A Truthful Mechanism for the Generalized Assignment Problem

We propose a truthful-in-expectation, $(1-1/e)$-approximation mechanism for a strategic variant of the generalized assignment problem (GAP). In GAP, a set of items has to be optimally assigned to a set of bins without exceeding the capacity of any singular bin. In the strategic variant of the problem we study, values for assigning items to bins are the private information of bidders and the mechanism should provide bidders with incentives to truthfully report their values. The approximation ratio of the mechanism is a significant improvement over the approximation ratio of the existing truthful mechanism for GAP. The proposed mechanism comprises a novel convex optimization program as the allocation rule as well as an appropriate payment rule. To implement the convex program in polynomial time, we propose a fractional local search algorithm which approximates the optimal solution within an arbitrarily small error leading to an approximately truthful-in-expectation mechanism. The presented algorithm improves upon the existing optimization algorithms for GAP in terms of simplicity and runtime while the approximation ratio closely matches the best approximation ratio given for GAP when all inputs are publicly known.Comment: 18 pages, Earlier version accepted at WINE 201

Mechanism Design without Money via Stable Matching

Mechanism design without money has a rich history in social choice literature. Due to the strong impossibility theorem by Gibbard and Satterthwaite, exploring domains in which there exist dominant strategy mechanisms is one of the central questions in the field. We propose a general framework, called the generalized packing problem (\gpp), to study the mechanism design questions without payment. The \gpp\ possesses a rich structure and comprises a number of well-studied models as special cases, including, e.g., matroid, matching, knapsack, independent set, and the generalized assignment problem. We adopt the agenda of approximate mechanism design where the objective is to design a truthful (or strategyproof) mechanism without money that can be implemented in polynomial time and yields a good approximation to the socially optimal solution. We study several special cases of \gpp, and give constant approximation mechanisms for matroid, matching, knapsack, and the generalized assignment problem. Our result for generalized assignment problem solves an open problem proposed in \cite{DG10}. Our main technical contribution is in exploitation of the approaches from stable matching, which is a fundamental solution concept in the context of matching marketplaces, in application to mechanism design. Stable matching, while conceptually simple, provides a set of powerful tools to manage and analyze self-interested behaviors of participating agents. Our mechanism uses a stable matching algorithm as a critical component and adopts other approaches like random sampling and online mechanisms. Our work also enriches the stable matching theory with a new knapsack constrained matching model

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. 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, however, we show give a 2-approximate strategyproof mechanism and show by a matching lowerbound that this is optimal. Next we study knapsack-like problems, which are APX-hard. For these problems, we develop a general LP-based technique that extends the ideas of Lavi and Swamy to reduce designing a truthful mechanism without money to designing such a mechanism for the fractional version of the problem, at a loss of a factor equal to the integrality gap in the approximation ratio. We use this technique to obtain strategyproof mechanisms with constant approximation ratios 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.Comment: Extended abstract appears in the 11th ACM Conference on Electronic Commerce (EC), 201

Expressiveness and Robustness of First-Price Position Auctions

Since economic mechanisms are often applied to very different instances of the same problem, it is desirable to identify mechanisms that work well in a wide range of circumstances. We pursue this goal for a position auction setting and specifically seek mechanisms that guarantee good outcomes under both complete and incomplete information. A variant of the generalized first-price mechanism with multi-dimensional bids turns out to be the only standard mechanism able to achieve this goal, even when types are one-dimensional. The fact that expressiveness beyond the type space is both necessary and sufficient for this kind of robustness provides an interesting counterpoint to previous work on position auctions that has highlighted the benefits of simplicity. From a technical perspective our results are interesting because they establish equilibrium existence for a multi-dimensional bid space, where standard techniques break down. The structure of the equilibrium bids moreover provides an intuitive explanation for why first-price payments may be able to support equilibria in a wider range of circumstances than second-price payments

Revenue Maximization and Ex-Post Budget Constraints

We consider the problem of a revenue-maximizing seller with m items for sale to n additive bidders with hard budget constraints, assuming that the seller has some prior distribution over bidder values and budgets. The prior may be correlated across items and budgets of the same bidder, but is assumed independent across bidders. We target mechanisms that are Bayesian Incentive Compatible, but that are ex-post Individually Rational and ex-post budget respecting. Virtually no such mechanisms are known that satisfy all these conditions and guarantee any revenue approximation, even with just a single item. We provide a computationally efficient mechanism that is a $3$-approximation with respect to all BIC, ex-post IR, and ex-post budget respecting mechanisms. Note that the problem is NP-hard to approximate better than a factor of 16/15, even in the case where the prior is a point mass \cite{ChakrabartyGoel}. We further characterize the optimal mechanism in this setting, showing that it can be interpreted as a distribution over virtual welfare maximizers. We prove our results by making use of a black-box reduction from mechanism to algorithm design developed by \cite{CaiDW13b}. Our main technical contribution is a computationally efficient $3$-approximation algorithm for the algorithmic problem that results by an application of their framework to this problem. The algorithmic problem has a mixed-sign objective and is NP-hard to optimize exactly, so it is surprising that a computationally efficient approximation is possible at all. In the case of a single item ($m=1$), the algorithmic problem can be solved exactly via exhaustive search, leading to a computationally efficient exact algorithm and a stronger characterization of the optimal mechanism as a distribution over virtual value maximizers