Keywords

JEL Classifications

A preordering (also called a weak ordering or a quasi-ordering) is a reflexive and transitive binary relation which is not necessarily 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 realtion R to y ∈ S. A preordering is a binary relation, R, which satisfies two properties: (i) reflexivity: for all x ∈ SxRx, and (ii) transitivity: for x, y, z ∈ S, if xRy and yRz, then xRz.

A simple example is given by the binary relation weak vector dominance which we denote V. Suppose S is Euclidean N-space, then xVy if and only if xn ≥ yn, n = 1,..., N. V is clearly reflexive and transitive; it is just as clearly not complete, that is, not all elements of S are ranked. For example if N = 2, x = (1, 2), and y = (2, 1) then it is not the case that xVy or that yVx.

Quasi-orderings have played their largest role in welfare economics where consistency in decision making is a desirable requirement but where one may be dubious about being able to rank all possible outcomes. Two examples follow for which the notion of a subrelation is useful. Suppose R and S are binary relations: S is a subrelation of R if xSy implies xRy. For example, strong vector dominance, \( \overline{V} \) is the binary relation which results when the above weak inequality is replaced with a strict inequality. Clearly, \( \overline{V} \) is a subrelation of V.

Interpreting the elements of N-space as vectors of utilities, it is possible to define a quasi-ordering which is a subrelation of both the utilitarian and the Rawls criteria: Define the binary relation M by xMy if and only if \( {\sum\limits}_{i=1}^N{x}_i\ge {\sum\limits}_{i=1}^N{y}_i \) and min{x1, … , xN} ≥ min {y1, … , yN}. M is clearly reflexive, transitive and not complete. The distributional insensitivity of the utilitarian principle is tempered by the Rawls’s difference principle.

As an alternative, consider evaluating social states by weighted utility sums where the weights represent utility comparisons but these comparisons are not precisely fixed. Instead, the weights are drawn from a subset of N-dimensional Euclidean space, say B. More formally define the quasi-ordering F by xFy if and only if \( {\sum\limits}_{i=1}^N{b}_i{x}_i\ge {\sum\limits}_{i=1}^N{b}_i{y}_i \) for all (b1, …, bN) ∈ B. Suppose that we try to evaluate the desirability of burning down Rome while Nero fiddles. The quasi-ordering F may show a gain for burning Rome only if the set of interpersonal weights is such that Nero is given extreme consideration. (These examples are taken from the articles listed below.)

See Also