01 May 2018

## What Does the Dictator Condition Mean in Arrow’s Theorem?

In the most recent Potpourri post a raging controversy began, which has the ability to rival the Great Debt Debate. The specific motivation was my recent appearance on the Tom Woods Show when I was promoting my Liberty Classroom course on the History of Economic Thought II, and I made a quick mention of Arrow’s Impossibility Theorem. In the comments of my Potpourri, Tel and Craw said that they thought something was fishy in the way Arrow defined “dictator.” (For what it’s worth, Arnold Kling thinks the same thing–calling it a “swindle.”)

In the interest of education, I am moving the discussion here. I think Arrow’s Impossibility Theorem is one of the most surprising and fun results in formal choice theory, because you don’t need any prior math training to understand the proof. (See Sen’s discussion.) Let me also say that when I first read Steve Landsburg talking about it in The Armchair Economist, I scoffed. “Give me a break,” I thought. “How could a formal proof actually discuss something real-world like elections?” I also spent about 24 hours in grad school thinking I had come up with a counterexample to Arrow’s Theorem. But, I realized my mistake.

So in light of that background, that’s why I am now such an ardent defender of it, because I have been through the crucible.

In one of his comments, Craw wrote:

There is an easy counterexample to your claim. Let there be a unique voter for each ranking and the rule is
Tel had that list so Tel was dictator. If Tel and Kirk swap lists then Kirk is dictator and Tel is not.
It is Kirk’s list which matches the social choice.

No. This is misunderstanding how Arrow’s framework works.

What Arrow is looking for are social choice rules that map from the vector of individual’s subjective ordinal preference rankings (over the elements of some choice set–we can neutrally call it “possible states of the world”) into a ranking of what “society’s” ordinal preferences are. One of Arrow’s conditions (which sounds quite reasonable in some settings but maybe not in others) is that the way Society ranks outcome x versus outcome y should only be a function of how each individual subjectively ranks x versus y.

So for example, a majority-rule social choice function would say, “Society ranks x higher than y if the number of people who subjectively rank x higher than y exceeds the number of people who rank y higher than x.” (Let’s say there is no indifference–where someone thinks x is as good as y–just for simplicity. But Arrow’s framework can handle that.)

Now there’s a problem with using majority-rule as the social choice function. Specifically, if you have at least three citizens and three possible outcomes in the choice set, then you can easily construct a possible list of subjective rankings that lead “Society” to have intransitive preferences. Namely, suppose Person #1 thinks x}y}z (where } means “is preferred to”), while Person #2 thinks y}z}x, and Person #3 thinks z}x}y. There’s nothing weird at the individual level with those preference rankings.

However, if we use a majority-rule social choice function, then we would say Society thinks x}y (because Persons #1 and #3 outweigh Person #2), and we would say Society thinks y}z (because Persons #1 and 2 outweigh Person #3). But oops, our rule also would say Society thinks z}x, because of Persons #2 and #3. So majority-rule social choice functions are in principle vulnerable to constructing intransitive Social preference rankings, and hence Arrow’s framework rules them out.

Another possible social choice rule would simply say, “If Person #2 thinks a}b, for any a,b in our social choice set, then Society thinks a}b also.” This rule would satisfy all of the requirements for a “reasonable” social ranking in Arrow’s framework, *except* it would make Person #2 a dictator. Arrow’s condition of non-dictatorship says that for each person in society, it should at least be possible for there to be a vector of individual preference rankings such that the social preference ranking of some a,b differs from that individual’s ranking. If this is NOT true–in other words, if there exists an individual such that his or her ranking of a,b matches what the Society ranking of a,b is, for all possible a,b in the social choice set and for all possible vectors of individual rankings–then that person is a “dictator” in Arrow’s framework.

So looking again at Craw’s statement above, we see the misunderstanding. Just because in some *particular* listing of ordinal preferences by individuals, the social choice ordering happens to exactly match one individual’s ranking, that alone does not establish that that individual is a dictator.

Let me illustrate by going back to majority rule. Suppose once again the Person #1 has ranking x}y}z. But now suppose Person #2 has ranking x}z}y, and Person #3 has ranking y}x}z. Taking the possible outcomes two at a time, and doing a majority-rule criterion, we determine that Society thinks x}y (Persons #1 and #2), Society thinks y}z (Persons #1 and #3), and finally that Society thinks x}z (Persons #1 and #3). Holy smokes! That means Society thinks x}y}z which is identical to Person #1’s ranking! Does that mean he’s a dictator?

