Proximity by Numbers

Imagine that everyone in a group chooses a real number and then these numbers are combined to produce a group number. Suppose that when everyone moves strictly closer to some individual’s number, the group number either stays where it is or moves closer to this number. We call this the proximity condition. Restricting attention to group choice rules that are homogeneous of degree one and constant-preserving, we show that the only rules satisfying this property are dictatorships.