Two-player envy-free multi-cake division

TL;DR

This paper explores a complex multi-cake division problem where two players select preferred pieces, revealing conditions for envy-free and Pareto-optimal allocations using a generalized Sperner's lemma.

Contribution

It introduces a generalized multi-cake division framework with linked preferences and characterizes existence conditions for envy-free solutions.

Findings

Envy-free disjoint selections may not exist for certain cake cuts.

Existence of envy-free solutions depends on the number of pieces per cake.

Solutions are Pareto-optimal with respect to the division.

Abstract

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.