the property as a strict order of preference: Let P be a binary relation on X and let R be non empty negation of P. Then: