We study the problem of finding an envy-free allocation of a cake to d + 1 players using d cuts. Two models are considered, namely, the oracle-function model and the polynomial-time function model. In the oracle-function model, we are interested in the number of times an algorithm has to query the players about their preferences to find an allocation with the envy less than ϵ. We derive a matching lower and upper bound of θ(1/ϵ)^{d − 1} for players with Lipschitz utilities and any d > 1. In the polynomial-time function model, where the utility functions are given explicitly by polynomial-time algorithms, we show that the envy-free cake-cutting problem has the same complexity as finding a Brouwer's fixed point, or, more formally, it is PPAD-complete. On the flip side, for monotone utility functions, we propose a fully polynomial-time algorithm (FPTAS) to find an approximate envy-free allocation of a cake among three people using two cuts.

We consider a sharing economy over a network in which each vertex agent allocates resources to its neighbors in response to their contributions. General equilibrium theory can be applied here to solve the problem of deciding how to fairly and efficiently allocate resources among agents as resource sharing over the network can be modeled as a pure exchange economy. It is known that proportional sharing dynamics converges to a market equilibrium solution. We are particularly interested in proportional sharing dynamics as a mechanism for network resource sharing. Our focus is on the key issue in internet market design: whether an agent may manipulate its report of its own private information to gain more resources under this mechanism. This work establishes the first mathematical proof that such a practical distributed network resource-sharing protocol is truthful against the manipulative strategies of feasible weight misreporting and edge deletion applied both individually and together. Funding: This work was supported by the National Nature Science Foundation of China [Grants 11871366, 62172012]; Beijing Outstanding Young Scientist Program [Grant BJJWZYJH012019100020098]; the Fundamental Research Funds for the Central Universities; the Research Funds of Renmin University of China [Grant 297522503709]; and the Intelligent Social Governance Platform, Major Innovation & Planning Interdisciplinary Platform for the “Double-First Class” Initiative, Renmin University of China.