3 Completeness
Another key assumption about rational decision makers is that they have opinions about all the alternatives. This means there are no alternatives \(x\) and \(y\) that are non-comparable 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 decision maker is indifferent.
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 indecision or incommensurability between alternatives, where neither strict preference nor indifference adequately captures their opinions about the alternatives. Indeed, some decision theorists consider the justification of completeness to be weaker than the justification of 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)