A protocol for honest compromise This question did not originate in my research, but I think it's interesting. It is somewhat underspecified. I'd be interested in an example in any particular case.

We have $n$ parties that between them must choose a value $x \in \mathbb{R}$. Each party has a preferred value $p_i$ for $x$, but we must find a compromise. Let's say we want the mean of the preferences.

Now, anybody who has been involved in politics will know that taking the middle way between stated preferences can be gamed by stating a more extreme position than the actual preference, so that the mean lands closer to it. Is there a protocol that lets us find (an approximation to) the mean of the true preferences? 

As an analogy, one can take the second-price auction, where everybody is best served by bidding their true valuation.
There's a trivial protocol whose expected value is the correct mean: everybody announces their preference, and one is chosen at random. No player (acting alone) can do better than to announce their true preference. I haven't yet looked at the collusion-resistance of this protocol (I would guess it depends on the exact payoff functions).