Truthful assignment without money

We study the design of truthful mechanisms that do not use payments for the generalized assignment problem (GAP) and its variants. An instance of the GAP consists of a bipartite graph with jobs on one side and machines on the other. Machines have capacities and edges have values and sizes; the goal is to construct a welfare maximizing feasible assignment. In our model of private valuations, motivated by impossibility results, the value and sizes on all job-machine pairs are public information; however, whether an edge exists or not in the bipartite graph is a job's private information. That is, the selfish agents in our model are the jobs, and their private information is their edge set. We want to design mechanisms that are truthful without money (henceforth strategyproof), and produce assignments whose welfare is a good approximation to the optimal omniscient welfare. We study several variants of the GAP starting with matching. For the unweighted version, we give an optimal strategyproof mechanism. For maximum weight bipartite matching, we show that no strategyproof mechanism, deterministic or randomized, can be optimal, and present a 2-approximate strategyproof mechanism along with a matching lowerbound. Next we study knapsack-like problems, which, unlike matching, are NP-hard. For these problems, we develop a general LP-based technique that extends the ideas of Lavi and Swamy [14] to reduce designing a truthful approximate mechanism without money to designing such a mechanism for the fractional version of the problem. We design strategyproof approximate mechanisms for the fractional relaxations of multiple knapsack, size-invariant GAP, and value-invariant GAP, and use this technique to obtain, respectively, 2, 4 and 4-approximate strategyproof mechanisms for these problems. We then design an O(log n)-approximate strategyproof mechanism for the GAP by reducing, with logarithmic loss in the approximation, to our solution for the value-invariant GAP. Our technique may be of independent interest for designing truthful mechanisms without money for other LP-based problems.