Equal split guarantee solution in economies with indivisible goods consistency and population monotonicity

We consider the problem of allocating a finite set of indivisible goods and a single infinitely divisible good among a group of agents, and we study a solution, called the Equal Split Guarantee solution, in the presence of consistency and population monotonicity properties. This solution is not consistent. We prove that its maximal consistent subsolution is the No-envy solution. Our main result is that the minimal consistent extension of the intersection of the Equal Split Guarantee solution with the Pareto solution is the Pareto solution. This result remains true in the restricted domain when all the indivisible goods are identical, but not when there is a unique indivisible good. Finally, we show that in the class of economies with a unique indivisible good, there is a selection from the Equal Split Guarantee solution that satisfies population monotonicity.