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This question is a follow-up to this one, and particular serves to clarify a potential error in one of the presented references.

The Context

When elaborating Arrow’s non-dictatorship criterion, the Stanford Encyclopedia of Philosophy presents the example of Zelig, which I here summarise in my own words:

Zelig is a member of a committee of three which uses majority voting to select between two options. Zelig always adopts the opinion of whoever is sitting next to him in the meeting.

The Encyclopedia then concludes:

Now suppose it so happens that Zelig strictly prefers one option x to another, y. Then someone else does too; that makes two of the three and so, when they vote, the result is a strict collective preference for x above y. The committee’s decision procedure is, in Arrow's sense, a dictatorship, and Zelig is the dictator.

My Take

I fail to make sense of this. The definition of a dictator (as per the Encyclopedia) is:

Person d is a dictator of f if for any alternatives x and y, and for any profile ⟨…,Rd,…⟩ in the domain of f: if xPdy, then xPy.

where:

  • f is the social welfare function, i.e., the voting system, which maps the individual orderings to the social ordering.
  • Rd is the ordering of choices by d,
  • Pd are the strict preferences of d (which is identical to Rd in this case as Zelig is not indifferent between any opinions),
  • P = R = f(⟨…,Rd,…⟩) is the social ordering, i.e., the final outcome of the vote according to the voting system,
  • xPy means that x is preferred to y in the ordering P.

Now, let the other committee members be Alice and Bob. Then 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 f.

Even if we consider Zelig’s peculiar behaviour part of the social welfare function, the dictator would be whoever is sitting next to Zelig (and not Zelig himself), as they get their vote boosted by Zelig copying it.

Question

Did I misunderstand or misinterpret anything or is the Encyclopedia indeed incorrect about this?

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  • What happens when Zelig strongly prefers x but both Alice and Bob are going to vote for y? That's the moment of truth of a dictatorship.
    – alamar
    Commented Jun 8, 2022 at 22:02
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    The two paragraphs in the encyclopedia seem to be in direct contradiction to each other. The first one says his opinion is identical to whatever opinion the person seated next to him has. The second one says he has his own opinion regardless of the opinion of the person seated next to him.
    – quarague
    Commented Jun 9, 2022 at 7:00
  • @alamar: Zelig does not have such preferences by construction.
    – Wrzlprmft
    Commented Jun 9, 2022 at 12:57

3 Answers 3

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The Encyclopedia is indeed incorrect. Where they are going wrong is that they are not considering arbitrary profiles of individual choices, but rather restrict to a particular subset: Those where Zelig always agrees with another committee member. Allowing arbitrary profiles would include those where Zelig alone is for some option, while the two others agree on a different one.

Whether or not someone is a dictator follows from the voting rule/social welfare function before any particular voting strategy/preference has been fixed for that person.

To give another example: If we are collectively chosing what fruit to eat for lunch, and the voting rule we are using is "doesn't matter, its apples anyway", then it would not make sense to call those of us who actually want apples the dictators.

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  • Regarding your last paragraph: If someone prefers apples 100% of the time, why wouldn't you want to call that person a dictator? It seems appropriate to me because their preference always matches the social choice (which is always apple). Commented Aug 16, 2018 at 22:17
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    @indigochild Again, once we have fixed a person to prefer apples 100% of the time, the moment to apply the definition has passed. We check whether someone is a dictator before we consider anyones preferences. Since we are considering the case where everyone prefers bananas, the rule "its apples, fullstop" has no dictator.
    – Arno
    Commented Aug 17, 2018 at 6:20
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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 f.

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 options.

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.

Summary

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.

