Proportional Fair Division of Multi-layered Cakes

We study the multi-layered cake cutting problem, where the multi-layered cake is divided among agents proportionally. This problem was initiated by Hosseini et al.(2020) under two constraints, one is contiguity and the other is feasibility. Basically we will show the existence of proportional multi-allocation for any number of agents with any number of preferences that satisfies contiguity and feasibility constraints using the idea of switching point for individual agent and majority agents. First we show that exact feasible multi-allocation is guaranteed to exist for two agents with two types of preferences. Second we see that we always get an envy-free multi-allocation that satisfies the feasibility and contiguity constraints for three agent with two types of preferences such that each agent has a share to each layer even without the knowledge of the unique preference of the third agent.