Some people interpret Arrow's theorem to mean that democracy is a futile exercise. For example, the anarchist Robert Paul Wolff uses an argument similar to Arrow's theorem to show that all democracies must be tyrannical to at least some of its members.
But Arrow's theorem is first and foremost an exercise in logic. It is grossly oversimplified, and therefore should not be treated as realistic simulation of real-world voting systems. We should be very careful when drawing political conclusions from logical proofs.
There are several reasons why most contemporary political theorists don't give a damn about Arrow's theorem, despite its logical plausibility.
1) Arrow's theorem assumes everyone's preferences to be fixed points, and only cares about finding a curve that fits all of those points. But people's preferences are not fixed. People are always changing their minds, often in response to the shifting preferences of others. Many political theorists in the "deliberative democracy" camp (the dominant model since the early 90s) argue that the whole point of a democratic discussion is to get people to reconsider their pre-existing preferences and find some sort of middle ground.
2) It's not even clear why an ideal procedure would need to satisfy all of the preferences, or even most of them. If making everybody happy were as simple as designing an election procedure, we would have gotten rid of politics a long time ago! You don't even need 3 or more preferences to arrive at a conflict. Two people with one preference each, that directly contradict each other, would be enough to produce a situation where no procedure can satisfy them all. In other words, there's nothing new here. Time to move on.
3) Arrow's theorem is somewhat effective in explaining how the actual share of seats in a lawmaking body can end up being very different from the number of votes that each party received in a first-past-the-post voting system with 3 or more major parties, such as UK and Canada. But there are much simpler, more intuitive ways to explain that.
All in all, Arrow's theorem was a neat response to the political theory of the mid-20th century, when people assumed democracy to be simply a matter of efficient curve-fitting. But political theory has come a long way since then, partly in response to problems like Arrow's theorem. In the new academic milieu, Arrow's theorem isn't as relevant as it used to be.
On the other hand, I can sort of imagine how Arrow's theorem might find a new use in designing distributed computer systems. Since computers aren't as fickle as human politicians, the logical conclusions of Arrow's theorem might be more relevant there. It's good to see that the HN thread so far focuses more on technical details than on grand, mostly irrelevant political narratives.
(1) Not quite. The "deliberative democracy" camp is not interested in the measurement of group preference, and are instead interested in consensus building, political "rationality" (in hopefully some eventually-stabilizing sense), and so on. That is not a response to Arrow and his associates, it is just a different topic.
(2) It is abundantly clear why a "procedure" should satisfy all of the requirements of the related theorems: they are trivial, intuitive, and absolutely spot-on. There is a reason why these results are surprising, and not just some arbitrary theorems concerning uninteresting axioms.
(3) Arrow's theorem has nothing to do with explaining anything. It is an impossibility result in mathematics.
The rest of your post seems just dismissive of the problem, rather than directly critical of it. ("All in all.." -- as if these results were just passing fads and now we've got our sense back??)
If we read early works (from the late 80s) in what is now called "deliberative democracy", we can see that it began as a response to certain models of democracy that emphasize preferences and procedures -- up to and including the participatory models of the 70s and early 80s. Although deliberative democracy is not a direct response to Arrow's impossibility theorem in particular, it was intended to sidestep its troubling implications as well as other problems with the older models.
There are two major factions within the deliberative democracy camp. One is indeed interested in consensus building and eventually-consistent rationality. This is the "Rawlsian" faction led by Gutmann and Thompson. The other faction, however, focuses more on actual practices of negotiation through which pre-existing preferences and power structures are transformed. This is the "critical theory" faction led by John Dryzek and the late Iris Marion Young. Personally, I think the latter remains closer to the original aims of deliberative democracy and presents a better contrast to the older models it was intended to transplant. The Rawlsians just took the opportunity to cram their own agenda into democratic theory, as they always do with everything they touch.
Your claims (2) and (3) seem to contradict each other. If it is so abundantly clear that a procedure that satisfies Arrow's conditions is desirable, why do you say that Arrow's theorem is just a mathematical result that doesn't explain anything IRL?
