Consider an arbitrary wealth distribution over a large (possibly infinite) set I of individuals. An individual i has wealth wi. Total wealth W is the sum over all individual wealth levels. Now, from set I a share phi of individuals are randomly put into group A and a complimentary share 1-phi is randomly put into group B.

The question is: Under what cirumstances (e.g. under what initial wealth distribution) is the total wealth in group A phi*W and the total wealth in group B (1-phi)*W. Obviously, this is the case if the initial distribution is uniform. But are there any other possibilities?

I have read the thread below, but that isn't quite what I'm looking for.


Thank you very much.