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Abstract
The one-dimensional facility location problem readily generalizes to many real world problems, including social choice, project funding, and the geographic placement of facilities intended to serve a set of agents. In these problems, each agent has a preferred point along a line or interval, which could denote their ideal preference, preferred project funding, or location. Thus each agent wishes the facility to be as close to their preferred point as possible. We are tasked with designing a mechanism which takes in these preferred points as input, and outputs an ideal location to build the facility along the line or interval domain. In addition to minimizing the distance between the facility and the agents, we may seek a facility placement which is fair for the agents. In particular, this thesis focusses on the notion of proportional fairness, in which endogenous groups of agents with similar or identical preferences have a distance guarantee from the facility that is proportional to the size of the group. We also seek mechanisms that are strategyproof, in that no agent can improve their distance from the facility by lying about their location.
We consider both deterministic and randomized mechanisms, in both the classic and obnoxious facility location settings. The obnoxious setting differs from the classic setting in that agents wish to be far from the facility rather than close to it. For these settings, we formalize a hierarchy of proportional fairness axioms, and where possible, characterize strategyproof mechanisms which satisfy these axioms. In the obnoxious setting where this is not possible, we consider the welfare-optimal mechanisms which satisfy these axioms, and quantify the extent at which the system efficiency is compromised by misreporting agents. We also investigate, in the classic setting, the nature of misreporting agents under a family of proportionally fair mechanisms which are not necessarily strategyproof. These results are supplemented with tight approximation ratio and price of fairness bounds which provide further insight into the compromise between proportional fairness and efficiency in the facility location problem. Finally, we prove basic existence results concerning possible extensions to our settings.
