The short answer is that Zelig is the dictator. His preferences always mirror the group's preferences, and because neither Alice nor Bob satisfy the conditions of being the dictator.
According to the Author: Zelig is the Only Dictator
I e-mailed Michael Morreau, the author of the SEP article linked to in the question. Aside from being the author of that article, he is a Professor of Philosophy in Norway and has published papers on social choice theory. His response (verbatim) is below.
Thanks for the question. In the example, Zelig is the dictator of the social welfare function and (provided he’s not sitting next to the same person every time) he’s the only dictator.
For a concrete example, let the other two people be Alice and Bob.
Every time they have their meeting with Zelig they call in dinner
afterwards, either x (say Chinese takeout) or y (pizza). They make
their decision whether it’s to be x or y, on any given occasion, by a
majority vote. Alice sometimes prefers x to y, and sometimes y to x;
the same is true for Bob; and these two have their preferences
independently of one another, so that sometimes Alice prefers x to y
but Bob prefers y to x, and sometimes Bob prefers x to y but Alice
prefers y to x. Zelig, meanwhile, sometimes sits next to Alice and
sometimes next to Bob and takes on the preference among x and y of
whomever he happens to be sitting next to. Furthermore, sometimes he
sits next to Alice when she and Bob have different preferences, and
sometimes he sits next to Bob when they have different preferences.
Now, Zelig is a dictator in Arrow’s sense. Whenever he prefers x to y
that’s because he’s sitting next to someone else who does, either
Alice or Bob, and so that’s two out of three, a majority. The group
also prefers x to y.
Alice is not a dictator. There are occasions on which she prefers x
to y, but Bob prefers y to x and happens to have Zelig sitting next to
him. On such occasions the majority preference is for y to x, which
does not agree with Alice’s preference. By identical reasoning, Bob
is not a dictator.
We could modify the example by stipulating a further domain
restriction, corresponding to the assumption that Zelig always sits
next to the same person, say Alice. Then both Zelig and Alice always
have the same preferences and both are dictators. It’s as if there
were just two people in the group, but one of them, Alice-Zelig has
two votes. I didn’t set up the example in this way because I wanted a
conformist to be the Arrovian dictator, and Zelig is the only
conformist: Alice and Bob have their preferences independently of one
another and of Zelig.
In the link you sent me, someone writes:
for the profile ⟨RZelig,RAlice,RBob⟩ with xRZeligy, yRAlicex, and
yRBobx, we have yPx, which contradicts the above definition of
dictatorship. Now, we never may find this profile in reality due to
Zelig’s voting behaviours, but that doesn’t affect the properties of
This profile as the writer realizes doesn’t arise “in reality”. In
technical terms, this means that it’s appropriate to let the social
welfare function f be majority rule on a domain that doesn’t include
this profile: this is a “restricted domain”. Since its domain is part
of the definition of a function, including a social welfare function,
this does contrary to what this author writes “affect the properties
of f”. This sort of mistake is easily made as we move back and forth
between speaking informally of “majority rule” and the technical
realization of this idea in Arrow’s framework, in which the functional
f is defined for a particular set of individuals (here just Alice, Bob
and Zelig), a particular set of options (here specified to include x
and y, but really this should be pinned down completely) and a
particular domain of preference profiles for these individuals and
The writer seems to be aware of this, suggesting that we could
“consider Zelig’s peculiar behavior part of the social welfare
function”. The writer’s following claim that then the dictator is
“whoever is sitting next to Zelig …as they get their vote boosted by
Zelig” is true, as I’ve illustrated, only if Zelig is always sitting
next to the same person, on every occassion. Even in this case,
though, and contrary to the writer’s claim, Zelig himself is also a
dictator of f in Arrow’s technical sense.
To be a dictator, either Alice or Bob's preferences would always have to match the group's preferences. This is not the case, since Alice or Bob's preference only match the group's preference when they sit closest to Zelig. Therefore, they are not dictators. Zelig is, because his preference always matches the preference of the group.
A dictator must be a single instance of a human being. The "person sitting next to Zelig" is not the same single person in all cases, so they cannot be a dictator. This changes if you assume consistent seating.