Cutting a Cake Fairly for Groups Revisited

TL;DR

This paper revisits fair division in cake cutting, exploring group envy-free partitions, characterizing when such divisions are algorithmically feasible, and extending results to chore division and mixed cakes.

Contribution

It provides a characterization of group sizes allowing finite algorithms for envy-free cake division and extends the analysis to chores and mixed resources.

Findings

Envy-free division is possible for any group sizes when one group is a singleton.

Finite algorithms exist for two-group divisions under specific size conditions.

The results do not extend to mixed cake scenarios.

Abstract

Cake cutting is a classic fair division problem, with the cake serving as a metaphor for a heterogeneous divisible resource. Recently, it was shown that for any number of players with arbitrary preferences over a cake, it is possible to partition the players into groups of any desired size and divide the cake among the groups so that each group receives a single contiguous piece and every player is envy-free. For two groups, we characterize the group sizes for which such an assignment can be computed by a finite algorithm, showing that the task is possible exactly when one of the groups is a singleton. We also establish an analogous existence result for chore division, and show that the result does not hold for a mixed cake.