Refinements of the no-envy solution in economies with indivisible goods

We consider the problem of fair allocation in economies with indivisible goods. Our primary concept is that of an envy-free allocation, that is, an allocation such that no agent would prefer anyone else's bundle to his own. Since there typically is a large set (a continuum) of such allocations, the need arises to identify well-behaved selections from the no-envy solution. First we establish the non-existence of population monotonic selections. Then we propose a variety of selections motivated by intuitive considerations of fairness.