5 Completeness
Another key assumption in a rational choice model is that decision makers have opinions about all the objects of choice. This means there are no objects \(x\) and \(y\) that 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. In other words, for all \(x, y\in X\), not-\(x\mathrel{N} y\).
Completeness implies that if neither \(x \mathrel{P} y\) nor \(y \mathrel{P} x\) holds, then the decision maker must be indifferent between \(x\) and \(y\) (i.e., \(x \mathrel{I} y\)). In other words, for any two options, the decision maker either has a strict preference for one over the other, or the options are viewed as equally desirable.
This property of completeness greatly simplifies the characterization of a rational preference. Instead of needing to explicitly define both the strict preference relation \(P\) and the indifference relation \(I\), we only need to specify the strict preference \(P\). The indifference relation \(I\) can then be inferred directly from \(P\)—specifically, \(x \mathrel{I} y\) holds if and only if neither \(x \mathrel{P} y\) nor \(y \mathrel{P} x\) is true.
However, it is important to recognize that while completeness simplifies the model, it represents a significant idealization. In reality, decision makers may experience ambiguity, indecision, or incommensurability between options, where neither strict preference nor indifference adequately captures their opinions about the choices. Indeed, many decision theorists consider the justification for assuming completeness to be weaker than the justification for assuming 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)