The classic fair cake-cutting problem [Steinhaus, 1948] is extended by introducing geometric constraints on the allocated pieces. Specifically, agents may demand to get their share as a square or a rectangle with a bounded length/width ratio. This is a plausible constraint in realistic cake-cutting applications, notably in urban and agricultural economics where the cake is land. Geometric constraints greatly affect the classic results of the fair division theory. The existence of a proportional division, giving each agent 1/n of his total cake value, is no longer guaranteed. We prove that it is impossible to guarantee each agent more than 1/(2n-1) of his total value. Moreover, we provide procedures implementing partially proportional division, giving each agent 1/(An-B) of his total value, where A and B are constants depending on the shape of the cake and its pieces. Fairness and social welfare implications of these procedures are analyzed in various scenarios.
fair division cake cutting land division geometry non-additive utilities social welfare