Here is a proof of Arrow's Impossibility Theorem:
I can’t shake the feeling that there is something wrong with this proof, and that there is some kind of subtle logical fallacy. This isn’t because I feel like IIA and Pareto Efficiency are so important, in fact I think they are both somewhat peculiar to begin with, as is the requirement of an output that is a strict total ordering, and I’m not trying to “save” them per se. I have not read Arrow’s own formal proof (yet) or other formal proofs of his result on paper, but if the result follows the same logic as the video, then I have some doubts, and I’m trying to articulate what those doubts are about here.
As explained on the Wikipedia page on Kenneth Arrow's paper "Social Choice and Individual Values," where his impossibility theorem was proved, "Arrow himself expresses hope at the end of his Nobel prize lecture that, though the philosophical and distributive implications of the paradox of social choice were 'still not clear,' others would 'take this paradox as a challenge rather than as a discouraging barrier.'"
In line with this observation, I believe actually that there may be a fundamental issue with the establishment of the identity of the dictator as formally proved by Arrow. If my reasoning is correct, then the formal impossibility theorem has been misinterpreted, and whether a voting system is a "dictatorship" or "non-dictatorship” may actually be inconsequential. In particular, the so-called “dictator” as formally defined by Arrow may very well exist, but this does not actually imply that the “dictator” is in control of the election. Rather, it might just as well be that the election is in control of the “dictator.” Here is a narrative to illustrate the paradox:
Suppose that four children named Alice, Bob, Charlie and Dale have a game night every Thursday at Alice’s house, where they play one game each of Chess, Yahtzee, and Poker. They use a deterministic ranked-order voting system to decide the ordering of the games, and vote at the beginning of each game night, since each week, each of them is more eager to play their more preferred games. They all have agreed on a particular voting system, and are pleased to know for a fact that the system they have chosen to use satisfies the independence of irrelevant alternatives and Pareto efficiency criteria, and will return a strict ordering of the games. They use this system many times, and accumulate an extensive record of their votes and the outcomes of each election. The way they do this is in a specific order: firstly (1) they submit their ballots; secondly (2) they observe the result of the system; and then thirdly (3) they record the result and the corresponding set of ballots in their record book.
On the day following one game night, another child named Kenny moves into the neighborhood. He is invited to Alice’s house that day and introduced to the other children, who mention their voting system and their record book. They also invite Kenny to join them on the game night of the following week. Kenny is intrigued by their voting system and the record they have kept of it, and asks to borrow the record for the rest of the week.
At home, he examines the record. However, he notices that, peculiarly, the other children have not indicated whose ballot is whose! Rather, the ballots are simply ordered from left to right. Tacitly, he assumes that the ballots have been arranged in alphabetical order, and using the exact logic in the proof in the video of Arrow's Impossibility Theorem and the record of the ballots, he concludes that Charlie is a dictator—that is, his conclusion is that their voting system simply ignores every ballot except for Charlie’s.
Before voting on the following game night, Kenny remarks that he thinks the children's "voting system" joke was very funny, since he has concluded that their "democracy" is in fact just a dictatorship controlled by Charlie. The children seem puzzled though, and thinking they are stringing him along, Kenny takes out the record book and demonstrates his proof that Charlie is a dictator. Charlie still seems puzzled, however, and states,
"But Kenny, we actually didn't put the ballots in alphabetical order. In fact, since the order we put the ballots in doesn't change the result of the election, we agreed to record our ballots in a totally random order from left to right each week!"
Kenny is now puzzled. "But... then what have I shown? Is it that the voter who is placed third from the left in this record is a dictator?"
Alice answers, "Well, I think that would be rather absurd, wouldn't it? The 'third from the left' voter does not even exist until after we have already seen the result of our system, and then the ‘third from the left’ voter will be chosen from among us totally at random. Are you suggesting that, somehow, our voting system can predict the future?”
Do you agree that it seems like something is wrong somewhere? That “something” could just be me making some kind of logical error, but I’m not having an easy time finding it if it’s there, and in any case I’m not having an easy time reconciling this narrative. What I mean to illustrate by this narrative is that there seems perhaps to be a causal misinterpretation of the formalism of Arrow’s theorem. Each time the ballots change, it is actually not clear that the voter identities “behind” each ballot have not changed as well. It is therefore not necessarily possible to “fix” the “dictator’s” ballot while simultaneously altering every other ballot, since the identity of the dictatorial ballot is potentially contingent on all of the ballots. If that’s true, then the specific identity of the dictator may not be predetermined by the system, but rather may be contingent on the election results.
I understand that the formality would have us associate the “first” ballot with “voter 1,” and the “second” ballot with “voter 2,” but again, voters 1 and 2 are indistinguishable except for their ballots and the arbitrary numbering. What is stopping us from applying the theorem to conclude that the “third from the left” voter is a dictator?
Just to be clear, I am quite confident that I fully understand all of the logic in the proof of the theorem. It is a theorem and the proof is correct. My skepticism is about how that formal theorem is being interpreted.