Violation of completeness axiom (simple everyday examples) Simple everyday examples of violations of transitivity are not difficult to come up with, but I'm having trouble thinking of some for the completeness axiom.
One possible formulation of the completeness axiom in plain English:

Given any two alternatives A and B, exactly one of the following is true: (1) I strictly prefer A to B; (2) I strictly prefer B to A; (3) I am indifferent between A and B.


One example that I was hoping could work (but didn't) came from the book/movie Sophie's Choice, where Sophie is made to choose between the following two alternatives:

A: Send her son to the gas chambers (and her daughter to the children's camp).
B: Send her daughter to the gas chambers (and her son to the children's camp).

She is initially very reluctant to express a preference for either alternative. So one possible interpretation is that she is exactly indifferent between A and B. Another is that she does not obey the completeness axiom.
However, she is then pressed further and told that if she doesn't choose, she'll simply be given alternative C: &quot;Send both children to the gas chambers.&quot;
Faced with this alternative C that is clearly inferior to both alternatives A and B, she then does indeed choose one of the alternatives (B: Send her daughter to the gas chambers), hence strongly suggesting that she does indeed obey the completeness axiom.
The above is of course merely my interpretation of the story in Sophie's Choice. Maybe someone can give an alternative interpretation where the above turns out to be an example of the completeness axiom being violated.
(One possibility might be along these lines: Sophie's preference relation is incomplete over {A,B}. When faced only with A and B, she doesn't prefer A to B, doesn't prefer B to A, and isn't indifferent between A and B. However, when given an additional option C that is vastly inferior to A or B and which she must default to in the absence of a choice, she now prefers B to A.)

For those who seem to believe the completeness axiom is inviolable and are  unaware that &quot;within economic theory, criticism of the axioms of transitivity and completeness have quite a long history&quot; (Putnam, 2002, p. 163), please see the following three examples of economists criticizing the axiom of completeness:
Von Neumann and Morgenstern (1953, Theory of Games and Economic Behavior, 3rd edition, p. 630):

The axiom (3:A)—or, more specifically, (3:A:a)—expresses the completeness of the ordering of all utilities, i.e. the completeness of the individual's system of preferences. It is very dubious, whether the idealization of reality which treats this postulate as a valid one, is appropriate or even convenient.

Aumann (1962, p. 446):

Of all the axioms of utility theory, the completeness axiom is perhaps the
most questionable. Like others of the axioms, it is inaccurate as a description of real life; but unlike them, we find it hard to accept even from the normative viewpoint.

Anand (1987, p. 190):

Whilst completeness is one of the first assumptions used in any&quot; formal&quot;
theory of rational choice, it is probably less acceptable as an axiom of
rationality than either transitivity or independence.
There is a problem in how you translate completeness into behavior. Let $R$ be any binary relation, representing preferences, on a set $X$ of alternatives and $A\subseteq X$ be a nonempty set of alternatives available. The usual assumption is that the decisionmaker chooses an alternative $a\in A$ optimally according to the relation $R$. Here are three ways to interpret this:


$a R b$ for all $b\in A$.
$b R a$ for no $b\in A$. 
$b R a$ implies $a Rb$ for all $b\in A$.


One usually uses 1. for weak (reflexive) preference relations, 2. for strict (irreflexive) preference relations, and 3. works in both cases. If we define $R'$ by $xR'y$ iff $\neg(yRx)\vee(xRy)$, then $R'$ is complete and satisfies 1. iff $R$ satisfies 3. If $R$ is irreflexive, $R'$ satisfies 1. iff $R$ satisfies 2. So completeness in itself has no behavioral implications from this perspective. It can interact with other properties. For example, transitive and complete is more restrictive than transitivity alone- completeness forces transitive to be applicable to more comparisons.

Now one can try to make a distinction between indifference and incomparability. But such a distinction necessarily uses a different framework than the usual maximization paradigm.