No, it doesn’t. This (I believe) is what Tel and Craw *think* Arrow’s framework implies, and I agree that if it *did*, that would be a pretty screwy definition. (For one thing, if everyone happened to have the same ranking, then every single person would simultaneously be a dictator.) But nope, that’s not what it means to be a “dictator” in Arrow’s framework. With the majority-rule function, we could tweak the individual rankings, and then find that Society ranks some a,b differently from how Person #1 ranks them, and so Person #1 is not a dictator.

Here is Steve Landsburg’s attempt to explain Arrow’s Theorem in simple terms, but again, I point you to Sen’s actual proof that doesn’t require any prior mathematical training. And of course, if you subscribe to Liberty Classroom you can see me explain it–one in plain English, and then again walking through a formal proof using symbolic notation.

#### 30 Responses to “What Does the Dictator Condition Mean in Arrow’s Theorem?”

1. Rory says:

Libertarianism: when you observe that people think they’re dictators just because their votes happen to match an electoral outcome, but the anarcho-capitalist whose blog you comment on explains where you went wrong in understanding the greater proof of why elections are a farce.

2. Michael says:

Keshav spelled this out at least a dozen times in the other thread, e.g:

” It’s not enough for the social preference list to match Tel’s preference list in one case and Kirk’s preference list in another case. The social preference lists has to match the same person’s preference list in all cases.”

That’s as plain english as it gets. I don’t know why the same repeated arguments kept getting made without any reference to this point, over and over. It’s not even an interesting point of clarification.

• Bob Murphy says:

Michael wrote: “That’s as plain english as it gets. I don’t know why the same repeated arguments kept getting made without any reference to this point, over and over. It’s not even an interesting point of clarification.”

Michael, ironically enough, literally the next comment (by Rick Hull) is also getting tripped up on this point. So I think it makes sense for me to try to clarify it.

• Michael says:

for sure, wasn’t a criticism of you. I think you have the patience of a saint, that’s all.

3. Rick Hull says:

I like Landsburg’s explication but I think he’s wrong or misleading in his conclusion:

> The conclusion remains the same — if you have a system that translates a collection of individual preference orderings into a single “social preference ordering”, and if that system is designed to have certain features that strike many people as reasonable, then the system anoints a dictator.

Not every time. It’s only for very certain states in a massive state space that a “dictator” is inevitable or even possible. Moreso, the fact of dictatorship is coincidence, literally, and this sort of dictatorship is not what people fear.

• Keshav Srinivasan says:

Rick, unfortunately you are having the same misunderstanding of Arrow’s theorem that Tel and Craw are having, the misunderstanding that was the entire reason for this blog post. What dictator means in Arrow’s theorem is someone who gets his way over the entire state space, it does NOT mean someone who coincidentally happens to get his way in one particular state. (Where “state” means a set of preference lists, one for each individual.)

If you don’t believe me, look at what the non-dictatorship condition says: “there is no i ∈ {1, …, N} such that for all (R_1, …, R_N) ∈ L(A)^N, a ranked strictly higher than b by R_i implies a ranked strictly higher than b by F(R1, R2, …, RN), for all a and b.” It does not say “for each (R_1, …, R_N) ∈ L(A)^N, a ranked strictly higher than b by R_i implies a ranked strictly higher than b by F(R1, R2, …, RN), for all a and b”.

Note the difference between these two statements.

• Keshav Srinivasan says:

Sorry, the end of the comment should say “It does not say “for each (R_1, …, R_N) ∈ L(A)^N, there is no i such that a ranked strictly higher than b by R_i implies a ranked strictly higher than b by F(R1, R2, …, RN), for all a and b”.”

• Rick Hull says:

Thanks Keshav. I was vaguely worried this was the case shortly after I posted. I’ll review further.

• Rick Hull says:

It seems to me that we can easily construct a set of preference lists for which no dictator is possible. Furthermore, it seems to me that a random state (set of preference lists) is unlikely to allow a dictator. What is coincidental is when a random state happens to allow a dictator.

If Alice, Bob, and Charlie happen to have identical preference lists, isn’t a dictator impossible? My understanding is that Murphy is saying that actual dictators are rarer than Tel and Craw think. I suspect actual dictators are ultra-rare. But they are nonetheless possible as we can construct preference lists that allow them. Landsburg seems to conclude they are inevitable, which seems to be the wrong takeaway.

