Is there a known example of a voting system that does not satify the dictator fairness criterion but does satisfy the others?

Question

As Wikipedia says, Arrow's impossibility theorem states that no rank-order electoral system can be designed that always satisfies these three "fairness" criteria:

1. If every voter prefers alternative X over alternative Y, then the
   group prefers X over Y.
2. If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain
   unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change).
3. There is no "dictator": no single voter possesses the power to always determine the group's preference.

Is there a known example of a voting system which fulfills the first two criteria, (and therefore fails to fulfill the third,) for which an example set of votes can be shown to contain one or more dictators?

Edit:

Voting systems that literally, directly, choose a dictator are not satisfying to me regarding this question. However if it can be demonstrated that directly choosing a dictator is the only way a voting system can fail the third criteria, then I would find that satisfying.

Answer 1

After asking this question and reading further, I have come to the conclusion that it is the case that if a voting system meets the first two criteria, then the voting system will be doing some form of directly choosing the dictator.

In particular, I found working through this proof of Arrow's Impossibility theorem (Wayback link) finally convinced me.

Answer 2

To get a better real-world feel for how the "dictator" in Arrow's impossibility theorem actually "dictates", imagine a hypothetical country that has some kind of "Supreme Court", with nine judges on it. This court regularly rules on legal cases of great political significance to the two major political parties, the "Red Party" and the "Blue Party". While in theory the judges are impartial, in practice, eight of the nine judges have well-known partisan alignments--four "Red", four "Blue"--and their votes on any given case are completely predictable: all the Reds vote the same way, and all the Blues vote the same way. There is only one judge who does not align with a particular party and whose preferences in any given case are unpredictable; they are considered the "swing judge".

Let's call the three conditions (I), (II), and (III) in OP's post "unanimity", "independence of irrelevant alternatives", and "non-dictatorship". I make two claims about how this voting system works in practice:

Claim 1: This system satisfies the conditions on unanimity and independence of irrelevant alternatives. In principle these conditions could be violated in this setting; but in practice they never are, because all four Red judges have identical preferences and vote identically, and so do all four Blue judges, so no "strategic voting" ever occurs.

Claim 2: The swing judge is the "dictator." Because the rest of the Court is evenly balanced, the swing judge's decision is also the Court's decision, in every case. Even though in principle, the swing judge's vote doesn't count more or less than anyone else's, in practice, their vote is always the decisive one.