Equivalence of Models of Cake-Cutting Protocols

The cake-cutting problem involves dividing a heterogeneous, divisible resource fairly between $n$ agents. Br\^{a}nzei et al. [6] introduced {\em generalised cut and choose} (GCC) protocols, a formal model for representing cake-cutting protocols as trees with "cut" and "choose" nodes corresponding to the agents' actions, and if-else statements. In this paper, we identify an alternative and simpler extensive-form game model for cake-cutting protocols, that we call {\em branch choice} (BC) protocols. We show that the class of protocols we can represent using this model is invariant under certain modifications to its definition. We further prove that any such protocol can be converted to a restricted form in which the agents first cut the cake and then get to choose between various branches leading to different allocations. Finally, we show that this model has the same expressive power as GCC protocols, i.e. they represent the same class of protocols up to a notion of equivalence involving the bounds on envy that each agent can guarantee for themselves. For this purpose, we introduce a new notion of envy-equivalence of protocols.