Correction: I now see that my formulation turns the question of completeness into a question of transitivity of indifference. An “incomplete” preference relation should not be understood as one in which allows strict preferences to go in both directions (which is what I interpret them as, above) but rather, a preference relation in which the ≤ relation (and hence the ∼ relation) is not transitive.
In this case, we can distinguish between ~ and “gaps”, IE, incomparable A and B. ~ might be transitive, but this doesn’t bridge across the gaps. So we might have a preference chain A>B>C and a chain X>Y>Z, but not have any way to compare between the two chains.
In my formulation, which lumps together indifference and gaps, we can’t have this two-chain situation. If A~X, then we must have A>Y, since X>Y, by transitivity of ≥.
So what would be a completeness violation in the wikipedia formulation becomes a transitivity violation in mine.
But notice that I never argued for the transitivity of ~ or ≥ in my comment; I only argued for the transitivity of >.
I don’t think a money-pump argument can be offered for transitivity here.
However, I took a look at the paper by Aumann which you cited, and I’m fairly happy with the generalization of VNM therein! Dropping uniqueness does not seem like a big cost. This seems like more of an example of John Wentworth’s “boilerplate” point, rather than a counterexample.
Correction: I now see that my formulation turns the question of completeness into a question of transitivity of indifference. An “incomplete” preference relation should not be understood as one in which allows strict preferences to go in both directions (which is what I interpret them as, above) but rather, a preference relation in which the ≤ relation (and hence the ∼ relation) is not transitive.
In this case, we can distinguish between ~ and “gaps”, IE, incomparable A and B. ~ might be transitive, but this doesn’t bridge across the gaps. So we might have a preference chain A>B>C and a chain X>Y>Z, but not have any way to compare between the two chains.
In my formulation, which lumps together indifference and gaps, we can’t have this two-chain situation. If A~X, then we must have A>Y, since X>Y, by transitivity of ≥.
So what would be a completeness violation in the wikipedia formulation becomes a transitivity violation in mine.
But notice that I never argued for the transitivity of ~ or ≥ in my comment; I only argued for the transitivity of >.
I don’t think a money-pump argument can be offered for transitivity here.
However, I took a look at the paper by Aumann which you cited, and I’m fairly happy with the generalization of VNM therein! Dropping uniqueness does not seem like a big cost. This seems like more of an example of John Wentworth’s “boilerplate” point, rather than a counterexample.