Oh, I see. I didn't take that problem into account, because it doesn't matter for expected utility, which is additive. But you're right that considering the entire probability distribution is the right thing to do, and under than assumption we're forced to be transitive.

The actual VNM axiom violated by median utility is independence: If you prefer X to Y, then a gamble of X vs Z is preferable to the equivalent gamble of Y vs Z. Consider the following two comparisons:

• Taking bet A, as above, versus the status quo.

• A 2/3 chance of taking bet A and a 1/3 chance of losing $5, versus a 2/3 chance of the status quo and a 1/3 chance of losing $5.

In the first case, bet A has median utility U($1) and the status quo has U($0), so you pick bet A. In the second case, a gamble with a possibility of bet A has median utility U(-$5) and a gamble with a possibility of the status quo still has U($0), so you pick the second gamble.

Of course, independence is probably the shakiest of the VNM axioms, and it wouldn't surprise me if you're unconvinced by it.