  We focus on a simple, one-dimensional collective decision problem (often
referred to as the facility location problem) and explore issues of
strategyproofness and proportionality-based fairness. We introduce and analyze
a hierarchy of proportionality-based fairness axioms of varying strength:
Individual Fair Share (IFS), Unanimous Fair Share (UFS), Proportionality (as in
Freeman et al, 2021), and Proportional Fairness (PF). For each axiom, we
characterize the family of mechanisms that satisfy the axiom and
strategyproofness. We show that imposing strategyproofness renders many of the
axioms to be equivalent: the family of mechanisms that satisfy proportionality,
unanimity, and strategyproofness is equivalent to the family of mechanisms that
satisfy UFS and strategyproofness, which, in turn, is equivalent to the family
of mechanisms that satisfy PF and strategyproofness. Furthermore, there is a
unique such mechanism: the Uniform Phantom mechanism, which is studied in
Freeman et al. (2021). We also characterize the outcomes of the Uniform Phantom
mechanism as the unique (pure) equilibrium outcome for any mechanism that
satisfies continuity, strict monotonicity, and UFS. Finally, we analyze the
approximation guarantees, in terms of optimal social welfare and minimum total
cost, obtained by mechanisms that are strategyproof and satisfy each
proportionality-based fairness axiom. We show that the Uniform Phantom
mechanism provides the best approximation of the optimal social welfare (and
also minimum total cost) among all mechanisms that satisfy UFS.
