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We study the multi-layered cake cutting problem, where the multi-layered cakeis divided among agents proportionally. This problem was initiated by Hosseiniet al.(2020) under two constraints, one is contiguity and the other isfeasibility. Basically we will show the existence of proportionalmulti-allocation for any number of agents with any number of preferences thatsatisfies contiguity and feasibility constraints using the idea of switchingpoint for individual agent and majority agents. First we show that exactfeasible multi-allocation is guaranteed to exist for two agents with two typesof preferences. Second we see that we always get an envy-free multi-allocationthat satisfies the feasibility and contiguity constraints for three agent withtwo types of preferences such that each agent has a share to each layer evenwithout the knowledge of the unique preference of the third agent.

Proportional Fair Division of Multi-layered Cakes