So, among other things, Arrow's Impossibility Theorem shows that if you've got individuals with double-peaked preferences, then a simple majority voting procedure will not yield consistent outcomes. While it's easy to construct someone with double-peaked preferences on a non-linear spectrum-- ranking favorite sports teams or colors, for example-- it's generally the case that double-peaked preferences don't exist in voting for candidates. It's vastly oversimplifying the voting process, but if you cared only about one dimension of politics-- say, political leaning-- then you probably won't prefer a radical republican and a radical democrat to someone in the middle. People could then argue that the assumption of double-peaked preferences is unrealistic, and hence, Arrow's situation doesn't describe the vast majority of real life. (Though assuming that all political decisions are made on a one-dimensional margin might be just as unrealistic.) Fair enough. It's not like there's a shortage of ways to cut down majority-rules voting systems.
What's interesting is that I've seem to run across a couple of situations recently where I (and a professor of mine) have, quite clearly, double-peaked preferences. Maybe I'm missing something. I also don't think that it matters that there's no voting involved in these scenarios-- the issue above was with the accuracy of double-peaked preferences in representing real life.
They're both similar-- one involving my car, the other involving a house that a professor of mine recently bought. My car's warranty expires next month, so I took it down for a routine checkup last week. With regards to the viability of what parts were under warranty, I either want them to be functioning very well or very poorly-- either way, I end up with a car running well (on its own or immediately fixed) at no additional cost to me. What I want least is my car to be running just well enough that they choose not to fix anything. So I want it in great shape or poor shape most, but least favorable to me is a moderately well running car. That's a double-peaked preference. (Principal/agent problems of the mechanic are tangential to my ultimate preference for the condition of my car.)
The same idea holds for the recently purchased house. New houses get checked out by an independent third party that determines if everything is in working order. If everything is fine, so be it-- if not, then the seller is responsible for the repairs. This creates the same incentives to the buyer of the house as to the driver of the car-- either you want the house in tip-top shape, or in poor condition such that it will get fixed by the seller. The worst-case scenario is a house that comes in just above the fix-it line. Again, we've got double-peaked preferences.
Maybe double-peaked preferences are a bit more common than I had previously thought.