• Rick Hull says:

OK, it does seem like the theorem predicts an actual dictator in any state, though possibly excluding certain states like “everyone has the same preferences”. It’s not clear that when Alice is the dictator for Monday’s state, she would remain dictator for Tuesday’s state, assuming some preferences change, perhaps radically.

• Keshav Srinivasan says:

Rick, Arrow’s theorem is about rules or methods of determining social preference lists. A dictatorship would be a rule like “Alice always gets her way.” If that’s the rule, then Alice will get her way in all possible situations, including the situation when Alice, Bob, and Charlie have identical preference lists. In that situation, the social preference list will much the individual preference list of Alice, which is the same as the preference list of Bob and Charlie. But Alice is only dictator, Bob and Charlie are not. Why is this? Because Bob and Charlie’s preference lists just happen to match the social preference list in this case, not all cases, whereas Alice’s preference list matches the social preference lists in all cases.

• Keshav Srinivasan says:

In that situation, the social preference list will match*

• Rick Hull says:

I see how a rule like “Alice always gets her way” results in Alice’s dictatorship necessarily. I can also imagine how choosing a selection function which satisfies the other 2 conditions constrains the allowable states, or how the allowable states constrain the selection function, perhaps down to the obviously poor “Alice always gets her way”.

My sense is that there exists at least one state (set of preferences) and selection function (selecting society’s preferences from the given individual preferences) for which there is no actual dictator (while meeting the other Arrow conditions).

When Alice, Bob, and Charlie have the same preferences, then any plausible selection function, e.g. “majority”, should result in society’s ranking matching that of Alice, Bob, and Charlie. I think we can say that Alice is not the dictator. Likewise Bob and Charlie.

If this is the case, then maybe there are other states and selection functions which meet the other 2 conditions yet do not produce a dictator. However, I’m starting to suspect that meeting the 2 other conditions severely limits the universe of selection functions in a way that is difficult to anticipate, perhaps yielding the surprising result that a dictator is always present, or that a dictatorial selection function is necessary.

It still seems to me that if Alice is the dictator on Monday, and the preference lists are scrambled on Tuesday, Alice is unlikely to remain the dictator.

• Andrew says:

It isn’t sufficient to come up with a selection function that satisfies Arrow’s theorem in a single state. To disprove Arrow’s theorem, you must devise a rule that satisfies all of Arrow’s conditions for every possible state.

• Andrew says:

This is correct:

[M]eeting the 2 other conditions severely limits the universe of selection functions in a way that . . . a dictatorial selection function is necessary.

In other words, dictatorship is the only selection function that satisfies the other 2 conditions.

• Bob Murphy says:

Rick Hull wrote: “It still seems to me that if Alice is the dictator on Monday, and the preference lists are scrambled on Tuesday, Alice is unlikely to remain the dictator.”

This doesn’t really make sense, even though I understand where you’re coming from.

There is no formal treatment of time in Arrow’s theorem. The state of all possible outcomes is a formal set, over which people have preferences.

So if you want to have something that repeats, like “What’s the tax rate on Monday, on Tuesday, on Wednesday…” then the way you’d handle that is to have each possible vector of tax rates per day defined as an outcome in the set. E.g. element A in the set of outcomes would be, “14% tax on Monday, 19% on Tuesday, 89% on Wednesday…”

So in that framework, the sole dictator (there can only be one) would have Society’s time-stamped tax rate preferences match his own personal preferences.

• Rick Hull says:

Andrew,

Can we come at this from a different angle? There seems to be a qualitative difference between the dictatorship condition (a single individual’s preferences happen to match society’s) and having a dictatorship selection function.

Can we enumerate maybe 5 or so plausible selection functions, none of which are obvious dictatorships, which seem like they will meet the other two conditions? I recall that “majority” functions will generally fail one of the 2 other conditions. But the devil is in the details as to how the function is exactly specified to take into account all preference rankings. Another plausible rule is to “score” each outcome according to its rank, so Pepperoni gets 0 points at the top of someone’s list, and 4 points for the bottom of someone’s list if it is 5 deep. Lowest score goes to the top of society’s ranking.

I’m not trying to disprove Arrow’s theorem, but to explore it from different angles to better understand its relevance.

• Bob Murphy says:

Rick Hull wrote: “There seems to be a qualitative difference between the dictatorship condition (a single individual’s preferences happen to match society’s) and having a dictatorship selection function.”

