Population monotonic rules for fair allocation problems

We consider the property of population monotonicity in the context of fair allocation problems in private good economies. It is already known that the property is compatible with the equal division lower bound. We show that if the equal division lower bound is replaced with no envy, the compatibility does not hold. We also show that the incompatibility persists even though no envy is weakened to no domination or ε-no domination. Hence, among the fairness criteria mentioned above, only the equal division lower bound is compatible with population monotonicity.