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The part of the impossibility proof that deals with dictators is subtle and neat. At first, I was convinced that the dictator was actually a "swing voter" whose

The part of the impossibility proof that deals with dictators is subtle and neat. At first, I was convinced that the dictator was actually a "swing voter" whose power was entirely conditioned on the other voter's preferences, but that's not actually the case. Still, my guess is that these types of edge cases could be handled by using some randomness and low probability arguments. For example, claiming that such dictators exist with negligible probability given a small-ish fraction of voters that are considered "random." While you would never want to inject random votes into an election, it's probably reasonable to assume that some fraction of voters behave naturally random (e.g., people get sick and don't vote, or are just politically fickle). This probably conflicts with the property of "unrestricted domain" but it seems like a reasonable concession in practice.

Wowfunhappy 2y ago
> The part of the impossibility proof that deals with dictators is subtle and neat. At first, I was convinced that the dictator was actually a "swing voter" whose power was entirely conditioned on the other voter's preferences, but that's not actually the case. Based on the video, I am still convinced of this. Why is it not the case?
- dirtdobber 2y ago
  The "no dictator" property states that a single voter should not override the preferences of the majority. For example, if every voter (excluding the dictator) prefers candidate A over candidate C, then the dictator should not be able to control the outcome so that candidate C is ranked higher than candidate A. If you follow the proof closely (it's very subtle) you can see that despite every candidate ranking A over candidate C, the actual preferential outcome will have candidate C ranked higher than candidate A (and this is entirely dependent on the dictator's choice). In other words, there's a counterexample where every single voter decides to rank A > C, and yet the dictator alters the outcome so that C > A. This is different from a swing voter that pushes one candidate over the top in a very close race.

Wowfunhappy 2y ago

> The part of the impossibility proof that deals with dictators is subtle and neat. At first, I was convinced that the dictator was actually a "swing voter" whose power was entirely conditioned on the other voter's preferences, but that's not actually the case. Based on the video, I am still convinced of this. Why is it not the case?

dirtdobber 2y ago
The "no dictator" property states that a single voter should not override the preferences of the majority. For example, if every voter (excluding the dictator) prefers candidate A over candidate C, then the dictator should not be able to control the outcome so that candidate C is ranked higher than candidate A. If you follow the proof closely (it's very subtle) you can see that despite every candidate ranking A over candidate C, the actual preferential outcome will have candidate C ranked higher than candidate A (and this is entirely dependent on the dictator's choice). In other words, there's a counterexample where every single voter decides to rank A > C, and yet the dictator alters the outcome so that C > A. This is different from a swing voter that pushes one candidate over the top in a very close race.