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We study the problem of fair and efficient mechanism design for allocating multiple resources in multiple servers among a set of users with Leontief utilities. This problem is motivated by a mobile edge computing environment where each mobile user cannot establish a wireless connection simultaneously to multiple edge servers. Each user is a selfish utility maximizing agent that chooses a single server for its job execution. When a server is shared by multiple users, a resource allocation rule decides the utility that each user must receive. Our goal is to design a mechanism that always admits a Nash Equilibrium (NE), i.e., a state where no user has incentive to change its server, that (1) can be reached in polynomial time and (2) provides fair and efficient resource allocation. We propose the Multi-resource Allocation Game Induced by Kalai-Smorodinsky bargaining solution (MAGIKS) and prove that under discrete resource demands it finds an NE in <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}({\textrm {poly}(n)})$ </tex-math></inline-formula> moves for any fixed server configuration, where <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is the number of users. Furthermore, MAGIKS satisfies envy-freeness, sharing incentive, and Pareto optimality on each server. Regarding fairness among users in different servers, we prove that MAGIKS satisfies 2-approximate envy-freeness and maximin share guarantee. Moreover, we show that 2-approximate envy-freeness is the best that any mechanism that satisfies local Pareto optimality can achieve at its NEs.

MAGIKS: Fair Multi-Resource Allocation Game Induced by Kalai-Smorodinsky Bargaining Solution