CONSISTENT EXTENSIONS

An allocation rule, or solution, is consistent if what it recommends for any economy is always 'in agreement' with what it recommends for associated 'reduced economies'. I propose here to evaluate the extent to which a solution may fail to be consistent by identifying its minimal consistent extension, that is, the smallest consistent solution that contains it. I calculate the minimal consistent extensions of two solutions that have played a central role in the literature on the problem of fair division. I also propose the concept of maximal consistent subsolution of a solution - this is the largest consistent solution that the solution contains - and show how it can be used to relate several solutions that have been discussed separately in that literature.