Social Choice is something that happens all the time. We can't stop it from happening. But does having a theory of Social Choice
a) improve decision making?
b) not quickly lead to nonsense?
On the evidence- no.
I guess in the old days Social Choice was mainly done by criminals, lunatics and sociopaths- the Robber Baron in his Castle did Fiscal policy (i.e. levying protection money) and Institutional infrastructure (organizing markets and chopping of hands and heads) ; the lunatics (i.e. religious nutjobs) did Capital formation (i.e. Temples and Monasteries and so on) and Division of Labor (enforcing a Caste system and Gender dimorphism and regulating Innovation and Technology Transfer) and the sociopaths (lawyers and merchants and whores) did the rest.
Have things really changed?
Sure. The sociopaths are now in charge. So a Theory is needed coz just cutting of hands and heads and burning heretics isn't cool anymore BECAUSE sociopaths are cowardly little shites.
So we need a Theory of Social Choice.
But why have a mathematical theory of Social Choice? After all, the U.S was dismantling its system of administered pricing just at the time Arrow was working his mojo. I suppose, one argument for Social Choice ties up with stuff related to Market failure- it made sense to show why, Mathematically, that was bound to happen just as it also made sense to show there were bound to be Government failures to balance out the picture.
Arrow showed that there was no benign way of aggregating preferences by means of a voting rule that did not yield perverse results. You're always bound to have some sort of Preference Revelation problem or you're going to be losing important information' or screwing up dynamically or something. Gibbard Satterthwaite & Myerson-Satterthwaite are like a reverse Coase's theorem showing why mechanism design can fuck up.
What if you combine mechanism design in a bargaining game such that 'the gains from trade' are being divided up between parties? Perhaps here the maths would become intractable as an element of impredicativity arises.
In any case, in ordinary life all sorts of people who can hardly count do a pretty good job of doing back of the envelope calculations of things like Shapley index of power for a single-issue voting group. People intuitively understand when preference diversity is too great for a voting type solution. So why should this stuff not just be left to the back-room boys? Why is Social Choice Theory intellectually sexy in a manner that Cost Accountancy or Actuarial Science isn't?
Why not just go back to good old fashioned Cardinal Utility and ad hoc comparisons of different people's Utility- coz that's what it's all gonna cash out as anyway? Either that, or just let different vested interest groups battle it out in the courts and in the media and through the political system? I mean that's the way the world has always worked and Social Choice is a fact of our day to day life BECAUSE ours is a swinish existence. I'm not saying we're all not swine in the ultimate reckoning but just that we don't like our nose rubbed in it.
Where's the harm in tarting up Social Choice with a bit of Math like maybe it wasn't just swinishness but something elegant and beautiful like theoretical physics?
Pigs might fly and mathematical swine soar skyward upon Platonic wings.
In addition to Social Choice Theory, the Dr.Stangeloves of the RAND Corp. had prepared for us a beautiful Armageddon of Nuclear missiles raining down on our heads like so many air-borne pigs.
Still better than being crapped on by angels? Ah, but angels don't exist.