# What does the non-dictatorship principle of the Arrow theorem mean exactly?

According to Wiki:

The property of non-dictatorship is satisfied if there is no single voter i with the individual preference order P, such that P is the societal ("winning") preference order, unless all voters have the same P. Thus, as long as there are voters in the society that have different preference orderings, the preferences of individual i should not always prevail.

What exactly does that mean in practice? What would be an example of a "dictator" in a real world election?

The property of non-dictatorship is satisfied if there is no single voter i with the individual preference order P, such that P is the societal ("winning") preference order, unless all voters have the same P.

This is very badly phrased: it's ambiguous and the most natural parsing is flatly wrong. It sounds like it's saying that the election result must never agree with the preference of a single voter: for example, if any voter ranks the candidates as A > B > C, then the result cannot be A > B > C unless every voter thinks A > B > C. This is obviously nonsense: consider any fair result when 999,999 voters think A > B > C and one voter thinks C > B > A.

What it's trying to say is that a dictator is a voter V who is chosen before the election and the result is pre-ordained to be "Whatever V chooses." In other words the quantifiers are the other way around: there's no single voter whose choice determines the election regardless of how anyone else votes. That one person would be a "dictator".

• It also doesn't state that there has to be more than one voter :P
Jul 26, 2018 at 13:03
• @JAD The non-dictatorship principle does, in fact, necessitate that there be more than one voter. If there is only one voter, then the impossibility theorem does not apply: we can just take whatever that voter wants. Jul 26, 2018 at 15:32
• "the most natural parsing is flatly wrong." No, it's not. "It sounds like it's saying that the election result must never agree" No, it says "The property of non-dictatorship is satisfied if there is no single voter i with the individual preference order P..." [bolding mine] It doesn't say this is a necessary condition, it says this is a sufficient condition to satisfy the non-dictatorship principle. Jul 26, 2018 at 15:37
• @Acccumulation I suppose I should check the context. The sentence reads like a definition, and definitions are usually phrased as "A thing is 'good' if [condition]", which is understood to mean "A thing is 'good' if and only if [condition]". Jul 26, 2018 at 15:57
• @Acccumulation: it says this is a sufficient condition – The context and particularly the following sentence (“Blind voting systems (with at least two voters) automatically satisfy the non-dictatorship property.”) both suggest a definition, where a sufficient condition would not make sense. Jul 27, 2018 at 21:24

## Non-Dictatorship

In Arrow's theorem, there are voters and candidates. Each voter makes a choice between the candidates (each voter can say for each pair of candidates which they prefer or if they are indifferent). A voting system, or "social-choice function" takes the collection of the individual choices and outputs a single "social choice".

It is possible to create a "voting" system like this:

Given voters' social choices P1, P2,... Pn, return as the social choice the P1 (ignore P2...Pn)

In other words, the social choice is always the same as the first voter's preferences. In this situation, the first voter is called a "dictator".

The condition "all voters are equal" is stronger than "there is no dictator". Non-dictatorship is one of the conditions of a fair voting system that Arrow places on the social choice function. He then goes on to prove that no such function can exist in general.

## Why is this important?

Arrow himself explained the necessity of this assumption in Chapter 1 of Social Choice and Individual Values (reference below). He was interested in exploring social choice theory, which provides a way to aggregate individual opinions. When there exists a dictator (whose preferences always mirror the aggregate of society) then there is no need for a vote: the outcome is always what the dictator would choose. Any such system is outside the scope of social choice theory.

For more information, see the non-dictatorship section on the Stanford Encyclopedia of Philosophy. For a primary source, the discussion in Social Choice and Individual Values is invaluable.

