Fairness and Stability for Shared Resource Allocation Problems

This paper investigates the problem of shared resource allocation, where a set of agents must be assigned to heterogeneous resources, with each agent allocated exactly one resource and each resource potentially shared by multiple agents. An agent’s utility for a given resource is jointly determined by the resource's type and the number of agents sharing it. We focus on two fundamental classes of monotone valuations: monotone nondecreasing and monotone nonincreasing, where an agent’s utility respectively increases or decreases with the number of agents sharing the resource. Within this shared resource framework, we examine classical notions of fairness and stability, including maximin-share fairness, envy-freeness, Nash stability, and two epistemic relaxations—epistemic envy-freeness and epistemic Nash stability—as well as swap stability. We propose formal definitions adapted to this setting and systematically analyze the relationships among these concepts. The primary contributions of this work consist of establishing existence and computational complexity results for each notion under both monotonicity assumptions and developing polynomial-time algorithms in cases where fair or stable allocations are guaranteed to exist.