One classic fair division allocation problem is to allocate indivisible goods among agents with respect to envy freeness (no agent envies another agent).
In my settings, all the goods are divisible (we can split the goods), but we restrict the number of shared goods to one. So only one good can be split, but the split good can be any good.
Is there any strategy to choose a good say G to be the split good such that it is optimal? In other words, is there any way to show that if there exists an EF allocation with any split good, then there exists an EF allocation with G as split good?
If the agent evaluations are identical, then it is easy to prove that the good to be split is the biggest good. Unfortunately, the proof doesn't work in the general case (when the evaluations are different).