Rick, believe me, I get the distinction you are making here. It is one thing if the social choice function literally says, “Whatever Individual #67 says about choices a,b, then that’s the Social ranking too,” on the one hand, versus some complicated mapping function that takes a math PhD to prove to you, after 8 steps, that “oh wow, the Social ranking will end up always matching Individual #67’s ranking, I didn’t see that coming.”

But either way, please be sure you realize, that to be a dictator, it’s not enough for the individual and Social ranking to be identical, in just one particular permutation of individual preferences. It has to be the case that for all possible rankings by everyone in society, that in all cases the Social ranking is identical to some particular individual’s.

• Rick Hull says:

Bob,

I don’t have time to review in depth right now, but I definitely understand the gist of your comment. I have no doubt that Arrow’s theorem is internally consistent and proven true. I am trying to judge its applicability. It seems to me that if we don’t have a dictatorship selection function (e.g. Alice always wins), and preferences can change over time, that we won’t end up with a “scary dictator”, even while remaining compatible with Arrow Impossibility.

Now if the first 2 conditions demand a dictatorial selection function, then this is the way to explain the theorem’s surprising result.

Gotta run…

4. Transformer says:

I read Steve’s post and at first I thought ‘that’s easy, what’s all the fuss about’. Then I thought about it again and I realized I didn’t get it at all and I’m hoping someone can clear up my confusion.

Steve says ‘On any day when Alice prefers Anchovies/Pepperoni/Mushrooms in that order, and everyone else prefers Pepperoni to Mushrooms, Anchovies must rank higher than Mushrooms.’ This makes sense because we already have a rule that Anchovies rank higher than Pepperoni, and the fact that everyone prefers Pepperoni to Mushrooms means the societal ranking must be Anchovies/Pepperoni/Mushrooms.

But then a new rule is stated: ‘Our preferences about Pepperoni should not affect the relative ranking of Anchovies and Mushrooms.’ and from this a conclusion is drawn ‘On any day when Alice prefers Anchovies to Mushrooms, Anchovies must rank higher than Mushrooms.’. That’s where I stopped getting it.

As far as I can see this is the equivalent of saying ‘On any day when Alice prefers Anchovies/Pepperoni/Mushrooms in that order Anchovies must rank higher than Mushrooms even if some people prefer mushroom to pepperoni’. And that doesn’t make any sense to me. If some people prefer mushroom to pepperoni , how can we just go ahead and declare that Anchovies/Pepperoni/Mushrooms is the correct ranking just because this would be true if everyone does happen to prefer pepperoni ?

Help, anyone ?

• Bob Murphy says:

Transformer,

I am saving time on my end by NOT going back and re-reading Steve’s whole post to remind myself of his proof. So I am taking your summary here for granted, assuming it is accurate. With that caveat…

You sound like you’re saying that you followed Steve up to this point:

(1) ==> If Alice happens to rank ‘Anchovies}Pepperoni}Mushrooms’ and everybody else thinks ‘Pepperoni}Mushrooms’ then that alone is enough to conclude that Society ranks ‘Anchovies}Pepperoni}Mushrooms.’

So now if that’s true (and it seems you agreed with it), then the IIA assumption means that if we just want to focus on Society’s rankings involving Anchovies and Mushrooms, then the only information we need from Alice is her feeling on those two. So from (1) above we can deduce (2):

(2) ==> If Alice thinks ‘Anchovies}Mushrooms’ (and we don’t know about Pepperoni) and everyone else happens to think ‘Pepperoni}Mushrooms’ then we can conclude Society thinks ‘Anchovies}Mushrooms’

So far so good? I’m effectively just throwing out information from (1) to say (2).

But now because of IIA, we realize that how the other people feel about Pepperoni vs. Mushrooms can’t be relevant in a Social ranking of Anchovies vs. Mushrooms. So from (2) we can conclude:

(3) ==> If Alice thinks ‘Anchovies}Mushrooms’ then we can conclude Society thinks ‘Anchovies}Mushrooms.’

I.e. we constructed (3) just by throwing out information from (2) that we agreed must be irrelevant.

5. Transformer says:

Still not quite getting it.

I see that we have a rule based on IIA that says Pepperoni vs. Mushrooms can’t be relevant in the social ranking of anchovies vs. Mushrooms and that sounds perfectly reasonable . But without knowledge that everyone has Pepperoni}Mushrooms I don’t see how we derive Anchovies}Mushrooms (in the context of Steve’s example) so appealing to IAA feels like cheating.

Let me think it though all again….

