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As Wikipedia says, Arrow's impossibility theorem states that no rank-order electoral system can be designed that always satisfies these three "fairness" criteria:

  1. If every voter prefers alternative X over alternative Y, then the group prefers X over Y.

  2. If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change).

  3. There is no "dictator": no single voter possesses the power to always determine the group's preference.

Let's say we decide that we really want a ranked-choice voting system, and that we prioritize the first two criteria over the third. But we also have some additional criteria which may or may not be realizable with a ranked-choice voting system:

  • We don't want someone to be a statically appointed dictator. That is, which voter is the dictator in a particular vote should be determined by voter preferences, AKA how voters voted in that vote itself. Ideally the dictator could be any of the voters depending upon which preferences voters have.
  • It should be hard, that is, cryptographically hard, to determine which voter is the dictator without knowing what some given fraction, say 50% for example, of the votes are.
  • As a strengthening of the previous criterion, we don't want someone who knows what less than the given fraction of the votes are to be able to narrow down who the dictator is to some given fraction of the voting population. Again, we could use 50% as an a example.

Is it known whether it possible to create a ranked choice voting system that meets these criteria? If so, is it possible or not?

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    $\begingroup$ I think you are misunderstanding what a "dictator" is. From your quote, "a single voter who possesses the power to always determine the group's preference" so that person cannot "be determined by voter preferences". A dictator is a dictator regardless of other voters preferences. $\endgroup$
    – user247327
    Sep 17, 2020 at 23:00
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    $\begingroup$ If the first two conditions are met by an ranked-choice voting system, there must be a specific voter whose preferences dictate the outcome no matter what the voters’ preference schedules are — what you’re calling a statically appointed dictator and what in this context is simply called a dictator. $\endgroup$ Sep 17, 2020 at 23:11
  • $\begingroup$ @BrianM.Scott Say we took all the votes, converted them to numbers, and created a function from this collection of numbers to a number indicating which voter to be the dictator. Is your claim that this function cannot possibly satisfy the other criteria? I suppose this may be so, but I find myself unconvinced. If I look at Arrow's impossibility theorem proofs where the dictatorship portion falls out as a consequence, they all seem to me to be talking about some voter whose choices happen to be pivotal in this case, which at least seems to me to be different to the definition you use. $\endgroup$
    – Ryan1729
    Sep 18, 2020 at 0:38
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    $\begingroup$ @Ryan1729: No, that is not what I’m saying. Apparently you still have not understood what dictator means in this context. And the definition that I use is precisely the one used in the theorem. The proof shows that there is a single voter who is pivotal for all pairwise choices no matter what the voters’ preference schedules are. (And there are proofs that don’t use the notion of pivotal voters at all.) $\endgroup$ Sep 18, 2020 at 0:53
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    $\begingroup$ @BrianM.Scott Can you please direct me to some resource that defines dictator as you have and uses it in a proof, and/or an example of a proof that doesn't use pivotal voters? I'm hoping one of those would help me understand. (It's hard to search for something that doesn't have something in it!) $\endgroup$
    – Ryan1729
    Sep 18, 2020 at 5:57

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After reading some of the literature mentioned in the comments, I am convinced that my original question was flawed. Specifically, I accept that there is no way to meet the first two fairness criteria without having a dictator that is completely determined by the voting system, independently of the votes cast.

I think the closest thing to what I had in mind, while being actually possible, would be to randomize what the voting system is itself, after the votes are counted, through cryptographically secure means. Since all voting systems meeting the first two criteria have system-determined dictators, all such systems would either not meet those two criteria, or be equivalent to randomly picking one of some non-empty subset of voters to be the dictator.

I am less certain about this, but I think that if any non-empty subset other than the entire voting population is chosen via looking at what they voted, then that would violate, the second fairness criteria, which is known as independence of irrelevant alternatives, hereafter IIA.

For example, choosing one of voters that had the most common top preference is equivalent to first-past-the-post voting which is known to violate IIA.

Similarly, choosing to remove from the random dictator pool the voters who voted the least common way, (leaving them all there in the unanimous case,) would violate IIA since adding alternatives can push voters in and out of that pool, therefore affecting the outcome under some randomizations.

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