# 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?

## 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 "there is no dictator" is stronger than "all voters are equal". It 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? – JonathanReez Jul 26 '18 at 8:40
• @JonathanReez: The theorem wouldn’t be true if dictators were allowed. Remember that this is mathematics. – chirlu Jul 26 '18 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. – origimbo Jul 26 '18 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. – chirlu Jul 26 '18 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. – David Schwartz Jul 26 '18 at 18:46

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 – JAD Jul 26 '18 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. – Acccumulation Jul 26 '18 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. – Acccumulation Jul 26 '18 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]". – David Richerby Jul 26 '18 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. – Wrzlprmft Jul 27 '18 at 21:24

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. – origimbo Jul 26 '18 at 9:26
• @origimbo who's the dictator in that szenario? – DonQuiKong Jul 26 '18 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. – indigochild Jul 26 '18 at 15:24
• @Bobson: Yes, that’s non-dictatorship. There is no single individual (dictator) that alone determines the result. – chirlu Jul 26 '18 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? – JonathanReez Jul 27 '18 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? – indigochild Jul 27 '18 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). – Wrzlprmft Jul 27 '18 at 21:14
• Thanks! I upvoted after that edit. Definition 5 and the other text are interesting. – indigochild Jul 27 '18 at 21:18
• @indigochild: I asked a follow-up question on the Zelig example. – Wrzlprmft Aug 16 '18 at 21:20