7  Completeness

Another key assumption in a rational choice model is that decision makers are opinionated about all the objects of choice. That is, there are no objects \(x\) and \(y\) such that \(x\) and \(y\) are incomparable for the decision maker.

Completeness

For all \(x, y\in X\), exactly one of \(x \mathrel{P} y\), \(y\mathrel{P} x\) or \(x \mathrel{I} y\) is true. I.e., for all \(x, y\in X\), not-\(x\mathrel{N} y\).

Completeness is a common simplifying assumption in many rational choice models. However, assuming completeness does not have the same type of justification as transitivity:

[O]f all the axioms of utility theory, the completeness axiom is perhaps the most questionable. Like others, it is inaccurate as a description of real life; but unlike them we find it hard to accept even from the normative viewpoint. (Aumann 1962, 446)

Aumann, Robert. 1962. “Utility Theory Without the Completeness Axiom.” Econometrica 30 (3): 445–62. https://doi.org/10.2307/1909888.