• So essentially the "no dictator" rule refers to a hypothetical election system which is never used in practice? What's the point of that condition? Jul 26, 2018 at 8:40
• @JonathanReez: The theorem wouldn’t be true if dictators were allowed. Remember that this is mathematics. Jul 26, 2018 at 8:43
• @cpast within the context of the theory, any system with equal numbers of party A, party B (forced to vote for their own party) and one independent has a dictator, since the independent's preference always wins. Jul 26, 2018 at 10:54
• @cpast: Sorry, I think I got confused with some negations … I’ve drawn a diagram to be sure and now agree that it’s stated backwards. I.e. “all voters are equal” is a stronger requirement than “there is no dictator”; or equivalently, “all voters are equal” implies “there is no dictator”, but the inverse isn’t true. (Always assuming more than one voter.) I’ve removed the erroneous comments. Jul 26, 2018 at 11:09
• @JonathanReez The theorem is about algorithms that take a set of preferences for individuals and from them generate a single preference. This is what voting systems like "majority rules" do. They take the preference of each individual as input and produce as output something we call the majority preference. One such algorithm could be "ignore everyone but Jeff". Such an algorithm would violate the non-dictatorship rule. Arrow is interested in systems that don't violate this rule because systems that do are trivial and uninteresting. Jul 26, 2018 at 18:46

It means, quite simply, that the outcome of the election depends on the vote of only one particular person (the dictator).

I don’t think there are any real-world examples of this; real dictators either don’t hold elections at all, or if they do, they want to make it at least appear as if the other voters actually can affect the outcome (perhaps they can choose among several candidates who all faithfully support the dictator).

Hypothetically, an election system with the dictatorship property could work like this: All people get to vote in the same way, but the dictator votes on a yellow ballot while the other voters receive white ballots. When determining the results, only the yellow ballot is actually considered; the others are ignored.

Note that it is a requirement for any such voting system that the dictator’s ballot can be distinguished from the others. If that is not the case (the voting system is then called “blind”; in particular, that applies to secret elections), there can’t be a dictator, as the Wikipedia article also states:

Blind voting systems (with at least two voters) automatically satisfy the non-dictatorship property.

• Something close to a real world example would be a parent and children deciding what to have to lunch. "Voting system" in this theorem is much more generic than an election and includes all kinds of deterministic decision making processes. Jul 26, 2018 at 9:26
• @origimbo who's the dictator in that szenario? Jul 26, 2018 at 14:59
• Can you back-up this answer? It contradicts the quote OP provided, SEP, as well as Arrow's work itself. So I'm curious what it's based on. Jul 26, 2018 at 15:24
• @Bobson: Yes, that’s non-dictatorship. There is no single individual (dictator) that alone determines the result. Jul 26, 2018 at 16:13
• @indigochild so would an election where we choose a random ballot and use it to resolve the outcome satisfy the non-dictatorship criteria? Jul 27, 2018 at 20:43

The non-dictatorship rule simply states no single individual (aka 'dictator') can determine the result of a vote by his/her choice alone.

Please note that the non-dictatorship rule isn't of much use just by itself. It is best considered part of the impossibility theorem, which states that a voting system can't fulfill all of certain criteria at once.

For voting systems, the criterion can be illustrated by its use in the more specific Gibbard–Satterthwaite theorem. It states:

Every non-dictatorial, non-imposing voting system for choosing a single candidate from more than two candidates is susceptible to tactical voting, i.e., voting against one’s preferences to effect one’s preferences.

Now, the boldfaced parts are the main insight to be had here for real elections. Still, the conditions non-dictatorial and non-imposing are necessary to cover the following pathological cases:

• A system where the choice of a single voter (the dictator) is given absolute preference, i.e., whoever that dictator chooses wins irrespective of the other votes.

• A system where some candidates cannot win irrespective of the votes, e.g., the system where the tallest candidate automatically wins. (The outcome is imposed by the voting system and not affected by the voters at all.)