Arrow's theorem is surprising and troubling only if you believe in some sort of sacred relationship between the trinity of democracy, voting, and satisfaction of all pre-existing preference. To the contrary, I find it both trivial and intuitive that it is impossible to satisfy all of the preferences of all human beings, and I would be very surprised and troubled if someone claimed to be able to do so.
I think you are a bit confused. What would Arrow's theorem explain? There is no empirical phenomena that we are puzzled about that Arrow's theorem solves (unless you are wondering: Why is it so hard to come up with a voting system that doesn't have the potential for goof-ball results? - Answer, because it is impossible..)
I don't know what you mean by the "trinity" in which "satisfaction of all pre-existing preference" is a part.. That's obviously not relevant; we aren't interested in a system that satisfies preferences. What we are interested in is a system that (1) isn't a dictatorship, (2) is fair [i.e., everyone counts equally], (3) allows us to decide any kind of potential matter _as a group_.
If you look at the conditions this way it should be glaringly obvious what this has to do with the (possibility) of democracy..
If we're not interested in a system that satisfies preferences, then Arrow's impossibility theorem is irrelevant.
The conditions, of course, sound obvious and intuitive. Of course we don't want a dictatorship, and of course we want everyone to count as equally as possible. But it takes a very specific interpretation of your third condition to bootstrap the rest of Arrow's theorem. You have to interpret it in a way that emphasizes translating fixed individual preferences into group preferences as straightforwardly as possible. If you care about that, then yes, you should be worried about Arrow's theorem. Otherwise, Arrow's theorem is just a cool thought experiment that helps explain why said interpretation is wrong.
Deliberative democracy currently happens to be the most popular model of democracy among political theorists, and they don't care about Arrow's theorem because whatever preferences people have before they enter the democratic "procedure" isn't worth jack shit to them. As a result, Arrow's theorem is much less relevant to the possibility and fate of democracy as currently understood than it was 60 years ago.
But Arrow's theorem is first and foremost an exercise in logic. It is grossly oversimplified, and therefore should not be treated as realistic simulation of real-world voting systems. We should be very careful when drawing political conclusions from logical proofs.
There are several reasons why most contemporary political theorists don't give a damn about Arrow's theorem, despite its logical plausibility.
1) Arrow's theorem assumes everyone's preferences to be fixed points, and only cares about finding a curve that fits all of those points. But people's preferences are not fixed. People are always changing their minds, often in response to the shifting preferences of others. Many political theorists in the "deliberative democracy" camp (the dominant model since the early 90s) argue that the whole point of a democratic discussion is to get people to reconsider their pre-existing preferences and find some sort of middle ground.
2) It's not even clear why an ideal procedure would need to satisfy all of the preferences, or even most of them. If making everybody happy were as simple as designing an election procedure, we would have gotten rid of politics a long time ago! You don't even need 3 or more preferences to arrive at a conflict. Two people with one preference each, that directly contradict each other, would be enough to produce a situation where no procedure can satisfy them all. In other words, there's nothing new here. Time to move on.
3) Arrow's theorem is somewhat effective in explaining how the actual share of seats in a lawmaking body can end up being very different from the number of votes that each party received in a first-past-the-post voting system with 3 or more major parties, such as UK and Canada. But there are much simpler, more intuitive ways to explain that.
All in all, Arrow's theorem was a neat response to the political theory of the mid-20th century, when people assumed democracy to be simply a matter of efficient curve-fitting. But political theory has come a long way since then, partly in response to problems like Arrow's theorem. In the new academic milieu, Arrow's theorem isn't as relevant as it used to be.
On the other hand, I can sort of imagine how Arrow's theorem might find a new use in designing distributed computer systems. Since computers aren't as fickle as human politicians, the logical conclusions of Arrow's theorem might be more relevant there. It's good to see that the HN thread so far focuses more on technical details than on grand, mostly irrelevant political narratives.