We introduce a generalized cake-cutting problem in which we seek to divide
multiple cakes so that two players may get their most-preferred piece
selections: a choice of one piece from each cake, allowing for the possibility
of linked preferences over the cakes. For two players, we show that disjoint
envy-free piece selections may not exist for two cakes cut into two pieces
each, and they may not exist for three cakes cut into three pieces each.
However, there do exist such divisions for two cakes cut into three pieces
each, and for three cakes cut into four pieces each. The resulting allocations
of pieces to players are Pareto-optimal with respect to the division. We use a
generalization of Sperner's lemma on the polytope of divisions to locate
solutions to our generalized cake-cutting problem.Comment: 15 pages, 7 figures, see related work at
http://www.math.hmc.edu/~su/papers.htm