In the first system, the dictator cannot vote tactically since the outcome is identical to their vote. Everybody else in those two cases cannot vote tactically since their vote has no effect at all. So, both systems are immune to tactical voting and thus the theorem would be wrong if it did not impose the respective conditions. Of course, the immunity to tactical voting does not outweigh the fact that those systems are so absurd that not even tinpot dictators would try to market them as fair voting systems.

The property of non-dictatorship is satisfied if there is no single voter i with the individual preference order P, such that P is the societal ("winning") preference order, unless all voters have the same P. Thus, as long as there are voters in the society that have different preference orderings, the preferences of individual i should not always prevail.

Like David Richerby, I find this formulation misleading. In particular the restriction “unless all voters have the same P” does not make sense to me in any interpretation of those words. Here is Arrow’s definition from A difficulty in the concept of social welfare:

In its pure form [dictatorship] means that social choices are to be based solely on the preferences of one man. That is whenever the dictator prefers x to y, so does society. […]

Definition 5: A social welfare function is said to be “dictatorial” if there exists an individual i such that for all x and y, xPiy implies xPy regardless of the orderings of all individuals other than i, where P is the social preference relation corresponding to those orderings.

Translated to the context of elections, “social welfare function” corresponds to the voting system, “social preference relation” corresponds to the voting outcome, and xPiy or xPy means that Pi or P prefers x to y.

No special attention is paid to the case that all voters have the same preferences. Instead the key feature of dictatorship is that the dictator’s preference prevail for all possible preferences of the other voters.

• This seems to contradict the original quote, the SEP, and Arrow's own published papers. Can you reconcile this to Arrow's work? Jul 27, 2018 at 20:00
• @indigochild: See my edit. The quote from Arrow seems to align with the definition in the SEP. The example in the SEP (Zelig) seems to be a misinterpretation of Arrow’s dictatorship, as in Zelig’s case \$P_i\$ and \$P\$ are not identical because of the voting system but due to Zelig’s own system of determining his choices (a weird form of tactical voting if you so wish). Jul 27, 2018 at 21:14
• Thanks! I upvoted after that edit. Definition 5 and the other text are interesting. Jul 27, 2018 at 21:18
• @indigochild: I asked a follow-up question on the Zelig example. Aug 16, 2018 at 21:20

The quote in question is a little loose because it's trying to avoid strict mathematical precision to reach a more general audience. As there are a number of comments that seem to be haggling over issues that should be resolved by the precise definition, I will give that here, and try to explain it as best I can. I will follow the outline given by Wikipedia.

NB: Notation will be slightly hampered by a lack of TeX implementation on this particular SE.

So we have a society with a number of issues, with various possible ways of dealing with them. We let A be the set of all possible outcomes to all of these issues. So if we have three issues X,Y,Z, then an element of A consists of three outcomes (X',Y',Z'), where X' is one potential outcome/resolution to X, etc. It does not matter what constrains what is allowed to be in A. When devising our collective process of making a decision, the system can permit or restrain the outcomes as desired. For example, if an issue is "Should we pass Law T?", then we may restrict the outcomes to that to "Yes" and "No", and exclude answers like "Let's amend it to TT and pass that" or "The only way to fix what Law T wants to fix is to nuke the entirety of our own country, so let's nuke our country".

Now let L(A) denote the set of all (full) linear orderings on A. Such a thing is like a generalization of how we compare real numbers via statements like "x <= y", read "x is less than or equal to y". But instead of numbers x,y we are using elements of A, outcomes to our issues.

The role of any one such linear ordering on A is as follows. We are imagining that given any voter, you can present to them any two possible outcomes x,y and that voter will tell you which one they prefer: x<y (they prefer y); y<x (they prefer x); or y=x (both are equally acceptable). We want these to have the same sort of nice properties that we have when comparing real numbers: if x,y,z are three outcomes and x<y and y<z then we require also x<z, etc.

So an element of L(A) is one possible voter's preferences of all possible outcomes, telling us which combination of outcomes he likes more than others.

