A Pie That Can't Be Cut Fairly

David Gale asked in [8, 1993] whether, when a pie is to be divided among n claimants, it is always possible to find a division that is both envy free and undominated. The pie is cut along n radii and the claimants’ preferences are described by separate measures. We answer Gale’s question in the negative for n = 3 by exhibiting three measures on a pie such that, when players have these measures, no division of the pie can be both envy free and undominated. The measures assign positive values to pieces with positive area.