# Orderings

**DOI:**https://doi.org/10.1057/978-1-349-95189-5_1516

## Abstract

An ordering (also called a complete preordering or a weak ordering) is a binary relation which is reflexive, transitive and complete, that is, it is a preordering that is complete.

### Keywords

Choice Compensation criteria Happiness Interpersonal utility comparisons Ordering Preference orderings Preferences Transitivity Utility measurement Welfare economics Well-beingAn ordering (also called a complete preordering or a weak ordering) is a binary relation which is reflexive, transitive and complete, that is, it is a preordering that is complete.

A binary relation *R* defined on a set *S* is a set of ordered pairs of elements of *S*, that is, a subset of the Cartesian product of *S* with itself, *S* × S. One writes *xRy* (or (*x*, *y*) ∈ *R*) to mean that *x* ∈ *S* stands in relation *R* to *y* ∈ *S.* An ordering is a binary relation, *R*, which satisfies three properties: (i) reflexivity: for all *x* ∈ *S, xRx;* (ii) transitivity: for *x*, *y*, *z* ∈ *S*, if *xRy* and *yRz*, then *xRz;* and (iii) completeness for all x, *y* ∈ *S, xRy* or *yRx*, where ‘or’ is used in its non-exclusive sense.

A simple example results from letting *S* be the real line and *R* the greater than or equal to relation so that *xRy* if and only if *x* ≥ *y.* The most common use of orderings in economics is in preference theory where *S* is a commodity space and *R* stands for ‘at least as desirable as’. Every ordering can be separated into its symmetric and asymmetric factors, respectively, as follows:

*xIy*if and only if

*xRy*and

*yRx*

*xPy*if and only if

*xRy*and not

*yRx.*

In the case of preference theory, these correspond to indifference and strict preference relations.

In consumer theory orderings first appeared in the work of Wold (1943–4). In an attempt to put utility theory on a more solid foundation, Wold posited the existence of an ordering with certain properties and demonstrated that this could be represented by a continuous real-valued function, thus making absolutely clear that this was an ordinal concept. Perhaps the most innovative and useful aspect of Wold’s argument was an insightful definition of a continuous ordering. (An ordering is continuous if the sets *x*|*xRy*,*y* ∈ *S* and *x|yRy,y* ∈ *S* are closed.)

The first modern treatment of the subject appears in Arrow (1951). Agents as well as society as a whole are characterized by their orderings over spaces of alternative. That the choices of society be consistent with an ordering, and understanding the implications of that requirement, has been particularly important in welfare economics. For example, various compensation criteria have been shown to fail transitivity (see Gorman 1955) and hence be unsuitable for public decision-making. In addition, by representing agents and society by their orderings, Arrow made the first step toward unravelling a long-standing confusion between the measurability of utility on the one hand and interpersonal comparability on the other. This step was critical if social decision-making was to rest on solid ground; for an accessible discussion of these issues see Blackorby et al. (1984).

It is common in economics to represent agents by their preference orderings. This leads to a set of complicated and somewhat unresolved issues: what are the relationships among the notions of preference, choice and happiness or well-being. Either a preference ordering or the choices of an individual may be viewed as a primitive and they may or may not be mutually consistent; the issues at stake can, however, be characterized quite precisely. The relationship between either of these and some notion of happiness or well-being is much less clear; for a good introduction to these problems see Sen and Williams (1982).

### Bibliography

- Arrow, K.J. 1951.
*Social choice and individual values*. New York: Wiley. 2nd edn, 1963.Google Scholar - Blackorby, C., D. Donaldson, and J. Weymark. 1984. Social choice with interpersonal utility comparisons: A diagrammatic introduction.
*International Economic Review*25: 327–356.CrossRefGoogle Scholar - Gorman, W. 1955. The intransitivity of certain criteria used in welfare economics.
*Oxford Economic Papers*7: 25–35.CrossRefGoogle Scholar - Sen, A., and B. Williams, eds. 1982.
*Utilitarianism and beyond*. Cambridge: Cambridge University Press.Google Scholar - Wold, H. 1943–4. A synthesis of pure demand analysis, I–III.
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