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Arrow's non-dictatorship principle states that "no single voter possesses the power to always determine the group's preference."

I am unsure if I am missing some nuance from this definition, so I was wondering if the following would be an example of a dictator:

There exists a simple majoritarian democracy, where a referendum asks voters to vote between policy A or B, and there are 101 voters. If 50 vote for A then the 51st voter will be a dictator, as their choice of A or B will determine which policy results, regardless of what the other voters vote for. Would this be a correct formulation?

If this is correct then what significance does Arrows theorem really have; is it not quite obvious that even the most basic democratic systems will have a dictator? In addition, why does this even matter? In most real world elections, it is basically impossible to "find" this dictator, since the location of the dictator itself can only be determined retrospectively, as it is dependent on what everyone else has voted for. Even at its most basic, for instance, in my example, the 51st voter would no longer be a dictator, if someone who originally voted for B now votes for A; this voter would now be the dictator. It seems then quite inconsequential that a dictator does exist, if it cannot be pinpointed, but more importantly, the specific "dictator" voter is dependent on what everyone else has voted for in a specific election.

Any help fixing my misunderstanding of the significance of Arrow's theorem would be much appreciated, as I realise it is quite important in the field.

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    "...the 51st voter..." It's not like they vote one after another, or do they? On the other hand it is trivial to show that with two options and 101 votes in total, there will be at least one option with 51 votes. Always. Commented Jan 8, 2023 at 17:29
  • I realise now that I have misunderstood the theorem. But I am still unsure if I have grasped its conclusion from the proof given on wikipedia. Assuming that the kth voter is always the dictator of the election given in the "pivotal voter" proof, is it not quite easy to create a situation where the kth voter is no longer the dictator, simply by flipping k for all other voters. Doesn't this trick only work because it is shown by sequentially flipping adjacent votes and applying IIA, which is not realistic and also means you get different results even if your end configurations are identical.
    – Ahmed
    Commented Jan 9, 2023 at 2:07
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    @Ahmed You're assuming that the voting system being analyzed is, in fact, a genuinely democratic voting system. The point of the proof is that IIA and unanimity principles cannot be achieved in such a system, and it proves this by showing that successfully achieving them implies that you actually have a dictatorship rather than a democracy.
    – Douglas
    Commented Jan 9, 2023 at 4:18

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The key word here is always. In your hypothetical example, let's say that the votes occur in succession, and the same person is always the 51st voter. Can this person always determine the group preference, regardless of the preference of the individual voters? No: if the 50 people before them vote for candidate A, and the 49 people after them vote for candidate A, candidate A will win even if the 51st voter chose candidate B.

Therefore, the 51st voter cannot always determine the preferences of the group regardless of how the rest vote, but rather can only do so in the rare case in which the vote is split just so—which, of course, is not "regardless of how the rest vote." They thus do not fit the definition of a dictator used in proving Arrow's impossibility theorem.

More formally, the non-dictatorship principle is defined as follows (from the corresponding Wikipedia article, rewritten a bit to avoid LaTeX characters):

There is no voter i with ordering P(i) in {1, ..., n} such that, for every set of orderings in the domain of the constitution, and every pair of social states x and y, x P(i) y implies x P y.

In other words, the single voter's preference has to be decisive in all cases. This applies to all pairs of social states—all candidates, for instance—for which P, the overall preference of the collective, might be anything. The fact that x P(i) y implies x P y for one specific pair of x, y and for one set of voter preferences regarding those states, as in the question, does not imply that this relation will hold for all pairs.

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  • I see. I am still a bit confused then with what is given on the wikipedia page. In part 2 of the proof, in the last paragraph, I understand why A>C due to IIA, but why must B>A, rather than C>B (if we assume the "worst case" and k picks B>A>C and everyone else will switch so that C>B)
    – Ahmed
    Commented Jan 9, 2023 at 1:55
  • Why would it not be that A>C due to IIA, and then C>B simply because more voted for C>B, then B>C (assuming all other non-k voters flip)
    – Ahmed
    Commented Jan 9, 2023 at 2:08
  • @Ahmed In part 2 of the proof, the result must include B>A because, except for the (irrelevant to A vs B) position of C, the voting is identical to the voting in part 1 of the proof. Effectively, B>A is part of the starting premise of part 2, because it takes the vote arrangement from part 1 (where voter k is pivotal for B>A) and changes it only in ways that should not affect A vs B due to IIA. B>A combined with A>C then implies B>C due to transitivity - the full ranking is B>A>C.
    – Douglas
    Commented Jan 9, 2023 at 4:05
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Based on your comments, I think you are getting confused by preconceptions and have missed the point of the proof. It's not really about defining the concept of a dictator in an unconventional way, or about proving that there is a dictator. It's about proving that, without a dictator, the principles of unanimity and independence of irrelevant alternatives are impossible for any ranked preferences voting system to satisfy.

Take the normal mainstream definition of a dictator: A person who has the authority to unilaterally decree all societal decisions.

Now assume that you have somehow devised a ranked preferences voting system that successfully guarantees the principles of unanimity and independence of irrelevant alternatives. That is:

  • If literally everyone votes for X, then X wins.
  • In a vote between options X, Y, and Z, changing people's opinions about Z will never affect the vote results about X vs Y.

The entire logic of the proof of Arrow's impossibility theorem is to show that, if you have indeed successfully guaranteed those two principles with a ranked preferences voting system, then the system you used to do so is in fact a dictatorship, in the conventional sense of the word.

Therefore, if you want to not have a dictatorship, you will necessarily have to make some kind of compromise on other principles (such as independence of irrelevant alternatives), or use a fundamentally different type of voting system.

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Here is an explanation of the dictator principle in very simple words.

Arrow's theorem says there can be no voting system such that properties a), b) and c) are all satisfied at the same time. But a careful reading of the rules shows there is indeed such a voting system, namely everybody votes and then we do exactly what 'Bob' wants, regardless of what every other voter said. This situation would be referred to as having a dictator.

This is not a very interesting counter example so Arrows theorem is formulated to exclude this situation. Otherwise it would not be a true theorem as stated.

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