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  • 2
    If that isn't satisfactory, just say so. – This isn’t satisfactory. 1) My understanding is that you are a dictator if the group’s preference always mirror your preferences, not vice versa. 2) Is Alice a dictator? Only if Zelig always votes yes - which means she is a dictator only if she always sits closest to Zelig. – Well, we only have one vote and assuming the sitting order is as imposing as assuming Zelig’s “strategy” (if you ask me). 3) I fail to make sense of your last paragraph. Where did I reconstruct something?
    – Wrzlprmft
    Commented Aug 16, 2018 at 22:18
  • I went ahead and opened up a chat. chat.stackexchange.com/rooms/81792/zelig Commented Aug 16, 2018 at 22:24
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    That Alice and Bob are not dictators does not imply that Zelig is.
    – Arno
    Commented Aug 17, 2018 at 6:18
  • @Arno Agreed. That was poorly articulated. I removed that entire statement. The answer now focuses on the response from Dr. Morreau. Commented Aug 17, 2018 at 18:26
  • 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” – I think this conclusion is where I disagree with Morreau: Applying a restricted domains is not appropriate here because the restriction does not arise from the nature of the choices (e.g., as in the temperature example from the SEP) or similar. (I here interpret appropriate in the sense that it gives relevant insights about social welfare functions in the respective application.)
    – Wrzlprmft
    Commented Sep 7, 2018 at 13:21
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The SEP's description of Zelig involves some rhetorical sleight of hand, but I don't think it's quite incorrect, in a formal, mathematical sense. I believe SEP resolves the problem with this paragraph:

Arrow imposed D in conjunction with the requirement U that the domain is completely unrestricted. Perhaps this condition expresses something closer to its intended meaning then. With an unrestricted domain, a dictator, unlike Zelig, is someone whose preferences conflict with everybody else’s in a range of cases, and it is in each instance his preferences that agree with social preferences, not theirs. However this may be, the example of Zelig shows that whether it is appropriate to impose D on social welfare functions depends on the details of the choice problem at hand. The name of this condition is misleading. Sometimes there is nothing undemocratic about having a “dictator”, in Arrow’s technical sense.

Let's unpack what U means. Informally, U means that each person can vote for any order of preferences they like. While this may appear to be the case with the example involving Zelig, the author has tacitly assumed that Zelig's behavior is part of the voting system itself, and not a voluntary behavior on Zelig's part. In other words, the author has assumed that Zelig doesn't actually get a choice at all, and instead is given a prefilled ballot that he immediately returns without modifying.

In Dr. Moreau's defense (the author of this article), there is no other reasonable way to formally describe Zelig's behavior, unless we want to deviate from Arrow's original formulation. Arrow doesn't allow for the notion that "a voter voluntarily limits his own choices," and so we can either have a voting system in which Zelig's vote is artificially restricted, or a voting system in which Zelig can vote for anything, but not a voting system in which Zelig voluntarily chooses to vote for "whatever the other guy wants." While we could probably find another way to formalize Zelig's behavior, the result is not a "voting system" in the sense that Arrow defines it, because it would contain additional mathematical structures that Arrow did not contemplate.

Arrow chose to formalize things in this manner, because Arrow doesn't really care about voters like Zelig. Arrow is trying to describe a system that works for all possible votes and all possible voters, and in that context, it's common sense that voters should be "allowed" to vote in any way they like. This is also why Arrow uses the term "dictator" - because Arrow didn't care about systems in which U is violated, such as the example with Zelig. In systems that fulfill U, the term "dictator" is largely accurate. On the other hand, Dr. Moreau points out that, in systems where U is violated, D may also be violated, even where the violation does not look much like a dictator at all (remember, Zellig is not given a choice!).

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  • the author has tacitly assumed that Zelig's behavior is part of the voting system itself – But in the introduction to this example, the author claims something else: “Even pairwise majority voting, that paradigm of a democratic procedure, is in Arrow’s sense sometimes a dictatorship. Consider Zelig. …” No mention of Zelig’s behaviour being part of the system. — there is no other reasonable way to formally describe Zelig's behavior – That may be, but why attempt this in the first place? Is the Zelig example anything more than a badly chosen example?
    – Wrzlprmft
    Commented Jun 9, 2022 at 13:15
  • @Wrzlprmft: In my opinion, yes, it is a poorly-chosen example. The problem is, to my estimation, that the author wants to construct a system that violates U but also "looks democratic," and in my subjective opinion ("looks democratic" is a subjective quality), you just can't do that. However, at the same time, the author is not wrong to point out that you can violate D without anyone being a "dictator" in an informal, intuitive sense.
    – Kevin
    Commented Jun 9, 2022 at 18:12

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