Truthful Facility Assignment with Resource Augmentation: An Exact Analysis of Serial Dictatorship

We study the truthful facility assignment problem, where a set of agents with private most-preferred points on a metric space are assigned to facilities that lie on the metric space, under capacity constraints on the facilities. The goal is to produce such an assignment that minimizes the social cost, i.e., the total distance between the most-preferred points of the agents and their corresponding facilities in the assignment, under the constraint of truthfulness, which ensures that agents do not misreport their mostpreferred points. We propose a resource augmentation framework, where a truthful mechanism is evaluated by its worstcase performance on an instance with enhanced facility capacities against the optimal mechanism on the same instance with the original capacities. We study a very well-known mechanism, Serial Dictatorship, and provide an exact analysis of its performance. Although Serial Dictatorship is a purely combinatorial mechanism, our analysis uses linear programming; a linear program expresses its greedy nature as well as the structure of the input, and finds the input instance that enforces the mechanism have its worst-case performance. Bounding the objective of the linear program using duality arguments allows us to compute tight bounds on the approximation ratio. Among other results, we prove that Serial Dictatorship has approximation ratio g/(g−2) when the capacities are multiplied by any integer g ≥ 3. Our results suggest that even a limited augmentation of the resources can have wondrous effects on the performance of the mechanism and in particular, the approximation ratio goes to 1 as the augmentation factor becomes large. We complement our results with bounds on the approximation ratio of Random Serial Dictatorship, the randomized version of Serial Dictatorship, when there is no resource augmentation. ∗Ioannis Caragiannis was partially supported by a Caratheodory research grant E.114 from the University of Patras. Aris Filos-Ratsikas was partially supported by the COST Action IC1205 on “Computational Social Choice” and by the ERC Advanced Grant 321171 (ALGAME). Aris Filos-Ratsikas, Søren Kristoffer Stiil Frederiksen, and Kristoffer Arnsfelt Hansen acknowledge support from the Danish National Research Foundation and The National Science Foundation of China (under the grant 61361136003) for the Sino-Danish Center for the Theory of Interactive Computation and from the Center for Research in Foundations of Electronic Markets (CFEM), supported by the Danish Strategic Research Council. †University of Patras, Greece. E-mail: [email protected] ‡University of Oxford, United Kingdom. E-mail: [email protected] §Aarhus University, Denmark. E-mail: [email protected] ¶Aarhus University, Denmark. E-mail: [email protected] ‖University of Chicago, Unites States. E-mail: [email protected]