Cutting a Cake with both Good and Bad Parts

There is a heterogeneous resource that contains both good parts and bad parts, for example, a cake with some parts burnt, a land-estate with some parts heavily taxed, or a chore with some parts fun to do. The resource has to be divided fairly among n agents, each of whom has a personal value-density function on the resource. The value-density functions can accept any real value positive or negative. Can standard cake-cutting procedures, developed for positive valuations, be adapted to this setting? This paper focuses on the question of envy-free cake-cutting with connected pieces. It is proved that such a division exists for 3 agents. The proof uses a generalization of Sperner’s lemma, which may be of independent interest.