Now suppose we have N people whose preferences (elements of L(A)) are to be used to decide what society does (the final/actual outcome). We'll call this our "vote", though since voting isn't formally required it's usually just called something more fancy like "social welfare function". The process of deciding the final outcome from N preferences is a function F whose valid inputs are the elements of L(A)^N, meaning all possible preferences among N people, and whose outputs are in L(A). The output is the final outcome that spits out the collective preference, the results of the vote, determined by F.

So if R1,R2,...,RN are the preferences (elements of A) determined by each of our N voters, then the final output is F(R1,R2,...,RN).

It is important to note that this function F, the process by which we decide what to do, must be defined on arbitrary choices of R1,R2,...,RN.

Now we can finally get to what "dictator(ship)" means here.

Fix a person i, with preferences Ri (an element of L(A)). We say that voter i is a dictator if (and only if) for any pair of outcomes a,b from A, then if i prefers a over b—meaning the linear ordering Ri says that b < a—, then the final outcome prefers a over b—meaning the linear ordering F(R1,R2,...,RN) says that b < a—as well. So i is a dictator if society always prefers what he does, when he has a preference.

This is almost but not quite the same thing as saying that F(R1,R2,...,RN)=Ri. For on any pair of events a,b where R_i says a=b, which means that our dictator doesn't really care between the two, then the final outcome's preferences are unconstrained: F(R1,R2,...,RN) can say any of a=b, a<b, or b<a. So F is permitted a sort of tie-breaker: when the dictator doesn't care, something else may be used to decide the final outcome's preferences (possibly a "vice-dictator", or a majority vote, etc.).

Again, though, note that we are not dealing with a fixed set of preferences. Fixing F (and N), it is entirely possible that if you take some specific, fixed set of preferences R1,...,RN and plug them into F you will get your particular choice of Ri for some i (up to not caring about options where Ri doesn't care). But Ri would be a dictator only if that happens for every possible choice of R1,...,RN (including Ri itself). So it's okay if Joe Schmoe randomly ends up having the same preferences as the final outcome in some specific instance of decision-making; it's only when our decision making process always prefers whatever Joe Schmoe does (when he has a preference) in every conceivable situation that we have a dictator.

What we would call a dictator in normal politics will usually be a dictator in the sense of Arrow's Theorem, though in some cases (see next paragraph) they would not be. Any government with one absolute authority for decision making will likely have a dictator in the sense of Arrow's Theorem. Most historical monarchies are of this sort.

It's worth noting that the origins of the term come from ancient Rome, where "dictator" was a specific role the Senate could create to endow someone with the full powers of the state for the purpose of resolving certain specific issues. Dealing with wars, stabilizing a crumbling nation, or major emergencies, for example. But this was not really a dictator in the sense of Arrow's Theorem, for the Senate and the people still reserved some powers of oversight and veto.

A less obvious example of a dictator in Arrow's sense would be the US President, when the set A is suitably constrained. He does not have absolute decision making powers on absolutely every issue, but he is the Commander in Chief of the military. Provided we restrict our issues and A to the "lawful and constitutional orders to the US military" (and we exclude anything that requires Congress to make a decision), and we limit our N people to the enlisted members of the military, then the President is in effect a dictator in Arrow's sense. If he orders something be done, then it shall be done, regardless of what anyone else desires.

The wiki page on dictatorship mechanism provides a few other non-standard examples. For example, suppose you are a university trying to allocate housing to students. A common system is to create a system of priority, where various attributes are used to determine who has "highest priority", such as age, grades, degree progress (Freshman often get lower priority than returning students, unless this is Freshman-specific housing). The person with highest priority then picks their room from whatever is available. This makes that person a dictator for the problem of students selecting rooms, under the assumption that once a person selects their room they no longer care about who gets the remaining rooms. Indeed, the system becomes a "serial dictatorship": after the first priority student gets his room, the second highest priority student then selects his, and so on and so forth.