4

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?

0
4

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.

2
  • 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). Aug 16 '18 at 22:17
  • 2
    @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
    Aug 17 '18 at 6:20
0

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.

5
  • 1
    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
    Aug 16 '18 at 22:18
  • I went ahead and opened up a chat. chat.stackexchange.com/rooms/81792/zelig Aug 16 '18 at 22:24
  • That Alice and Bob are not dictators does not imply that Zelig is.
    – Arno
    Aug 17 '18 at 6:18
  • @Arno Agreed. That was poorly articulated. I removed that entire statement. The answer now focuses on the response from Dr. Morreau. Aug 17 '18 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
    Sep 7 '18 at 13:21

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .