Strategic Facility Location via Predictions

The facility location with strategic agents is a canonical problem in the literature on mechanism design without money. Recently, Agrawal et. al. considered this problem in the context of machine learning augmented algorithms, where the mechanism designer is also given a prediction of the optimal facility location. An ideal mechanism in this framework produces an outcome that is close to the social optimum when the prediction is accurate (consistency) and gracefully degrades as the prediction deviates from the truth, while retaining some of the worst-case approximation guarantees (robustness). The previous work only addressed this problem in the two-dimensional Euclidean space providing optimal trade-offs between robustness and consistency guarantees for deterministic mechanisms. We consider the problem for \emph{general} metric spaces. Our only assumption is that the metric is continuous, meaning that any pair of points must be connected by a continuous shortest path. We introduce a novel mechanism that in addition to agents' reported locations takes a predicted optimal facility location $\hat{o}$. We call this mechanism $\texttt{Harmonic}$, as it selects one of the reported locations $\tilde{\ell}_i$ with probability inversely proportional to $d(\hat{o},\tilde{\ell}_i)+ \Delta$ for a constant parameter $\Delta$. While \harm \ mechanism is not truthful, we can \emph{characterize the set of undominated strategies} for each agent $i$ as solely consisting of the points on a shortest path from their true location $\ell_i$ to the predicted location $\hat{o}$. We further derive \emph{consistency and robustness guarantees on the Price of Anarchy (PoA)} for the game induced by the mechanism.