(not being published for some reason so trying again)

• Bob Murphy says:

Transformer, I’m happy to walk you through it, but please refer to my previous attempt. Where did I lose you? I.e. tell me if you agree with everything in step (1) but don’t like the jump to step (2), or if you actually never bought the stuff in step (1) either.

6. Transformer says:

It was at (3) where I stopped understanding.

You say ‘But now because of IIA, we realize that how the other people feel about Pepperoni vs. Mushrooms can’t be relevant in a Social ranking of Anchovies vs. Mushrooms.’ .

I agree this statement appears to be very reasonable.

But the statement ‘If Alice thinks ‘Anchovies}Mushrooms’ then we can conclude Society thinks ‘Anchovies}Mushrooms.’’ as far as I can cannot be validated without some ‘evidence’ that ‘Anchovies}Mushrooms is really true. .

And this ‘evidence’ is::

a. We have already decided (in an earlier part of Steve’s post) that Anchovies}Pepperoni

b. By unanimity Pepperoni}Mushrooms

c. Putting these 2 together we get Anchovies}Pepperoni}Mushrooms vita the application of some basic logic.

Therefore ‘Anchovies}Mushrooms’ depends upon b. and this dependence can’t be eliminated just by citing IIA as (f In understand things correctly) y then once we have the ‘Anchovies}Mushrooms’ rule it is will apply even in situations where there is not unanimity on Pepperoni}Mushrooms

• Bob Murphy says:

Transformer,

OK I’ll try to say it in different words any maybe it will click. I agree it’s a little bit weird but I think the logic is inescapable once you understand the procedure.

The argument is something like this:

“Suppose we can prove (with considerations I’m omitting here for brevity) that in the special case where Alice thinks x}y}z and everybody else thinks z}y, that even so Society thinks x}z.

Now then, what Society thinks about x}z can only depend on people’s ranking of x versus z. So if I tweak each person’s rankings while *not* changing his or her single comparison of x versus z, then Society’s ranking of x}z can’t change either, because of IIA.

So in the first step two paragraphs ago, I convinced you that when Alice says x}y}z and everybody else thinks z}y, just from those facts I can conclude Society thinks x}z. So I am now allowed to fill in all possible permutations of Alice and others’ rankings, so long as I respect that Alice thinks x}z and everybody else can have any rankings at all, and in all of these combinations I have to have Society think x}z. This is because in the original demonstration, the only relevant information about x and z we had from Alice was x}z, and we didn’t have ANY relevant information about x and z for anybody else.

Thus can have shown that for any possible permutations of individual preferences, whenever Alice thinks x}z, that is sufficient to conclude that Society thinks x}z. Thus Alice is decisive over x and z.”

If you wanted, Transformer, the arguments that we deployed in Step 1, could be reproduced for each possible permutation of rankings of the other, irrelevant preferences, in order to comprehensively show (invoking IIA each time) that Alice thinking x}z was enough to prove that Society thinks x}z.

• Transformer says:

I think an (admittedly low wattage) light bulb may have just come on in my head !

I have seen and understood (from the Wikipedia article) the example for the special case similar to what you describe in your first step.

I see now that you can then use IIA to generalize away from the special case (play around with all other choices) as long as you leave the order of the choices relevant for the special case in place for the ‘dictator’.

And to complete the proof you use the same methodology for all available combinations of choices and prove that the ‘dictator’ dominates everything.

So thanks for taking the time to explain !

My only concern is that this makes very little sense from an intuitive perspective (and for exactly the kinds of reasons people have brought up in the comments) – is it meant to ?

• Bob Murphy says:

Transformer can I email you my class notes on the proof? That might help. Should I use the email tied to your comment?

• Transformer says:

That would be great, thanks – its the email I use in the comments but without the number “2” (for some reason if I use the right one my comments don’t make it through the filter).

7. Harold says:

“My only concern is that this makes very little sense from an intuitive perspective (and for exactly the kinds of reasons people have brought up in the comments) – is it meant to ?”

I think this is why the result is so important. It reveals the limitations of “common sense.” If the result made sense from an intuitive perspective it would have little consequence. It is heartwarming to see an internet discussion that sheds both heat and light.

• Bob Murphy says:

Transformer wrote: ““My only concern is that this makes very little sense from an intuitive perspective (and for exactly the kinds of reasons people have brought up in the comments) – is it meant to ?””

I grant you that the PROOF of Arrow’s theorem is extremely unexpected and hard to understand at first–at least, the use of IIA in places. But my pushback against Tel (and to a much lesser extent, Craw) was his claim that the statement of the theorem and its applicability, was somehow misleading.